C₂²×#686906 #72321348

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰×C₄

Aliases

direct product

C₂²×#686906

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

1107236 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)(9,10)(11,12)(14,16)(19,24)(22,23);(6,8)(10,11)(17,18,21,20)(19,23,24,22);(13,14)(15,16)(17,19)(18,22)(20,23)(21,24);(1,2)(3,4);(17,19)(18,23)(20,22)(21,24);(5,6,7,8)(9,11,12,10)(13,14)(15,16)(17,20,21,18)(19,23,24,22)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72233877
maximal normal subgroup of	C₂²×#72233877
maximal quotient of	C₂²×#72278330

GAP

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

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

C22×#686906 #72321348