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Cayley diagram

C5×GL(2,𝔽3) #6742

order240
isoclinismisoclinic class of #226
automorphismAut(G) = C2×C4×S4 (order 192) · Inn(G) = S4 · Out(G) = C2×C4
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×GL(2,𝔽3)

Splittings

FittingC5×Q8 ⋊ D3
derivedSL(2,𝔽3) ⋊ C2×C5

Subgroups

subgroupG/H
center, socle, hypercenterC2×C5S4
commutator, cosocle, hypocenterSL(2,𝔽3)C2×C5
FrattiniC2C5×S4
FittingC5×Q8D3
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#91892, #91894, #91896, …
maximal normal subgroup of#91892, #91894, #91896, …
maximal quotient ofC5×#859, C2×C5×GL(2,𝔽3), C5×#3689, …
outer automorphism group ofC32×D11, C3⋊D33

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 103);