Finite Group Databasebeta

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C5×#66635 #13132335

order1680
isoclinismisoclinic class of #66632
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×#66635

Splittings

HallC3×C5×C7 ⋊ Q16
HallC5×C7 ⋊ #216
HallC3×C7 ⋊ C5×Q16
HallC3×C5 ⋊ #1144
SylowC7 ⋊ C5×#216
SylowC3 ⋊ C5×#1144

Subgroups

subgroupG/H
centerC2×C5#4268
commutator, cosocleC3×C4×C7C22×C5
FrattiniC4C2×C5×D21
FittingC3×C5×C7×Q8C2
socle, prime radicalC2×C3×C5×C7D4
hypercenterC5×Q8D21
hypocenterC3×C7C5×Q16
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×Q8

72 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)(6,7,9,11,12,10,8)(13,14,15)(16,18,20,26)(17,21,23,30)(19,28,29,25)(22,31,27,24);(7,8)(9,10)(11,12)(14,15)(16,17,20,23)(18,24,26,31)(19,27,29,22)(21,28,30,25)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 355);