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C5×#3722 #10623789

order720
isoclinismisoclinic class of #625
automorphism?
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#3722

Splittings

HallC32×C5 ⋊ C22⋊C4
SylowC32 ⋊ C5×C22⋊C4

Subgroups

subgroupG/H
center, hypercenterC22×C5D32
commutatorC2×C32C2×C4×C5
FrattiniC22C5×D32
Fitting, socleC22×C32×C5C22
prime radicalC22×C5×C3⋊D3C2
cosocleC22×C32C22×C5
hypocenterC32C5×C22⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C5

168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×#63738, #13085591, D5×#3722, …
maximal normal subgroup ofC5×#63738, #13085591, D5×#3722, …
maximal quotient ofC5×#63557, C5×#63559, C5×#63561, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 258);