C₂²×#7737413 #72301072

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

Aliases

direct product

C₂²×#7737413

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

1599752 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)(11,12)(13,14)(15,16)(17,18)(19,22)(20,24)(21,23);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,23)(22,24);(17,19)(18,22)(20,23)(21,24);(10,13)(11,15)(18,23)(20,22);(9,12)(10,11)(13,15)(14,16)(17,18)(19,22)(20,24)(21,23);(9,13,14,10)(11,16,15,12)(17,20,21,22)(18,19,23,24);(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, C₂²×#72236233
maximal normal subgroup of	C₂²×#72232033, C₂²×#72236233

GAP

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

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

C22×#7737413 #72301072