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Cayley diagram

C32×D25 #90640

order450
isoclinismisoclinic class of D25
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC32×D25
unambiguous semidirect productC25⋊(C2×C32)

Splittings

Fitting, HallC32×C25 ⋊ C2
derived, SylowC25 ⋊ C2×C32

Subgroups

subgroupG/H
center, hypercenterC32D25
commutator, cosocle, hypocenterC25C2×C32
FrattiniC5C32×D5
FittingC32×C25C2
socleC32×C5D5
LCSG ▷ C25
UCSC1 ◁ C32

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 31 points:

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

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC32×#1035, C3×C3⋊#1035, C3⋊(C3⋊#1035), …
maximal normal subgroup ofC32×#1035, C3×C3⋊#1035, C3⋊(C3⋊#1035), …
maximal quotient ofC32×Dic25, C2×C32×D25, C3×C9×D25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(450, 5);