C₂²×#7642257 #72337198

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂²×#7642257

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1621736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72235095
maximal normal subgroup of	C₂²×#72235095

GAP

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

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

C22×#7642257 #72337198