C₂³×#47942 #72325170

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

Aliases

direct product

C₂³×#47942

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

1763392 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7918389, C₂³×#7792761
maximal normal subgroup of	C₂³×#7918389, C₂³×#7792761
maximal quotient of	C₂³×#1259281

GAP

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

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

C23×#47942 #72325170