C₂³×#37650 #72289503

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶×C₄

Aliases

direct product

C₂³×#37650

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

2348176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 22 points:

(11,12,13,14)(15,16)(17,20)(18,21)(19,22);(11,12)(13,14)(15,17)(16,19)(18,22)(20,21);(7,8,9,10)(11,12,13,14)(15,17)(16,20)(18,22)(19,21);(8,10)(11,12)(13,14)(15,17)(16,20)(18,22)(19,21);(7,8)(9,10)(15,16)(17,19)(18,21)(20,22);(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×#7684106
maximal normal subgroup of	C₂³×#7684106

GAP

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

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

C23×#37650 #72289503