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C5×C3⋊(Dic13⋊C3) #14556754

order2340
isoclinismisoclinic class of C3⋊(D13⋊C3)
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×C3⋊(Dic13⋊C3)
unambiguous semidirect productC3⋊(C5×Dic13⋊C3)
unambiguous semidirect productDic39⋊(C3×C5)
unambiguous semidirect product(C5×Dic39)⋊C3

Splittings

derivedC3×C13 ⋊ C3×C4×C5
HallC3×C5×C13⋊C3 ⋊ C4
HallC3×C13⋊C3 ⋊ C4×C5
HallC5×C13 ⋊ C3×Dic3
SylowC13 ⋊ C3×C5×Dic3

Subgroups

subgroupG/H
center, hypercenterC2×C5C3⋊(D13⋊C3)
commutator, hypocenterC3×C13C3×C4×C5
FrattiniC2C5×C3⋊(D13⋊C3)
Fitting, socleC2×C3×C5×C13C2×C3
prime radicalC2×C3×C5×C13⋊C3C2
cosocleC2×C3×C13C2×C3×C5
LCSG ▷ C3×C13
UCSC1 ◁ C2×C5

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 780

A minimal faithful permutation representation — 2 generators on 25 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC25×C3⋊(Dic13⋊C3), C52×C3⋊(Dic13⋊C3), C5×C3⋊(Dic65⋊C3), …
maximal normal subgroup ofC25×C3⋊(Dic13⋊C3), C52×C3⋊(Dic13⋊C3), C5×C7×C3⋊(Dic13⋊C3), …
maximal quotient ofC25×C3⋊(Dic13⋊C3), C52×C3⋊(Dic13⋊C3), C5×C3⋊(Dic65⋊C3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2340, 58);