C₂³×#8195808 #72301722

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

Aliases

direct product

C₂³×#8195808

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

14266811 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);(7,8,11,12)(9,10,13,14)(15,16,19,21)(17,20,22,18)(23,24,27,29)(25,26,30,28);(7,9)(8,10)(11,13)(12,14)(15,17)(16,18)(19,22)(20,21)(23,25,27,30)(24,28,29,26);(9,13)(10,14)(15,17)(16,20)(18,21)(19,22)(23,25)(24,26)(27,30)(28,29);(15,17,19,22)(16,20,21,18);(7,9)(8,10)(11,13)(12,14)(15,17)(16,18)(19,22)(20,21);(7,10)(8,9)(11,14)(12,13)(15,18)(16,17)(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), (7,8,11,12)(9,10,13,14)(15,16,19,21)(17,20,22,18)(23,24,27,29)(25,26,30,28), (7,9)(8,10)(11,13)(12,14)(15,17)(16,18)(19,22)(20,21)(23,25,27,30)(24,28,29,26), (9,13)(10,14)(15,17)(16,20)(18,21)(19,22)(23,25)(24,26)(27,30)(28,29), (15,17,19,22)(16,20,21,18), (7,9)(8,10)(11,13)(12,14)(15,17)(16,18)(19,22)(20,21), (7,10)(8,9)(11,14)(12,13)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28)(29,30));

C23×#8195808 #72301722