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Cayley diagram

C2×C5×SL(2,𝔽3) #6792

order240
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C2×C4×S4 (order 192) · Inn(G) = A4 · Out(G) = C22×C4
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC2×C5×SL(2,𝔽3)
unambiguous semidirect product(C2×Q8)⋊(C3×C5)
unambiguous semidirect product(C2×C5×Q8)⋊C3

Splittings

Fitting, HallC2×C5×Q8 ⋊ C3
derivedQ8 ⋊ C2×C3×C5
SylowC2×Q8 ⋊ C3×C5

Subgroups

subgroupG/H
center, socle, hypercenterC22×C5A4
commutator, cosocle, hypocenterQ8C2×C3×C5
FrattiniC2C2×C5×A4
FittingC2×C5×Q8C3
LCSG ▷ Q8
UCSC1 ◁ C22×C5

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11751590, #11751592, #11751593, …
maximal subgroup ofC5×#859, #91184, Dic5×SL(2,𝔽3), …
maximal normal subgroup ofC5×#859, #91184, Dic5×SL(2,𝔽3), …
maximal quotient ofC4×C5×SL(2,𝔽3), C22×C5×SL(2,𝔽3), C2×C5×Q8⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 153);