C₃×#13312 #10631919

order	768
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13312

Subgroups

	subgroup	G/H
center	C₃×C₄	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂×C₃	#1999
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#131 ◁ G

520 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 67
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 67 points:

(1,2,3)(4,5)(6,18)(7,24)(8,29)(9,16)(10,15)(11,17)(12,20)(13,27)(14,32)(19,48)(21,54)(22,42)(23,47)(25,57)(26,59)(28,40)(30,45)(31,44)(33,43)(34,41)(35,64)(36,52)(37,53)(38,49)(39,62)(46,50)(51,67)(55,58)(56,61)(60,65)(63,66);(4,6)(5,12)(7,25)(8,30)(9,21)(10,22)(11,23)(13,38)(14,42)(15,36)(16,45)(17,37)(18,29)(19,27)(20,32)(24,55)(26,60)(28,50)(31,53)(33,52)(34,51)(35,40)(39,46)(41,65)(43,47)(44,54)(48,56)(49,62)(57,59)(58,61)(63,64)(66,67);(4,7)(5,13)(6,19)(8,26)(9,34)(10,27)(11,28)(12,35)(14,39)(15,40)(16,24)(17,41)(18,46)(20,49)(21,25)(22,50)(23,51)(29,56)(30,58)(31,63)(32,61)(33,62)(36,65)(37,55)(38,45)(42,67)(43,66)(44,59)(47,57)(48,54)(52,64)(53,60)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6300);