C₂³×#16632 #72325673

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰×C₄

Aliases

direct product

C₂³×#16632

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁶×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

876616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 7 generators on 30 points:

(8,10)(12,14)(24,27)(28,29);(8,10)(12,14)(15,16,18,20)(17,19,22,21)(23,24,26,27)(25,29,30,28);(3,4);(5,6);(8,10)(11,12,13,14)(15,16)(17,21)(18,20)(19,22)(23,25,26,30)(24,28,27,29);(1,2)(3,4)(5,6);(7,8)(9,10)(11,12)(13,14)(15,17)(16,19)(18,22)(20,21)(23,24)(25,28)(26,27)(29,30)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#850465, C₂³×#1262353
maximal normal subgroup of	C₂³×#850465, C₂³×#1262353

GAP

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

gap> G := Group((8,10)(12,14)(24,27)(28,29), (8,10)(12,14)(15,16,18,20)(17,19,22,21)(23,24,26,27)(25,29,30,28), (3,4), (5,6), (8,10)(11,12,13,14)(15,16)(17,21)(18,20)(19,22)(23,25,26,30)(24,28,27,29), (1,2)(3,4)(5,6), (7,8)(9,10)(11,12)(13,14)(15,17)(16,19)(18,22)(20,21)(23,24)(25,28)(26,27)(29,30));

C23×#16632 #72325673