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C3×#33657 #10652264

order768
nilpotency class4
isoclinismisoclinic class of #33657
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C24

Aliases

direct productC3×#33657

Subgroups

subgroupG/H
center, socleC2×C3C2×#481
commutator, Frattini, cosocleC2×C8C24×C3
prime radicalC3×#3058C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

566 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)(6,14)(7,17)(8,20)(10,23)(12,27)(13,26)(15,31)(16,19)(18,33)(21,34)(22,32)(24,30)(25,28)(29,35);(4,5)(6,10)(7,18)(8,13)(9,11)(12,21)(14,30)(15,24)(16,25)(17,33)(19,35)(20,26)(22,29)(23,31)(27,34)(28,32);(4,6,8,22,9,15,20,16)(5,10,13,29,11,24,26,25)(7,19,17,31,21,32,34,14)(12,28,27,30,18,35,33,23);(5,11)(10,24)(12,18)(13,26)(23,30)(25,29)(27,33)(28,35)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#33657, #72270320, #72270628
maximal subgroup ofC3×#500349, D3×#33657, #72270320, …
maximal normal subgroup ofC3×#500349, D3×#33657, #72270320, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 26645);