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C5×#3750 #10623890

order720
isoclinismisoclinic class of #634
automorphism?
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2×C3

Aliases

direct productC5×#3750

Splittings

HallC32×C5 ⋊ C4⋊C4
SylowC32 ⋊ C5×C4⋊C4

Subgroups

subgroupG/H
centerC22×C5C2×C3⋊D3
commutatorC2×C32C2×C4×C5
FrattiniC22C2×C5×C3⋊D3
FittingC2×C32×C4×C5C2
socle, prime radicalC22×C32×C5C22
cosocleC22×C32C22×C5
hypercenterC2×C4×C5C3⋊D3
hypocenterC32C5×C4⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C5 ◁ C2×C4×C5

156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 720

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13086767, #13086773, #13086776, …
maximal normal subgroup of#13086767, #13086773, #13086776, …
maximal quotient ofC5×#63634, C5×#63635, C5×#63641, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 359);