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Cayley diagram

C2×C3×C52⋊C3 #90660

order450
isoclinismisoclinic class of C52⋊C3
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3×C5

Aliases

direct productC2×C3×C52⋊C3
unambiguous semidirect product(C2×C52)⋊C32
unambiguous semidirect product(C2×C3×C52)⋊C3

Splittings

Fitting, socleC2×C3×C52 ⋊ C3
derived, SylowC52 ⋊ C2×C32
HallC2×C52 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC2×C3C52⋊C3
commutator, cosocle, hypocenterC52C2×C32
FrattiniC1G
Fitting, socleC2×C3×C52C3
LCSG ▷ C52
UCSC1 ◁ C2×C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 90

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2)(6,7,9)(8,11,16)(10,14,18)(12,17,13)(15,19,20);(3,4,5)(6,12,20,8,18)(9,14,16,19,13)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

derived subgroup of#14268144, #14268147, #14268149, …
maximal subgroup ofC3×#64470, C3⋊#64470, C3×#1039⋊C3, …
maximal normal subgroup ofC3×#64470, C3⋊#64470, C3×#1039⋊C3, …
maximal quotient ofC3×C4×C52⋊C3, C22×C3×C52⋊C3, C2×C9×C52⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(450, 25);