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Cayley diagram

C5×C22⋊SL(2,𝔽3) #92057

order480
automorphism?
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C23

Aliases

direct productC5×C22⋊SL(2,𝔽3)
unambiguous semidirect productC22⋊(C5×SL(2,𝔽3))
unambiguous semidirect product(C22×C5)⋊SL(2,𝔽3)

Splittings

Fitting, HallC22×C5×Q8 ⋊ C3
derived, SylowC22×Q8 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC2×C5C22⋊A4
commutator, cosocle, hypocenterC22×Q8C3×C5
FrattiniC2C5×C22⋊A4
FittingC22×C5×Q8C3
socleC23×C5A4
LCSG ▷ C22×Q8
UCSC1 ◁ C2×C5

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC5⋊#5930, C5⋊#5931, #14511535, …
maximal subgroup ofC5⋊(#395⋊C3), C5⋊(#393⋊C3), C5×#393⋊C3, …
maximal normal subgroup ofC5⋊(#395⋊C3), C5⋊(#393⋊C3), C5×#393⋊C3, …
maximal quotient ofC5×#543⋊C3, C5×Q8⋊SL(2,𝔽3), C2×C5×C22⋊SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1133);