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C3×#33533 #10652140

order768
nilpotency class4
isoclinismisoclinic class of #33531
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC3×#33533

Subgroups

subgroupG/H
center, socleC2×C3C2×#447
commutator, Frattini, cosocleC2×C8C24×C3
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

1102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72270144, #72270148, D3×#33533
maximal subgroup ofC3×#499752, #72270144, #72270148, …
maximal normal subgroup ofC3×#499752, #72270144, #72270148, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 26521);