C₂²×#9093 #72308497

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#9093

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#386
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
prime radical	C₂⁴×D₈	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

11369 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,14,16,22)(15,18,23,21)(17,20,25,27)(19,26,28,24);(1,2)(3,4)(7,11)(10,12)(14,18)(15,23)(19,20)(21,22)(24,26)(27,28);(1,2);(3,4);(5,7,8,12,9,11,6,10)(13,15,16,23)(14,19)(17,26,25,24)(18,27)(20,21)(22,28)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#196123
maximal normal subgroup of	C₂²×#196123
maximal quotient of	#72309257

GAP

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

gap> G := Group((1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,14,16,22)(15,18,23,21)(17,20,25,27)(19,26,28,24), (1,2)(3,4)(7,11)(10,12)(14,18)(15,23)(19,20)(21,22)(24,26)(27,28), (1,2), (3,4), (5,7,8,12,9,11,6,10)(13,15,16,23)(14,19)(17,26,25,24)(18,27)(20,21)(22,28));

C22×#9093 #72308497