C₂×#7716724 #72323775

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7716724

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

95285 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 22 points:

(15,16)(17,22)(18,20)(19,21);(4,6)(8,9)(12,13);(3,4)(5,6)(7,8,11,9)(10,12,14,13)(15,17,19,18)(16,20,21,22);(8,9)(12,13)(15,16)(17,20)(18,22)(19,21);(1,2);(7,9,11,8)(10,12,14,13);(7,10,11,14)(8,12,9,13)(15,18,19,17)(16,20,21,22)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72238224, C₂×#72232032, C₂×#72323293
maximal normal subgroup of	C₂×#72238224, C₂×#72232032, C₂×#72323293
maximal quotient of	C₂×#72323250, C₂×#72232052, C₂×#72323396, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((15,16)(17,22)(18,20)(19,21), (4,6)(8,9)(12,13), (3,4)(5,6)(7,8,11,9)(10,12,14,13)(15,17,19,18)(16,20,21,22), (8,9)(12,13)(15,16)(17,20)(18,22)(19,21), (1,2), (7,9,11,8)(10,12,14,13), (7,10,11,14)(8,12,9,13)(15,18,19,17)(16,20,21,22));

C2×#7716724 #72323775