C₃×#13612 #10632219

order	768
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#13612

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#2127
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₄⋊Q₈	C₂³×C₃
prime radical	C₂×C₃×#41	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#536 ◁ G

304 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 86 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)(4,5,8,14,10,16,28,43)(6,17,19,41,21,34,48,60)(7,12,24,37,23,38,52,65)(9,15,30,42,31,13,27,40)(11,26,33,29,35,47,61,20)(18,44,45,63,22,49,50,66)(25,36,56,62,53,39,55,64)(32,57,58,51,59,54,67,46);(4,6,10,21)(5,11,16,35)(7,22,23,18)(8,26,28,47)(9,29,31,20)(12,32,38,59)(13,34,15,17)(14,41,43,60)(19,27,48,30)(24,54,52,57)(25,46,53,51)(33,42,61,40)(36,44,39,49)(37,63,65,66)(45,56,50,55)(58,62,67,64);(4,7)(5,12)(6,18)(8,24)(9,25)(10,23)(11,32)(13,39)(14,37)(15,36)(16,38)(17,44)(19,45)(20,46)(21,22)(26,57)(27,55)(28,52)(29,51)(30,56)(31,53)(33,58)(34,49)(35,59)(40,64)(41,63)(42,62)(43,65)(47,54)(48,50)(60,66)(61,67)

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6600);