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Cayley diagram

C5×C24⋊C5 #87919

order400
isoclinismisoclinic class of C24⋊C5
automorphism?
homologyAbelianization (H1) = C52 · Schur multiplier (H2) = C22×C5

Aliases

direct productC5×C24⋊C5
unambiguous semidirect productC24⋊C52
unambiguous semidirect product(C24×C5)⋊C5

Splittings

Fitting, socleC24×C5 ⋊ C5
derived, SylowC24 ⋊ C52

Subgroups

subgroupG/H
center, hypercenterC5C24⋊C5
commutator, cosocle, hypocenterC24C52
FrattiniC1G
Fitting, socleC24×C5C5
LCSG ▷ C24
UCSC1 ◁ C5

42 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 50

A minimal faithful permutation representation — 2 generators on 15 points:

(1,2,4,6,8)(3,5,7,9,10);(1,3)(8,10)(11,12,13,14,15)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC5⋊((C24⋊C5)⋊C2), #13107522, #13107903, …
maximal subgroup ofC5×(C24⋊C5)⋊C2, C5⋊((C24⋊C5)⋊C2), D5×C24⋊C5, …
maximal normal subgroup ofC5×(C24⋊C5)⋊C2, C5⋊((C24⋊C5)⋊C2), D5×C24⋊C5, …
maximal quotient ofC5×2−1+4⋊C5, C2×C5×C24⋊C5, C3×C5×C24⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 213);