C₂²×#7643002 #72337199

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

Aliases

direct product

C₂²×#7643002

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

1836712 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72235095, C₂×#72339320
maximal normal subgroup of	C₂²×#72235095, C₂×#72339320

GAP

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

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

C22×#7643002 #72337199