Finite Group Databasebeta

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C7×C3⋊#672 #13131990

order1680
isoclinismisoclinic class of D15
automorphism?
homologyAbelianization (H1) = C7×C16 · Schur multiplier (H2) = C1

Aliases

direct productC7×C3⋊#672
unambiguous semidirect productC3⋊(C7×#672)

Splittings

derived, HallC3×C5 ⋊ C7×C16
HallC3×C5×C7 ⋊ C16
HallC5×C7 ⋊ C3⋊C16
HallC3×C7 ⋊ #672
SylowC5 ⋊ C7×C3⋊C16
SylowC3 ⋊ C7×#672

Subgroups

subgroupG/H
center, hypercenterC7×C8D15
commutator, hypocenterC3×C5C7×C16
FrattiniC8C7×D15
FittingC3×C5×C7×C8C2
socle, prime radicalC2×C3×C5×C7C8
cosocleC3×C5×C8C2×C7
LCSG ▷ C3×C5
UCSC1 ◁ C7×C8

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 1680

A minimal faithful permutation representation — 2 generators on 31 points:

(1,2,3,4,5,6,7)(9,10)(11,12)(14,15)(16,17,18,21,19,22,24,27,20,23,25,28,26,29,30,31);(8,9,11,12,10)(13,14,15)

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(1680, 10);