Finite Group Databasebeta

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Cayley diagram

C5×#986 #91945

order480
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×#986

Splittings

FittingC5×#41 ⋊ D3
derivedSL(2,𝔽3) ⋊ C22×C5

Subgroups

subgroupG/H
center, socleC2×C5C2×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C22×C5
FrattiniC2C2×C5×S4
FittingC5×#41D3
hypercenterC4×C5S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C5 ◁ C4×C5

84 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:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11751619, #11751621, #11751625, …
maximal normal subgroup of#11751619, #11751621, #11751625, …
maximal quotient ofC5×#5391, C5×C4⋊GL(2,𝔽3), C5×#5394, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1021);