C₂²×#7855 #72280414

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄²

Aliases

direct product

C₂²×#7855

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×C₄²
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂⁴	C₂×#95
prime radical	C₂⁶	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

16045 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#233610, #72280514, C₂×#72216619, …
maximal normal subgroup of	C₂²×#233610, #72280514, C₂×#72216619, …
maximal quotient of	C₂×#72261543

GAP

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

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

C22×#7855 #72280414