C₃×#129173 #72226282

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#129173

Subgroups

	subgroup	G/H
center	C₃×C₄	#1776
commutator	C₂×C₈	C₂×C₃×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³×C₃
socle	C₂×C₃	#8286
prime radical	C₂³×C₃	#444
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₃×C₄×M₃₂ ◁ G

430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,4,11,6,10,5,21,7,12,16,25,23,13,22,8)(3,14,18,27,20,24,19,9,17,26,32,29,15,30,31,28)(33,34,35);(2,8,10,11,12,21,13,25)(3,15,17,20)(4,16)(5,22)(9,26,27,24,28,14,29,30)(18,19,32,31);(1,3,16,32,6,20,22,31,7,17,4,18,23,15,5,19)(2,9,21,30,13,27,11,26,12,28,8,24,10,29,25,14)

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#72214092
maximal normal subgroup of	C₃×#72214092

GAP

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

gap> G := Group((1,2,4,11,6,10,5,21,7,12,16,25,23,13,22,8)(3,14,18,27,20,24,19,9,17,26,32,29,15,30,31,28)(33,34,35), (2,8,10,11,12,21,13,25)(3,15,17,20)(4,16)(5,22)(9,26,27,24,28,14,29,30)(18,19,32,31), (1,3,16,32,6,20,22,31,7,17,4,18,23,15,5,19)(2,9,21,30,13,27,11,26,12,28,8,24,10,29,25,14));

C3×#129173 #72226282