C₃×#13078 #10631685

order	768
nilpotency class	4
isoclinism	isoclinic class of #13077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13078

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#1893
commutator	C₂×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₂²×C₈	C₂³×C₃
prime radical	C₂×C₃×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#131 ◁ G

454 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 146 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)(4,5,9,16,11,15,10,34)(6,17,21,47,20,44,32,48)(7,22,24,43,25,13,33,42)(8,26,29,36,31,58,30,14)(12,35,38,19,40,54,39,27)(18,46,51,66,50,63,52,67)(23,56,57,45,28,59,60,41)(37,62,64,53,55,61,65,49);(4,6,8,27,10,21,30,35,11,20,31,19,9,32,29,54)(5,12,36,17,16,39,58,48,15,40,14,44,34,38,26,47)(7,18,28,49,24,52,60,61,25,50,23,53,33,51,57,62)(13,37,41,46,43,64,59,66,22,55,45,63,42,65,56,67);(4,7)(5,13)(6,18)(8,28)(9,33)(10,24)(11,25)(12,37)(14,45)(15,22)(16,43)(17,46)(19,53)(20,50)(21,52)(23,31)(26,56)(27,49)(29,57)(30,60)(32,51)(34,42)(35,61)(36,41)(38,65)(39,64)(40,55)(44,63)(47,67)(48,66)(54,62)(58,59)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6066);