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C3×#13310 #10631917

order768
nilpotency class4
isoclinismisoclinic class of #13310
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C24

Aliases

direct productC3×#13310

Subgroups

subgroupG/H
center, socleC2×C3#1999
commutator, Frattini, cosocleC22×C8C23×C3
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(4,5)(6,17)(7,11)(8,16)(9,14)(10,15)(12,19)(13,21)(18,32)(20,31)(22,27)(23,26)(24,30)(25,33)(28,35)(29,34);(5,11)(7,21)(8,9)(12,28)(13,16)(14,27)(15,26)(17,32)(18,33)(19,24)(22,23)(25,30)(29,31)(34,35);(4,6)(5,12)(7,18)(8,24)(9,19)(10,20)(11,25)(13,29)(14,31)(15,30)(16,17)(21,33)(22,35)(23,34)(26,28)(27,32)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72271409, #72271419, D3×#13310, …
maximal subgroup ofC3×#500001, #72271409, #72271419, …
maximal normal subgroup ofC3×#500001, #72271409, #72271419, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6298);