C₂²×#613649 #72295720

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²²

Aliases

direct product

C₂²×#613649

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1033832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72232559
maximal normal subgroup of	C₂²×#72232559

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

C22×#613649 #72295720