Finite Group Databasebeta

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C3×C7×#672 #13131986

order1680
isoclinismisoclinic class of D5
automorphism?
homologyAbelianization (H1) = C3×C7×C16 · Schur multiplier (H2) = C1

Aliases

direct productC3×C7×#672

Splittings

derived, SylowC5 ⋊ C3×C7×C16
HallC3×C5×C7 ⋊ C16
HallC5×C7 ⋊ C3×C16
HallC3×C5 ⋊ C7×C16

Subgroups

subgroupG/H
center, hypercenterC3×C7×C8D5
commutator, hypocenterC5C3×C7×C16
FrattiniC8C3×C7×D5
FittingC3×C5×C7×C8C2
socle, prime radicalC2×C3×C5×C7C8
cosocleC5×C8C2×C3×C7
LCSG ▷ C5
UCSC1 ◁ C3×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)(4,5,6,7,8,9,10)(12,13)(14,15)(16,17,18,21,19,22,24,27,20,23,25,28,26,29,30,31);(11,12,14,15,13)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 6);