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Cayley diagram

C4×C5×SL(2,𝔽3) #91579

order480
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C22×C4×S4 (order 384) · Inn(G) = A4 · Out(G) = C23×C4
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC4×C5×SL(2,𝔽3)
unambiguous semidirect product(C4×Q8)⋊(C3×C5)
unambiguous semidirect product(C4×C5×Q8)⋊C3

Splittings

Fitting, HallC4×C5×Q8 ⋊ C3
derivedQ8 ⋊ C3×C4×C5
SylowC4×Q8 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC2×C4×C5A4
commutator, hypocenterQ8C3×C4×C5
FrattiniC22C2×C5×A4
FittingC4×C5×Q8C3
socleC22×C5C2×A4
prime radicalC2×C5×SL(2,𝔽3)C2
cosocleC2×Q8C2×C3×C5
LCSG ▷ Q8
UCSC1 ◁ C2×C4×C5

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 160

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

derived subgroup of#14511642, #14511643, #14511644, …
maximal subgroup ofC5×SL(2,𝔽3)⋊C8, SL(2,𝔽3)⋊#168, SL(2,𝔽3)×#168, …
maximal normal subgroup ofC5×SL(2,𝔽3)⋊C8, SL(2,𝔽3)⋊#168, SL(2,𝔽3)×#168, …
maximal quotient ofC5×C8×SL(2,𝔽3), C2×C4×C5×SL(2,𝔽3), C4×C5×Q8⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 655);