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Cayley diagram

C22×C3×C5×D3 #67228

order360
isoclinismisoclinic class of D3
automorphismAut(G) = C2×C4×D3×S4 (order 1152) · Inn(G) = D3 · Out(G) = C2×C4×S4
homologyAbelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C23

Aliases

direct productC22×C3×C5×D3
unambiguous semidirect productC3⋊(C23×C3×C5)
unambiguous semidirect product(C2×C3)⋊(C22×C3×C5)

Splittings

Fitting, socleC22×C32×C5 ⋊ C2
derivedC3 ⋊ C23×C3×C5
HallC32×C5 ⋊ C23
SylowC32 ⋊ C23×C5

Subgroups

subgroupG/H
center, hypercenterC22×C3×C5D3
commutator, cosocle, hypocenterC3C23×C3×C5
FrattiniC1G
Fitting, socleC22×C32×C5C2
LCSG ▷ C3
UCSC1 ◁ C22×C3×C5

138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×(C2×D3)⋊Dic5, C5×(C2×D3)⋊Dic3, (C2×D3)⋊Dic15, …
maximal normal subgroup ofC3×(C2×D3)⋊Dic5, C5×(C2×D3)⋊Dic3, (C2×D3)⋊Dic15, …
maximal quotient ofC2×C3×C5×Dic6, C2×C3×C4×C5×D3, C2×C3×C5×D12, …
automorphism group ofC22×C7×C11, Dic3×C31, C2×D3×C31, …
outer automorphism group ofC22×C7×C11, C22×C3×C31, C22×C9×C11, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 158);