C₂³×#14171 #72325774

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²³

Aliases

direct product

C₂³×#14171

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁴×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1685256 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#1418391, C₂³×#1420724
maximal normal subgroup of	C₂³×#1418391, C₂³×#1420724

GAP

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

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

C23×#14171 #72325774