C₂³×#2858 #72339125

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄²

Aliases

direct product

C₂³×#2858

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁴×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

185461 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#681049, C₂³×#35274, C₂³×#36121, …
maximal normal subgroup of	C₂²×#681049, C₂³×#35274, C₂³×#36121, …

GAP

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

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

C23×#2858 #72339125