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Then we specify which ones are capable. Moreover, we give an upper bound for the order of the Schur multiplier, the exterior product and the tensor square of a $d$-generator special $p$-group $ G $ when $ |G'|=p^{\\frac{1}{2}d(d-1)-1}$ for $ d\\geq 3 $ and $ p\\neq 2. $ Addition"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1807.04959","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-07-13T07:56:21Z","cross_cats_sorted":[],"title_canon_sha256":"07597a4c8bbb480a4dda541368c24f101411a3c3d950c76049659d728ada6a69","abstract_canon_sha256":"1616328fcf02657ffea160b2b2e1bf286fcacc28ed53262f24291b93e5adf4c9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:49.349157Z","signature_b64":"ScS3BIhM87Gc81cJksGNLhC9P+Gq3MO4lH2/k+DjCM0tlJFewcb8FyZbBVGADv9lsytEV5pHXLS1kHLyspUdDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5a8c1e8e413d258e720d10f6a422d4a5f46b8e4958f468a5c955d0c3ae999cfa","last_reissued_at":"2026-05-18T00:10:49.348585Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:49.348585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on some special $p$-groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Farangis Johari, Peyman Niroomand","submitted_at":"2018-07-13T07:56:21Z","abstract_excerpt":"Recently Rai obtained an upper bound for the order of the Schur multiplier of a $d$-generator special $p$-group when its derived subgroup has the maximum value $ p^{\\frac{1}{2}d(d-1)}$ for $ d\\geq 3 $ and $ p\\neq 2. $ Here we try to obtain the Schur multiplier, the exterior square and the tensor square of such $p$-groups. 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