Finite Group Databasebeta

← Prev (#90643) Next (#90645) →

Cayley diagram

C3⋊D75 #90644

order450
isoclinismisoclinic class of C3⋊D75
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊D75
unambiguous semidirect productC25⋊(C3⋊D3)

Splittings

Fitting, derived, HallC32×C25 ⋊ C2
SylowC25 ⋊ C3⋊D3
SylowC32 ⋊ D25

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC32×C25C2
FrattiniC5C3⋊D15
socleC32×C5D5
LCSG ▷ C32×C25
UCSC1

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 225

A minimal faithful permutation representation — 3 generators on 31 points:

(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,12)(14,20)(17,21)(18,22)(19,24)(23,25)(27,28)(30,31);(1,21,12,24,9,7,14,5,23,4,16,6,22,18,13,8,2,25,10,20,3,15,19,11,17)(29,30,31);(26,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11744621, C3⋊D3×D25, D3×D75, …
maximal normal subgroup of#11744621, C3⋊D3×D25, D3×D75, …
maximal quotient ofC3⋊Dic75, C2×C3⋊D75, C3⋊D225, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(450, 9);