C₃×#93759 #72245502

order	1536
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#93759

Subgroups

	subgroup	G/H
center	C₃×C₄	#1333
commutator	C₁₆	C₂×C₃×C₁₆
Frattini, cosocle	#1311	C₂²×C₃
socle	C₂×C₃	#7070
prime radical	C₃×#508	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×C₄×C₁₆ ◁ C₃×#1311 ◁ G

174 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 35 points:

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

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#93759, #72254965
maximal subgroup of	C₃×#72224647, D₃×#93759, #72254965
maximal normal subgroup of	C₃×#72224647, D₃×#93759, #72254965

GAP

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

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