C₃×#13647 #10632254

order	768
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#13647

Subgroups

	subgroup	G/H
center	C₃×C₃₂	D₄
commutator	C₄	C₂²×C₃×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₃
socle	C₂×C₃	D₄×C₁₆
prime radical	C₃×#135	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₃₂ ◁ C₂×C₃×C₃₂ ◁ G

186 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 118 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)(4,5,9,15,10,16,31,45,8,17,28,46,29,48,59,66,11,18,32,47,33,49,61,67,30,14,41,43,60,44,35,62)(6,19,21,50,22,25,39,56,23,37,52,63,54,27,40,58,24,42,53,65,55,57,38,64,20,12,34,36,51,7,13,26);(4,6)(5,12)(7,16)(8,23)(9,21)(10,22)(11,24)(13,35)(14,42)(15,36)(17,19)(18,37)(20,30)(25,48)(26,45)(27,49)(28,52)(29,54)(31,39)(32,53)(33,55)(34,41)(38,61)(40,59)(43,65)(44,57)(46,50)(47,63)(51,60)(56,66)(58,67)(62,64);(4,7,9,26,10,19,31,50,8,25,28,56,29,37,59,63,11,27,32,58,33,42,61,65,30,57,41,64,60,12,35,36)(5,13,15,6,16,21,45,22,17,39,46,23,48,52,66,54,18,40,47,24,49,53,67,55,14,38,43,20,44,34,62,51)

Conjugacy classes & character table

336 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#11711660
maximal quotient of	C₃×SD₁₆×C₃₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6635);