C₂³×#15670 #72325715

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

Aliases

direct product

C₂³×#15670

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁸	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

723816 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)(15,16,19,21)(17,22,18,20)(23,24,27,29)(25,30,28,26);(8,10)(12,14)(15,17,19,18)(16,20,21,22)(23,24,27,29)(25,26,28,30);(3,4);(5,6);(11,12)(13,14)(15,18,19,17)(16,20,21,22)(23,25)(24,26)(27,28)(29,30);(1,2)(3,4)(5,6);(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,22)(19,21)(23,26)(24,28)(25,29)(27,30)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#1420724
maximal normal subgroup of	C₂³×#1420724

GAP

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

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

C23×#15670 #72325715