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Cayley diagram

C5×C42⋊C3 #6671

order240
isoclinismisoclinic class of C42⋊C3
automorphismAut(G) = C4×#4626⋊C2 (order 1536) · Inn(G) = C42⋊C3 · Out(G) = C4×D4
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C4

Aliases

direct productC5×C42⋊C3
unambiguous semidirect productC42⋊(C3×C5)
unambiguous semidirect product(C42×C5)⋊C3

Splittings

Fitting, HallC42×C5 ⋊ C3
derived, SylowC42 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC5C42⋊C3
commutator, cosocle, hypocenterC42C3×C5
FrattiniC22C5×A4
FittingC42×C5C3
socleC22×C5A4
LCSG ▷ C42
UCSC1 ◁ C5

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC5⋊#857, #11746595, #11746598, …
maximal subgroup ofC5×#857, C5⋊#857, C5⋊(#127⋊C3), …
maximal normal subgroup ofC5×#857, C5⋊#857, C5⋊(#127⋊C3), …
maximal quotient ofC5×#95⋊C3, C2×C5×C42⋊C3, C5×C42⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 32);