Finite Group Databasebeta

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Cayley diagram

D5×GL(2,𝔽3) #91898

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

Aliases

direct productD5×GL(2,𝔽3)

Splittings

FittingC5×Q8 ⋊ C2×D3
derivedC5×SL(2,𝔽3) ⋊ C22
SylowC5 ⋊ C2×GL(2,𝔽3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D5×S4
commutator, cosocle, hypocenterC5×SL(2,𝔽3)C22
FittingC5×Q8C2×D3
socleC2×C5C2×S4
LCSG ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C2

102 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)(9,10,12,13,11);(2,5,8)(3,6,7)(10,11)(12,13)

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11756705, AGL(1,𝔽5)×GL(2,𝔽3), C2×D5×GL(2,𝔽3), …
maximal normal subgroup of#11756705, AGL(1,𝔽5)×GL(2,𝔽3), C2×D5×GL(2,𝔽3), …
maximal quotient of#11751608, Dic5×GL(2,𝔽3), #11751610, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 974);