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Cayley diagram

C5⋊D25 #6952

order250
isoclinismisoclinic class of C5⋊D25
automorphismAut(G) order 250000
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C5

Aliases

unambiguous semidirect productC5⋊D25
unambiguous semidirect productC25⋊D5

Splittings

Fitting, derived, SylowC5×C25 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC5×C25C2
FrattiniC5C5⋊D5
socleC52D5
LCSG ▷ C5×C25
UCSC1

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 125

A minimal faithful permutation representation — 3 generators on 30 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#92536, #92538, D5×D25, …
maximal normal subgroup of#92536, #92538, D5×D25, …
maximal quotient of#92524, C2×C5⋊D25, C3×C5⋊D25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(250, 7);