C₂²×#7626202 #72296079

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷×C₄ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂²×#7626202

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator	C₂²	C₂⁷×C₄
Frattini, cosocle	C₂³	C₂⁸
prime radical	C₂⁵×#120	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3456616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 8 generators on 24 points:

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

Conjugacy classes & character table

608 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#72217511
maximal normal subgroup of	C₂²×#72217511

GAP

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

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

C22×#7626202 #72296079