Finite Group Databasebeta

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Cayley diagram

C5×#709 #87890

order400
isoclinismisoclinic class of Dic10
automorphism?
homologyAbelianization (H1) = C23×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#709

Splittings

FittingC2×C4×C52 ⋊ C2
SylowC52 ⋊ #41

Subgroups

subgroupG/H
centerC4×C5C2×D5
commutator, cosocleC2×C5C23×C5
FrattiniC2C22×C5×D5
FittingC2×C4×C52C2
socleC2×C52C23
hypercenterC2×C4×C5D5
hypocenterC5C5×#41
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C4×C5 ◁ C2×C4×C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11716609, #11716610, C5×#3937, …
maximal normal subgroup of#11716609, #11716610, C5×#3937, …
maximal quotient ofC4×C5×Dic10, C5×#4016, C5×#4018, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 184);