C₂³×#14722 #72325803

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²³

Aliases

direct product

C₂³×#14722

Subgroups

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

1776472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 1024

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#1418391, C₂³×#1082737, C₂³×#1085256
maximal normal subgroup of	C₂³×#1418391, C₂³×#1082737, C₂³×#1085256

GAP

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

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

C23×#14722 #72325803