Finite Group Databasebeta

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C3×#89301 #13051165

order1344
isoclinismisoclinic class of D112
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#89301

Splittings

HallC3×C7 ⋊ #368
SylowC7 ⋊ C3×#368

Subgroups

subgroupG/H
centerC3×C4D56
commutatorC7×C8C2×C3×C4
FrattiniC2×C8C2×C3×D7
FittingC2×C3×C7×C16C2
socleC2×C3×C7#107
prime radicalC22×C3×C7Q16
cosocleC2×C7×C8C22×C3
hypercenterC2×C3×C16D7
hypocenterC7C3×#368
LCSG ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C2×C3×C16

80 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)(11,13,23,39,16,26,30,40,17,27,28,32,14,24,15,25)(12,18,29,36,19,33,35,38,21,34,42,41,31,37,20,22);(5,6)(7,8)(9,10)(11,12,14,19,17,21,16,31)(13,22,24,36,27,38,26,41)(15,29,28,35,30,42,23,20)(18,32,33,40,34,39,37,25)

Conjugacy classes & character table

354 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 454);