C₂³×#60045 #72301715

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

Aliases

direct product

C₂³×#60045

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

1141208 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 30 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#8934366, C₂³×#7667013
maximal normal subgroup of	C₂³×#8934366, C₂³×#7667013

GAP

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

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

C23×#60045 #72301715