Finite Group Databasebeta

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C7×#4689 #13055724

order1344
isoclinismisoclinic class of Dic6
automorphism?
homologyAbelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C2×C4

Aliases

direct productC7×#4689

Splittings

FittingC3×C4×C7×C8 ⋊ C2
HallC3×C7 ⋊ #405
SylowC3 ⋊ C7×#405

Subgroups

subgroupG/H
centerC2×C7×C8C2×D3
commutatorC2×C3C2×C42×C7
FrattiniC2×C4C22×D3×C7
FittingC3×C4×C7×C8C2
socleC22×C3×C7C22×C4
prime radicalC22×D3×C7C2×C4
cosocleC2×C3×C4C23×C7
hypercenterC4×C7×C8D3
hypocenterC3C7×#405
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7×C8 ◁ C4×C7×C8

260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 672

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3,4,5,6,7)(8,9,10)(15,17,18,23,19,24,25,29)(16,21,22,28,20,26,27,30);(11,12,13,14)(16,20)(21,26)(22,27)(28,30);(9,10)(15,16)(17,21)(18,22)(19,20)(23,28)(24,26)(25,27)(29,30)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5013);