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C5×C22⋊#10620252 #27189262

order13020
automorphism?
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×C22⋊#10620252
unambiguous semidirect productC22⋊(C5×#10620252)
unambiguous semidirect product(C22×C5)⋊#10620252

Splittings

Fitting, socle, HallC22×C5×C7×C31 ⋊ C3
derived, HallC22×C7×C31 ⋊ C3×C5
HallC5×C7×C31 ⋊ A4
HallC22×C5×C31 ⋊ ΓL(1,𝔽8)
HallC7×C31 ⋊ C5×A4
HallC5×C31 ⋊ C22⋊ΓL(1,𝔽8)
HallC22×C5×C7 ⋊ C31⋊C3
HallC22×C31 ⋊ C5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C22⋊(C31⋊C3)
SylowC31 ⋊ C5×C22⋊ΓL(1,𝔽8)
HallC22×C7 ⋊ C5×C31⋊C3
HallC22×C5 ⋊ #10620252
SylowC7 ⋊ C5×C22⋊(C31⋊C3)
SylowC22 ⋊ C5×#10620252

Subgroups

subgroupG/H
center, hypercenterC5C22⋊#10620252
commutator, cosocle, hypocenterC22×C7×C31C3×C5
FrattiniC1G
Fitting, socleC22×C5×C7×C31C3
LCSG ▷ C22×C7×C31
UCSC1 ◁ C5

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 47
transitive degreeμt(G) = 4340

A minimal faithful permutation representation — 2 generators on 47 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,21)(11,15,18)(13,19,16)(17,24,27)(20,26,25)(22,28,23)(29,31,30)(33,34,35)(36,38,37)(40,41,42)(43,44,45,46,47);(1,2,4,8,15,3,6,11,17,24,7,13,20,21,27,14,22,29,28,12,18,25,30,31,19,5,9,16,23,26,10)(32,33,34,36,35,37,38)(39,42)(40,41)

Conjugacy classes & character table

1460 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 229);