C₂³×#2857 #72339159

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂³×#2857

Subgroups

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

156373 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:

(15,16,18,20)(17,19,22,21);(11,12,13,14);(3,4);(5,6);(8,10)(15,17)(16,19)(18,22)(20,21);(7,8,9,10)(11,12)(13,14)(15,16)(17,21)(18,20)(19,22);(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×#612911, C₂³×#36121, C₂³×#36154, …
maximal normal subgroup of	C₂²×#612911, C₂³×#36121, C₂³×#36154, …

GAP

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

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

C23×#2857 #72339159