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C3×#13318 #10631925

order768
nilpotency class4
isoclinismisoclinic class of #13318
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C24

Aliases

direct productC3×#13318

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

532 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)(7,21)(8,16)(9,14)(10,15)(11,18)(12,19)(13,25)(20,33)(22,34)(23,29)(24,30)(26,32)(27,31)(28,35);(4,6)(5,11)(7,22)(8,19)(9,27)(10,20)(12,30)(13,31)(14,28)(15,29)(16,34)(17,21)(18,25)(23,24)(26,35)(32,33);(4,6)(5,12)(7,23)(8,27)(9,19)(10,20)(11,13)(14,33)(15,31)(16,17)(18,26)(21,28)(22,24)(25,35)(29,30)(32,34)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72270166, #72270170, D3×#13318, …
maximal subgroup ofC3×#499752, #72270166, #72270170, …
maximal normal subgroup ofC3×#499752, #72270166, #72270170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6306);