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C7×#5464 #13060800

order1344
isoclinismisoclinic class of #5464
automorphism?
homologyAbelianization (H1) = C3×C7 · Schur multiplier (H2) = C22×C42

Aliases

direct productC7×#5464

Splittings

Fitting, HallC7×H(3,𝔽4) ⋊ C3
derived, SylowH(3,𝔽4) ⋊ C3×C7

Subgroups

subgroupG/H
center, hypercenterC7#5464
commutator, cosocle, hypocenterH(3,𝔽4)C3×C7
FrattiniC22C7×C22⋊A4
FittingC7×H(3,𝔽4)C3
socleC22×C7C22⋊A4
LCSG ▷ H(3,𝔽4)
UCSC1 ◁ C7

140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3,4,5,6,7)(9,10,13)(11,12,20)(14,18,23)(15,19,21)(16,17,22);(9,16)(10,19)(13,23)(14,15)(17,18)(21,22);(8,13)(9,10)(11,22)(12,21)(14,19)(15,17)(16,18)(20,23)

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#5464, C7×#3532⋊C3, D7×#5464, …
maximal normal subgroup ofΓL(1,𝔽8)×#5464, C7×#3532⋊C3, D7×#5464, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10089);