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Cayley diagram

C5×#41⋊C3 #6793

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 productC5×#41⋊C3
unambiguous semidirect product#41⋊(C3×C5)
unambiguous semidirect product(C5×#41)⋊C3

Splittings

Fitting, HallC5×#41 ⋊ C3
Sylow#41 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC4×C5A4
commutator, cosocle, hypocenterQ8C2×C3×C5
FrattiniC2C2×C5×A4
FittingC5×#41C3
socleC2×C5C2×A4
LCSG ▷ Q8
UCSC1 ◁ C4×C5

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,5,8)(3,7,11,14,4,13,10,16,9,6,12,15)(17,18,19,20,21);(1,9,5,11)(2,13,8,15)(3,12,10,4)(6,16,14,7)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11751619, #11751620, #11751621, …
maximal subgroup ofC5×GU(2,𝔽3), #91185, #4094⋊C3, …
maximal normal subgroup ofC5×GU(2,𝔽3), #91185, #4094⋊C3, …
maximal quotient ofC4×C5×SL(2,𝔽3), C2×C5×#41⋊C3, C5×#41⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 154);