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C5×#3674 #10623593

order720
isoclinismisoclinic class of #3672
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×#3674

Splittings

HallC5×C9 ⋊ Q16
SylowC9 ⋊ C5×Q16

Subgroups

subgroupG/H
centerC2×C5#611
commutator, cosocleC4×C9C22×C5
FrattiniC3×C4C2×C5×D3
FittingC5×Q8×C9C2
socle, prime radicalC2×C3×C5#67
hypercenterC5×Q8D9
hypocenterC9C5×Q16
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×Q8

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 720

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,3,4,5)(6,7,10,12,14,13,8,11,9)(15,17,19,25)(16,20,22,29)(18,27,28,24)(21,30,26,23);(7,9)(8,12)(10,11)(13,14)(15,16,19,22)(17,23,25,30)(18,26,28,21)(20,27,29,24)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×#3674, #13084261, #13084266, …
maximal normal subgroup ofD5×#3674, #13084261, #13084266, …
maximal quotient ofC5×#63366, C5×#63368, C5×#63395, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 62);