Finite Group Databasebeta

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Cayley diagram

C4⋊(C3⋊D15) #67182

order360
isoclinismisoclinic class of #67180
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2×C3

Aliases

unambiguous semidirect productC4⋊(C3⋊D15)

Splittings

FittingC32×C4×C5 ⋊ C2
HallC32×C5 ⋊ D4
SylowC32 ⋊ D20
SylowC5 ⋊ C4⋊(C3⋊D3)

Subgroups

subgroupG/H
center, FrattiniC2C2×C3⋊D15
commutator, socle, cosocleC2×C32×C5C22
FittingC32×C4×C5C2
hypercenterC4C3⋊D15
hypocenterC32×C5D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 180

A minimal faithful permutation representation — 3 generators on 15 points:

(1,2,4,5,3)(6,7,8,9)(13,14,15);(2,3)(4,5)(7,9)(11,12)(14,15);(10,11,12)

Polycyclic presentation:

2^3,3^2,5|b^2=c,b^a=bc,d^a=d^2,e^a=e^2,f^a=f^4

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10623758, #10623759, #10623763, …
maximal normal subgroup of#10623758, #10623759, #10623763, …
maximal quotient of#10623900, #10623901, #10623902, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 112);