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Cayley diagram

C3×(C5⋊D5)⋊C3 #90657

order450
isoclinismisoclinic class of (C5⋊D5)⋊C3
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3×C5

Aliases

direct productC3×(C5⋊D5)⋊C3
unambiguous semidirect product(C5⋊D5)⋊C32
unambiguous semidirect product(C3×C5⋊D5)⋊C3

Splittings

Fitting, socleC3×C52 ⋊ C2×C3
derived, SylowC52 ⋊ C2×C32
HallC3×C52⋊C3 ⋊ C2
HallC5⋊D5 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC3(C5⋊D5)⋊C3
commutator, cosocle, hypocenterC52C2×C32
FrattiniC1G
Fitting, socleC3×C52C2×C3
LCSG ▷ C52
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 45

A minimal faithful permutation representation — 2 generators on 18 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72276706, #72276825, #72276832, …
maximal subgroup ofC3×#64480, C3⋊#64480, C3×#1043⋊C3, …
maximal normal subgroup ofC3×#64480, C3⋊#64480, C3×#1043⋊C3, …
maximal quotient ofC3×#1039⋊C3, C2×C3×(C5⋊D5)⋊C3, C9×(C5⋊D5)⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(450, 22);