C₃×#13742 #10632349

order	768
nilpotency class	6
isoclinism	isoclinic class of #13742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#13742

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂×D₃₂
commutator, Frattini, cosocle	C₃₂	C₂³×C₃
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×C₁₆ ◁ C₂×C₃×C₃₂ ◁ G

134 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 58 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)(6,11)(7,22)(8,26)(9,29)(10,12)(13,17)(14,18)(15,23)(16,27)(19,40)(20,41)(21,30)(24,57)(25,53)(28,59)(31,43)(32,42)(33,50)(34,45)(35,48)(36,56)(37,49)(38,52)(39,58)(44,60)(46,55)(47,54)(51,64)(61,66)(62,65)(63,67);(5,11)(6,17)(7,18)(8,23)(9,27)(12,30)(13,14)(15,40)(16,41)(19,49)(20,52)(21,45)(22,31)(24,58)(25,48)(26,32)(28,56)(29,33)(34,35)(36,64)(37,55)(38,54)(39,60)(42,43)(44,50)(46,47)(51,67)(53,61)(57,63)(59,62)(65,66);(5,12)(11,30)(13,34)(14,35)(15,36)(16,29)(22,53)(26,59)(31,61)(32,62)(33,41)(37,63)(38,60)(39,54)(40,64)(55,57)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#13742, #72255021
maximal subgroup of	C₃×#153598, D₃×#13742, #72255021
maximal normal subgroup of	C₃×#153598, D₃×#13742, #72255021

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6730);