Extraspecial groups p_±¹⁺²ⁿ

An extraspecial group is a p-group G with center Z of order p such that G/Z is (non-trivial and) elementary abelian. There are exactly two such groups for every prime power of the form p¹⁺²ⁿ, denoted p₊¹⁺²ⁿ and p_-¹⁺²ⁿ. For odd p, the '+' one has exponent p, and the '-' one exponent p². When n=1, p₊¹⁺²=He_p is the Heisenberg group, and p_-¹⁺²=He_p=C_p²⋊C_p the other non-abelian group of order p³.

D₄=2₊¹⁺² Q₈=2_-¹⁺² He₃=3₊¹⁺² 3_-¹⁺² 2₊¹⁺⁴ 2_-¹⁺⁴ He₅=5₊¹⁺² 5_-¹⁺² 2₊¹⁺⁶ 2_-¹⁺⁶ 3₊¹⁺⁴ 3_-¹⁺⁴ He₇=7₊¹⁺² 7_-¹⁺²

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Extraspecial groups p±1+2n

Extraspecial groups p_±¹⁺²ⁿ