C₃×#7227 #10625834

order	768
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7227

Subgroups

	subgroup	G/H
center	C₃×C₄	H(3,ℤ/4)
commutator	C₂×C₈	C₃×C₄²
Frattini, cosocle	C₂×#131	C₂²×C₃
socle	C₂×C₃	#1289
prime radical	C₂×C₃×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₈ ◁ G

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

normal subgroups only

This group has 248 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 — 2 generators on 67 points:

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

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, 215);