C₂²×#7733603 #72291833

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

Aliases

direct product

C₂²×#7733603

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

1650600 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)(18,23)(20,21);(17,18,22,23)(19,21,24,20);(9,10)(11,14)(12,15)(13,16)(17,19)(18,20)(21,23)(22,24);(9,11)(10,13)(12,16)(14,15)(17,20,22,21)(18,19,23,24);(5,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,21,22,20)(18,19,23,24);(9,10)(11,13)(12,15)(14,16)(17,19)(18,21)(20,23)(22,24);(3,4);(1,2)(3,4)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72232033
maximal normal subgroup of	C₂²×#72232033
maximal quotient of	C₂²×#72231160, C₂²×#72231349

GAP

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

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

C22×#7733603 #72291833