C₂³×#13956 #72325799

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁴

Aliases

direct product

C₂³×#13956

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁶×#41	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4994440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 7 generators on 26 points:

(8,9)(11,12)(13,14)(15,16,17,18)(19,20,22,24)(21,23,26,25);(7,8,10,9)(11,12)(13,14)(15,16)(17,18);(3,4);(5,6);(12,14)(16,18)(19,21)(20,23)(22,26)(24,25);(1,2)(3,4)(5,6);(7,9,10,8)(11,12)(13,14)(15,16)(17,18)(19,20)(21,25)(22,24)(23,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#850465, C₂³×#1420724, C₂³×#4928938
maximal normal subgroup of	C₂³×#850465, C₂³×#1420724, C₂³×#4928938

GAP

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

gap> G := Group((8,9)(11,12)(13,14)(15,16,17,18)(19,20,22,24)(21,23,26,25), (7,8,10,9)(11,12)(13,14)(15,16)(17,18), (3,4), (5,6), (12,14)(16,18)(19,21)(20,23)(22,26)(24,25), (1,2)(3,4)(5,6), (7,9,10,8)(11,12)(13,14)(15,16)(17,18)(19,20)(21,25)(22,24)(23,26));

C23×#13956 #72325799