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Cayley diagram

C5×C22⋊A4 #6843

order240
isoclinismisoclinic class of C22⋊A4
automorphismAut(G) = C4×AΓL(2,𝔽4) (order 23040) · Inn(G) = C22⋊A4 · Out(G) = C4×S5
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C24

Aliases

direct productC5×C22⋊A4
unambiguous semidirect productC22⋊(C5×A4)
unambiguous semidirect product(C22×C5)⋊A4

Splittings

Fitting, socle, HallC24×C5 ⋊ C3
derived, SylowC24 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC5C22⋊A4
commutator, cosocle, hypocenterC24C3×C5
FrattiniC1G
Fitting, socleC24×C5C3
LCSG ▷ C24
UCSC1 ◁ C5

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC5⋊#1020, #11746597, #11751559, …
maximal subgroup ofC5⋊(#120⋊C3), C5×#120⋊C3, C5×#1020, …
maximal normal subgroup ofC5⋊(#120⋊C3), C5×#120⋊C3, C5×#1020, …
maximal quotient ofC5×C22⋊SL(2,𝔽3), C5×#997, C2×C5×C22⋊A4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 204);