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Cayley diagram

Dic25⋊C5 #92523

order500
isoclinismisoclinic class of D25⋊C5
automorphismAut(G) = C2×AGL(1,ℤ/25) (order 1000) · Inn(G) = D25⋊C5 · Out(G) = C22
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productDic25⋊C5

Splittings

derivedC25 ⋊ C4×C5
Sylow5−1+2 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2D25⋊C5
commutator, hypocenterC25C4×C5
Frattini, socleC2×C5C5×D5
FittingC2×5−1+2C2
prime radicalC2×C52D5
cosocleC2×C25C2×C5
LCSG ▷ C25
UCSC1 ◁ C2

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 29 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6016⋊C5, 5−1+2⋊Q8, C4×D25⋊C5, …
maximal normal subgroup of#6016⋊C5, 5−1+2⋊Q8, C4×D25⋊C5, …
maximal quotient of#6014⋊C5, C2×Dic25⋊C5, C3×Dic25⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 9);