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C7×C4⋊Dic15 #13132346

order1680
isoclinismisoclinic class of Dic30
automorphism?
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×C4⋊Dic15
unambiguous semidirect productC4⋊(C7×Dic15)

Splittings

HallC3×C5×C7 ⋊ C4⋊C4
HallC5×C7 ⋊ C4⋊Dic3
HallC3×C7 ⋊ C4⋊Dic5
HallC3×C5 ⋊ C7×C4⋊C4
SylowC5 ⋊ C7×C4⋊Dic3
SylowC3 ⋊ C7×C4⋊Dic5

Subgroups

subgroupG/H
centerC22×C7C2×D15
commutatorC2×C3×C5C2×C4×C7
FrattiniC22C2×C7×D15
FittingC2×C3×C4×C5×C7C2
socle, prime radicalC22×C3×C5×C7C22
cosocleC22×C3×C5C22×C7
hypercenterC2×C4×C7D15
hypocenterC3×C5C7×C4⋊C4
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C7 ◁ C2×C4×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)(8,9,11,12,10)(13,14,15)(20,21,22,23);(9,10)(11,12)(14,15)(16,17,18,19)(21,23)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 366);