C₂²×#7670211 #72325844

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

Aliases

direct product

C₂²×#7670211

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

1482456 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72232072
maximal normal subgroup of	C₂²×#72232072

GAP

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

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

C22×#7670211 #72325844