Finite Group Databasebeta

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C7×#4623 #13051532

order1344
isoclinismisoclinic class of #857
automorphism?
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×#4623

Splittings

derivedC42⋊C3 ⋊ C4×C7

Subgroups

subgroupG/H
center, hypercenterC2×C7#857
commutator, hypocenterC42⋊C3C4×C7
FrattiniC23C7×S4
FittingC2×C42×C7D3
socleC23×C7S4
prime radicalC2×C7×C42⋊C3C2
cosocleC2×C42⋊C3C2×C7
LCSG ▷ C42⋊C3
UCSC1 ◁ C2×C7

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 84

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2,3,4,5,6,7)(11,18,17,15,14,16,13,19);(8,11,16)(9,13,15)(10,14,18)(12,17,19)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 821);