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C3×#13417 #10632024

order768
nilpotency class4
isoclinismisoclinic class of #13417
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC3×#13417

Subgroups

subgroupG/H
center, socleC2×C3#2041
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC3×#3155C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#405 ◁ G

454 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)(6,18)(7,12)(8,24)(9,15)(10,14)(11,17)(13,21)(16,26)(19,29)(20,32)(22,30)(23,34)(25,31)(27,33)(28,35);(5,12)(6,19)(7,10)(8,9)(13,30)(14,17)(15,34)(18,33)(20,21)(22,25)(23,26)(27,28)(29,32)(31,35);(4,6,7,20,11,22,10,28)(5,13,14,31,17,33,12,29)(8,25,23,21,26,19,9,27)(15,32,16,18,34,35,24,30)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#13417, #72271634, #72271688, …
maximal subgroup ofC3×#499269, D3×#13417, #72271634, …
maximal normal subgroup ofC3×#499269, D3×#13417, #72271634, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6405);