C₃×#32949 #10651556

order	768
nilpotency class	3
isoclinism	isoclinic class of #32947
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃×#32949

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#2847
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₃×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ G

1278 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 1278 conjugacy classes of subgroups — too many to draw. Only its 290 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 — 4 generators on 67 points:

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

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, 25937);