C₃×#13362 #10631969

order	768
nilpotency class	4
isoclinism	isoclinic class of #13360
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13362

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₃
prime radical	C₃×#585	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×#41 ◁ G

670 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 94 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)(4,5,11,16)(6,17,21,43)(7,22,27,47)(8,28,32,14)(9,33,10,15)(12,29,39,19)(13,26,42,25)(18,50,52,66)(20,55,34,49)(23,48,56,67)(24,59,30,57)(31,45,35,46)(36,60,65,58)(37,53,38,54)(40,51,64,63)(41,61,44,62);(4,6)(5,12)(7,23)(8,29)(9,20)(10,34)(11,21)(13,40)(14,43)(15,37)(16,39)(17,28)(18,26)(19,32)(22,36)(24,60)(25,52)(27,56)(30,58)(31,54)(33,38)(35,53)(41,50)(42,64)(44,66)(45,55)(46,49)(47,65)(48,59)(51,61)(57,67)(62,63);(4,7,11,27)(5,13,16,42)(6,18,21,52)(8,30,32,24)(9,26,10,25)(12,36,39,65)(14,44,28,41)(15,47,33,22)(17,48,43,67)(19,51,29,63)(20,56,34,23)(31,61,35,62)(37,40,38,64)(45,57,46,59)(49,66,55,50)(53,60,54,58)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6350);