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Cayley diagram

C5×#1039 #92552

order500
isoclinismisoclinic class of C5⋊D5
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C5

Aliases

direct productC5×#1039

Splittings

derivedC52 ⋊ C4×C5
SylowC53 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2×C5C5⋊D5
commutator, hypocenterC52C4×C5
FrattiniC2C5×C5⋊D5
Fitting, socle, prime radicalC2×C53C2
cosocleC2×C52C2×C5
LCSG ▷ C52
UCSC1 ◁ C2×C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 100

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

derived subgroup of#14522749, #14522750, #14522751, …
maximal subgroup ofC5×#6033, C5×#6034, #11758740, …
maximal normal subgroup ofC5×#6033, C5×#6034, #11758740, …
maximal quotient ofC5×#6029, C2×C5×#1039, C3×C5×#1039, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 38);