C₂³×#7985294 #72301720

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

Aliases

direct product

C₂³×#7985294

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

14062491 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

808 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), (7,8,10,12)(9,11,13,14)(15,16)(17,20)(18,22)(19,21)(23,24,26,29)(25,28,30,27), (7,9)(8,11)(10,13)(12,14)(15,17)(16,18)(19,22)(20,21)(23,25,26,30)(24,27,29,28), (9,13)(11,14)(17,22)(18,20)(23,25)(24,28)(26,30)(27,29), (15,17,19,22)(16,20,21,18), (7,9)(8,11)(10,13)(12,14)(15,17)(16,18)(19,22)(20,21), (7,8)(9,11)(10,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,24)(25,27)(26,29)(28,30));

C23×#7985294 #72301720