C₃×#13608 #10632215

order	768
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#13608

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#2127
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	#124	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

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

normal subgroups only

This group has 308 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)(6,16,19,43)(7,12)(8,24,26,58)(9,28,30,14)(10,15)(11,18,32,49)(13,27)(17,42,46,64)(20,39,29,44)(21,52,25,57)(22,54,56,37)(23,38)(31,48,62,67)(33,50,34,51)(35,45,63,66)(36,53)(40,59)(41,60)(47,61,55,65);(4,6,10,20)(5,11,15,34)(7,17,23,47)(8,18,26,51)(9,29,30,19)(12,35,38,31)(13,39,41,16)(14,32,28,33)(21,45,25,48)(22,55,56,46)(24,44,58,43)(27,49,60,50)(36,61,40,42)(37,63,54,62)(52,65,57,64)(53,66,59,67);(4,7,10,23)(5,12,15,38)(6,17,20,47)(8,25,26,21)(9,22,30,56)(11,31,34,35)(13,40,41,36)(14,37,28,54)(16,42,39,61)(18,48,51,45)(19,46,29,55)(24,57,58,52)(27,59,60,53)(32,62,33,63)(43,64,44,65)(49,67,50,66)

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, 6596);