Finite Group Databasebeta

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C3×#89315 #13051180

order1344
isoclinismisoclinic class of #89314
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×#89315

Splittings

HallC3×C7 ⋊ SD64
SylowC7 ⋊ C3×SD64

Subgroups

subgroupG/H
centerC2×C3#6400
commutator, cosocleC7×C16C22×C3
FrattiniC16C2×C3×D7
Fitting, prime radicalC3×C7×D16C2
socleC2×C3×C7D16
hypercenterC3×D16D7
hypocenterC7C3×SD64
LCSG ▷ C7×C16 ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×C16 ◁ C3×D16

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 672

A minimal faithful permutation representation — 2 generators on 42 points:

(1,2,3)(4,5,7,9,10,8,6)(12,17)(13,22)(14,23)(15,27)(18,19)(20,38)(21,32)(24,41)(25,33)(26,30)(28,36)(29,31)(34,42)(35,37)(39,40);(5,6)(7,8)(9,10)(11,12,16,21)(13,17,25,32)(14,26,28,42)(15,29,27,20)(18,33,35,22)(19,36,37,23)(24,38,39,31)(30,41,34,40)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 469);