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Cayley diagram

C22×C5⋊D5 #6064

order200
isoclinismisoclinic class of C5⋊D5
automorphismAut(G) order 288000
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23×C5

Aliases

direct productC22×C5⋊D5

Splittings

Fitting, socleC22×C52 ⋊ C2
derived, SylowC52 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC22C5⋊D5
commutator, cosocle, hypocenterC52C23
FrattiniC1G
Fitting, socleC22×C52C2
LCSG ▷ C52
UCSC1 ◁ C22

128 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:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13099989, #13099990, #13100010, …
maximal subgroup of#87779, #87808, C22⋊#1043, …
maximal normal subgroup of#87779, #87808, C22⋊#1043, …
maximal quotient ofC2×#6045, C2×C4×C5⋊D5, C2×C4⋊(C5⋊D5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 51);