C₃×#12981 #10631588

order	768
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#12981

Subgroups

	subgroup	G/H
center	C₃×C₈	#120
commutator	C₂³	C₂²×C₃×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂×C₃	#1869
prime radical	C₃×#525	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ C₂²×C₃×C₈ ◁ G

488 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)(4,5,8,13,9,14,22,28,10,6,11,16,23,17,24,30)(7,15,19,27,20,29,32,35,18,12,21,25,31,26,33,34);(4,6,8,16,9,17,22,30,10,5,11,13,23,14,24,28)(7,15,21,25,20,29,33,34,18,12,19,27,31,26,32,35);(4,7)(5,12)(6,15)(8,21)(9,20)(10,18)(11,19)(13,27)(14,26)(16,25)(17,29)(22,33)(23,31)(24,32)(28,35)(30,34)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#12981, #72270380
maximal subgroup of	C₃×#495505, D₃×#12981, #72270380
maximal normal subgroup of	C₃×#495505, D₃×#12981, #72270380

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5969);

C3×#12981 #10631588

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×#12981 #10631588