C₂²×#777314 #72321340

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

Aliases

direct product

C₂²×#777314

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

1143096 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72233877, C₂²×#72219674
maximal normal subgroup of	C₂²×#72233877, C₂²×#72219674
maximal quotient of	C₂²×#72278357

GAP

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

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

C22×#777314 #72321340