Finite Group Databasebeta

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Cayley diagram

C22×D25 #6026

order200
isoclinismisoclinic class of D25
automorphismAut(G) = S4×AGL(1,ℤ/25) (order 12000) · Inn(G) = D25 · Out(G) = C2×C5×S4
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC22×D25
unambiguous semidirect productC25⋊C23
unambiguous semidirect product(C2×C25)⋊C22

Splittings

FittingC22×C25 ⋊ C2
derived, SylowC25 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC22D25
commutator, cosocle, hypocenterC25C23
FrattiniC5C22×D5
FittingC22×C25C2
socleC22×C5D5
LCSG ▷ C25
UCSC1 ◁ C22

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 100

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#87720, C22⋊#1035, C2×C4×D25, …
maximal normal subgroup of#87720, C22⋊#1035, C2×C4×D25, …
maximal quotient ofC2×Dic50, C2×C4×D25, C2×D100, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 13);