C₂²×#3862142 #72291906

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂²×#3862142

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

At least 7836 conjugacy classes of subgroups; only the 41104 normal subgroups are shown.

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72231160
maximal normal subgroup of	C₂²×#72231160

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

C22×#3862142 #72291906