C₂³×#36737 #72325486

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

Aliases

direct product

C₂³×#36737

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

1574888 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)(11,12,15,16)(13,17,18,14)(19,20,23,25)(21,24,26,22);(8,10)(12,16)(14,17)(19,21)(20,24)(22,25)(23,26);(3,4);(5,6);(20,25)(21,26);(1,2)(3,4)(5,6);(8,10)(11,13)(12,14)(15,18)(16,17)(19,21)(20,22)(23,26)(24,25);(7,8,9,10)(11,14)(12,13)(15,17)(16,18)(19,22)(20,21)(23,24)(25,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7792761
maximal normal subgroup of	C₂³×#7792761
maximal quotient of	C₂³×#4312024

GAP

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

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

C23×#36737 #72325486