Finite Group Databasebeta

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C5×C7×C4⋊Dic3 #13132314

order1680
isoclinismisoclinic class of Dic6
automorphism?
homologyAbelianization (H1) = C2×C4×C5×C7 · Schur multiplier (H2) = C2

Aliases

direct productC5×C7×C4⋊Dic3
unambiguous semidirect productC4⋊(C5×C7×Dic3)

Splittings

HallC3×C5×C7 ⋊ C4⋊C4
HallC3×C7 ⋊ C5×C4⋊C4
HallC3×C5 ⋊ C7×C4⋊C4
SylowC3 ⋊ C5×C7×C4⋊C4

Subgroups

subgroupG/H
centerC22×C5×C7C2×D3
commutatorC2×C3C2×C4×C5×C7
FrattiniC22C2×C5×D3×C7
FittingC2×C3×C4×C5×C7C2
socle, prime radicalC22×C3×C5×C7C22
cosocleC22×C3C22×C5×C7
hypercenterC2×C4×C5×C7D3
hypocenterC3C5×C7×C4⋊C4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5×C7 ◁ C2×C4×C5×C7

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 1680

A minimal faithful permutation representation — 2 generators on 23 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 334);