C₂³×#14473 #72325804

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²¹×C₄

Aliases

direct product

C₂³×#14473

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁶×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1002952 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:

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

Conjugacy classes & character table

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

Relationships

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

GAP

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

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

C23×#14473 #72325804