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Cayley diagram

C2×C5⋊GL(2,𝔽3) #91952

order480
isoclinismisoclinic class of C5⋊#226
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5⋊GL(2,𝔽3)
unambiguous semidirect product(C2×C5)⋊GL(2,𝔽3)

Splittings

FittingC2×C5×Q8 ⋊ D3
derivedC5×SL(2,𝔽3) ⋊ C22
SylowC5 ⋊ C2×GL(2,𝔽3)

Subgroups

subgroupG/H
center, hypercenterC22C5⋊S4
commutator, cosocle, hypocenterC5×SL(2,𝔽3)C22
FrattiniC2C2×C5⋊S4
FittingC2×C5×Q8D3
socleC22×C5S4
LCSG ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22

102 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:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11751602, #11751603, #11751604, …
maximal normal subgroup of#11751602, #11751603, #11751604, …
maximal quotient of#11752105, C4×C5⋊GL(2,𝔽3), C4⋊(C5⋊GL(2,𝔽3)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1028);