C₂²×#14173 #72337041

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂²×#14173

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

80157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#1418394, C₂²×#1260050, C₂²×#1419356, …
maximal normal subgroup of	C₂²×#1418394, C₂²×#1260050, C₂²×#1419356, …

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

C22×#14173 #72337041