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C3×#33594 #10652201

order768
nilpotency class4
isoclinismisoclinic class of #2212
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC3×#33594

Subgroups

subgroupG/H
centerC3×C4#459
commutator, Frattini, cosocleC2×C8C24×C3
socleC2×C3C2×#459
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G

696 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)(7,20)(8,14)(9,24)(10,25)(11,16)(12,18)(13,17)(19,32)(21,35)(22,31)(23,27)(26,28)(29,33)(30,34);(4,6,9,18,11,19,8,10)(5,12,14,15,16,25,24,32)(7,17,22,33,23,34,21,26)(13,27,28,31,30,20,29,35);(4,7)(5,13)(6,17)(8,21)(9,22)(10,26)(11,23)(12,27)(14,28)(15,31)(16,30)(18,33)(19,34)(20,25)(24,29)(32,35);(4,8,11,9)(5,14,16,24)(6,10,19,18)(7,21,23,22)(12,15,25,32)(13,28,30,29)(17,26,34,33)(20,35,27,31)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#7623665, C3×#490485, C3×#488869
maximal normal subgroup ofC3×#7623665, C3×#490485, C3×#488869

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 26582);