C₂²×#7685422 #72325873

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

Aliases

direct product

C₂²×#7685422

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

At least 10146 conjugacy classes of subgroups; only the 449296 normal subgroups are shown.

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:

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

Conjugacy classes & character table

452 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((12,16)(14,15)(17,18)(19,22)(20,23)(21,24), (12,16)(14,15)(17,19)(18,21)(20,24)(22,23), (5,6)(7,8)(9,10,13,11)(12,14,16,15), (3,4), (6,8)(10,11)(12,16)(17,19)(18,22)(20,24)(21,23), (1,2)(3,4), (5,6)(7,8)(9,11,13,10)(12,14,16,15), (5,6)(7,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,21)(20,23)(22,24));

C22×#7685422 #72325873