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Cayley diagram

#1039⋊C5 #92522

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

Aliases

unambiguous semidirect product#1039⋊C5

Splittings

derivedC52 ⋊ C4×C5
SylowH(3,𝔽5) ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2(C5⋊D5)⋊C5
commutator, hypocenterC52C4×C5
Frattini, socleC2×C5C5×D5
Fitting, prime radicalC2×H(3,𝔽5)C2
cosocleC2×C52C2×C5
LCSG ▷ C52
UCSC1 ◁ C2

29 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 100

A minimal faithful permutation representation — 2 generators on 29 points:

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

Polycyclic presentation:

2,5,2,5^2|a^2=c,d^a=d^4,e^a=e^4,d^b=de

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6033⋊C5, #11758645, #11758657, …
maximal normal subgroup of#6033⋊C5, #11758645, #11758657, …
maximal quotient of#6029⋊C5, C2×#1039⋊C5, C3×#1039⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 8);