Finite Group Databasebeta

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Cayley diagram

C5×#997 #92058

order480
automorphism?
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C23

Aliases

direct productC5×#997

Splittings

Fitting, HallC5×H(5,𝔽2) ⋊ C3
derived, SylowH(5,𝔽2) ⋊ C3×C5

Subgroups

subgroupG/H
center, socle, hypercenterC2×C5C22⋊A4
commutator, cosocle, hypocenterH(5,𝔽2)C3×C5
FrattiniC2C5×C22⋊A4
FittingC5×H(5,𝔽2)C3
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C2×C5

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC5⋊#5932, C5⋊#5933, C5⋊#5934, …
maximal subgroup ofC5⋊(#458⋊C3), C5⋊(B(4,𝔽2)⋊C3), C5×B(4,𝔽2)⋊C3, …
maximal normal subgroup ofC5⋊(#458⋊C3), C5⋊(B(4,𝔽2)⋊C3), C5×B(4,𝔽2)⋊C3, …
maximal quotient ofC5×#543⋊C3, C5×Q8⋊SL(2,𝔽3), C5×#5465, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1134);