C₃×#12011 #10630618

order	768
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#12011

Subgroups

	subgroup	G/H
center	C₃×C₈	C₂×D₈
commutator	C₈	C₂²×C₃×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₃
socle	C₂×C₃	C₂×#335
prime radical	C₂³×C₃	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×C₄×C₈ ◁ G

220 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:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72228277
maximal subgroup of	C₃×#483512, D₃×#12011, #72226258, …
maximal normal subgroup of	C₃×#483512, D₃×#12011, #72226258, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 4999);

C3×#12011 #10630618

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×#12011 #10630618