C₂³×#8934366 #72301723

order	4096
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁴

Aliases

direct product

C₂³×#8934366

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁹
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

21057531 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 9 generators on 30 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#72236189
maximal normal subgroup of	C₂³×#72236189

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

C23×#8934366 #72301723