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Cayley diagram

C5×C3⋊Dic3 #4371

order180
isoclinismisoclinic class of C3⋊D3
automorphismAut(G) = C2×C4×AGL(2,𝔽3) (order 3456) · Inn(G) = C3⋊D3 · Out(G) = C2×C4×S4
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×C3⋊Dic3
unambiguous semidirect productC3⋊(C5×Dic3)

Splittings

derived, SylowC32 ⋊ C4×C5
HallC32×C5 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2×C5C3⋊D3
commutator, hypocenterC32C4×C5
FrattiniC2C5×C3⋊D3
Fitting, socle, prime radicalC2×C32×C5C2
cosocleC2×C32C2×C5
LCSG ▷ C32
UCSC1 ◁ C2×C5

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 180

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

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup of#10624006, #10624007, #10624008, …
maximal subgroup ofC5×#622, #67126, D5×C3⋊Dic3, …
maximal normal subgroup ofC5×#622, #67126, D5×C3⋊Dic3, …
maximal quotient ofC5×C3⋊(C3⋊C8), C2×C5×C3⋊Dic3, C5×C3⋊Dic9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 16);