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C3×#13396 #10632003

order768
nilpotency class4
isoclinismisoclinic class of #13396
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC3×#13396

Subgroups

subgroupG/H
center, socleC2×C3#2034
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC3×#3058C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

422 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)(6,15)(7,20)(8,9)(11,25)(12,27)(13,24)(16,28)(17,32)(18,33)(19,21)(22,23)(26,35)(29,34)(30,31);(4,5)(6,16)(7,12)(8,24)(9,13)(10,14)(11,18)(15,31)(17,30)(19,26)(20,27)(21,25)(22,34)(23,29)(28,32)(33,35);(4,6,9,18,10,17,8,19)(5,11,13,30,14,26,24,16)(7,21,20,32,23,33,22,15)(12,28,27,35,29,31,34,25)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#13396, #72270347, #72271642, …
maximal subgroup ofC3×#500349, D3×#13396, #72270347, …
maximal normal subgroup ofC3×#500349, D3×#13396, #72270347, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6384);