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C5×#3755 #10623895

order720
isoclinismisoclinic class of #3753
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×#3755

Splittings

FittingC32×C5×Q8 ⋊ C2
HallC32×C5 ⋊ SD16
SylowC32 ⋊ C5×SD16

Subgroups

subgroupG/H
centerC2×C5#638
commutator, cosocleC32×C4C22×C5
FrattiniC4C2×C5×C3⋊D3
FittingC32×C5×Q8C2
socleC2×C32×C5D4
prime radicalC5×C4⋊(C3⋊D3)C2
hypercenterC5×Q8C3⋊D3
hypocenterC32C5×SD16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×Q8

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 360

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

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

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×#3755, #13086985, #13086988, …
maximal normal subgroup ofD5×#3755, #13086985, #13086988, …
maximal quotient ofC5×#63635, C5×#63636, C5×#63662, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 364);