C₂³×#43118 #72300230

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂³×#43118

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1687680 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7849701, C₂³×#7792761
maximal normal subgroup of	C₂³×#7849701, C₂³×#7792761
maximal quotient of	C₂³×#1262353, C₂³×#1418391

GAP

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

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

C23×#43118 #72300230