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C5×C4⋊Dic9 #10623589

order720
isoclinismisoclinic class of Dic18
automorphism?
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×C4⋊Dic9
unambiguous semidirect productC4⋊(C5×Dic9)

Splittings

HallC5×C9 ⋊ C4⋊C4
SylowC9 ⋊ C5×C4⋊C4

Subgroups

subgroupG/H
centerC22×C5C2×D9
commutatorC2×C9C2×C4×C5
FrattiniC22×C3C2×C5×D3
FittingC2×C4×C5×C9C2
socle, prime radicalC22×C3×C5C2×D3
cosocleC22×C9C22×C5
hypercenterC2×C4×C5D9
hypocenterC9C5×C4⋊C4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C5 ◁ C2×C4×C5

78 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 720

A minimal faithful permutation representation — 2 generators on 22 points:

(1,2,3,4,5)(6,11,13,12,7,9,8,14,10)(19,20,21,22);(7,9)(8,12)(10,11)(13,14)(15,16,17,18)(20,22)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13083713, #13083715, #13083728, …
maximal normal subgroup of#13083713, #13083715, #13083728, …
maximal quotient ofC5×C4⋊(C9⋊C8), C5×#63372, C5×#63377, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 58);