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Cayley diagram

C4×C5⋊D5 #6046

order200
isoclinismisoclinic class of C5⋊D5
automorphismAut(G) = C22×AGL(2,𝔽5) (order 48000) · Inn(G) = C5⋊D5 · Out(G) = C22×A5⋊C4
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C5

Aliases

direct productC4×C5⋊D5

Splittings

FittingC4×C52 ⋊ C2
derived, SylowC52 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC4C5⋊D5
commutator, hypocenterC52C2×C4
FrattiniC2C2×C5⋊D5
FittingC4×C52C2
socle, cosocleC2×C52C22
prime radicalC2×C5⋊D5C2
LCSG ▷ C52
UCSC1 ◁ C4

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 100

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13103320, #13103321, #13103322, …
maximal subgroup of#87767, #87769, C8×C5⋊D5, …
maximal normal subgroup of#87767, #87769, C8×C5⋊D5, …
maximal quotient ofC8×C5⋊D5, #87799, C4×#1039, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 33);