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C5×C3⋊D609 #32912528

order18270
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×C3⋊D609
unambiguous semidirect productC3⋊(C5×D609)
unambiguous semidirect productC7⋊(C5×C3⋊D87)
unambiguous semidirect productC29⋊(C5×C3⋊D21)

Splittings

Fitting, socle, HallC32×C5×C7×C29 ⋊ C2
derived, HallC32×C7×C29 ⋊ C2×C5
HallC32×C5×C29 ⋊ D7
HallC5×C7×C29 ⋊ C3⋊D3
HallC32×C5×C7 ⋊ D29
HallC32×C29 ⋊ C5×D7
HallC7×C29 ⋊ C5×C3⋊D3
HallC5×C29 ⋊ C3⋊D21
HallC32×C7 ⋊ C5×D29
HallC32×C5 ⋊ D203
HallC5×C7 ⋊ C3⋊D87
SylowC29 ⋊ C5×C3⋊D21
SylowC32 ⋊ C5×D203
SylowC7 ⋊ C5×C3⋊D87

Subgroups

subgroupG/H
center, hypercenterC5C3⋊D609
commutator, cosocle, hypocenterC32×C7×C29C2×C5
FrattiniC1G
Fitting, socleC32×C5×C7×C29C2
LCSG ▷ C32×C7×C29
UCSC1 ◁ C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 47
transitive degreeμt(G) = 9135

A minimal faithful permutation representation — 3 generators on 47 points:

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

Conjugacy classes & character table

4575 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC5×D3×D609, C5×D21×D87, D15⋊D609, …
maximal normal subgroup ofC5×D3×D609, C5×D21×D87, D15⋊D609, …
maximal quotient ofC5×C3⋊Dic609, C2×C5×C3⋊D609

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18270, 66);