C₃×#20370 #10638977

order	768
nilpotency class	2
isoclinism	isoclinic class of #20338
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

direct product

C₃×#20370

Subgroups

	subgroup	G/H
center	C₃×C₄	C₂²×C₄²
commutator	C₄	C₂²×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂×C₃	C₂×#376
prime radical	C₂⁴×C₃	#41
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ G

908 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#20370, #72271752
maximal subgroup of	C₃×#375718, D₃×#20370, #72271752
maximal normal subgroup of	C₃×#375718, D₃×#20370, #72271752

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13358);

C3×#20370 #10638977

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×#20370 #10638977