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Cayley diagram

C3×2−1+4⋊C5 #91970

order480
automorphismAut(G) = C2×#66297 (order 640) · Inn(G) = C24⋊C5 · Out(G) = C2×C4
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC3×2−1+4⋊C5
unambiguous semidirect product(C3×2−1+4)⋊C5

Splittings

Fitting, HallC3×2−1+4 ⋊ C5
derived, Sylow2−1+4 ⋊ C3×C5

Subgroups

subgroupG/H
center, socle, hypercenterC2×C3C24⋊C5
commutator, cosocle, hypocenter2−1+4C3×C5
FrattiniC2C3×C24⋊C5
FittingC3×2−1+4C5
LCSG ▷ 2−1+4
UCSC1 ◁ C2×C3

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 35 points:

(1,2,3)(5,8,23,11,14)(6,12,27,16,20)(7,18,32,28,34)(10,13,25,35,19)(15,17,24,26,31)(21,22,29,30,33);(4,24,9,12)(5,25,13,8)(6,18,17,29)(7,28,22,33)(10,16,14,31)(11,21,19,34)(15,35,20,23)(26,32,27,30)

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC3⋊#66243, C3⋊#66244, C3⋊#10617016, …
maximal subgroup ofC3×#66243, C3×#66244, C3⋊#66243, …
maximal normal subgroup ofC3×#66243, C3×#66244, C3⋊#66243, …
maximal quotient ofC3×#564⋊C5, C2×C3×2−1+4⋊C5, C9×2−1+4⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1046);