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C5×C3⋊#859 #13086214

order1440
isoclinismisoclinic class of C3⋊#226
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×C3⋊#859
unambiguous semidirect productC3⋊(C5×#859)

Splittings

derivedC3×SL(2,𝔽3) ⋊ C4×C5

Subgroups

subgroupG/H
center, hypercenterC22×C5C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C4×C5
FrattiniC22C5×C3⋊S4
FittingC2×C3×C5×Q8D3
socleC22×C3×C5S4
prime radicalC2×C3×C5×SL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C2×C5
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C5

156 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 480

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 2531);