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C3×(C13⋊C3)⋊D53 #27137162

order12402
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×(C13⋊C3)⋊D53
unambiguous semidirect product(C13⋊C3)⋊(C3×D53)
unambiguous semidirect productC53⋊(C3×D13⋊C3)
unambiguous semidirect productD689⋊C32
unambiguous semidirect product(C3×D689)⋊C3

Splittings

Fitting, socleC3×C13×C53 ⋊ C2×C3
derived, HallC13×C53 ⋊ C2×C32
HallC3×C13⋊C3×C53 ⋊ C2
HallD689 ⋊ C32
HallC3×C13⋊C3 ⋊ D53
SylowC53 ⋊ C3×D13⋊C3
SylowC13 ⋊ C32×D53

Subgroups

subgroupG/H
center, hypercenterC3(C13⋊C3)⋊D53
commutator, cosocle, hypocenterC13×C53C2×C32
FrattiniC1G
Fitting, socleC3×C13×C53C2×C3
LCSG ▷ C13×C53
UCSC1 ◁ C3

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 69
transitive degreeμt(G) = 2067

A minimal faithful permutation representation — 2 generators on 69 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(55,59,60,56,57,61)(58,65,64,62,63,66);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,55,58,57,61,64,63,65,66,60,59,62,56)(67,68,69)

Conjugacy classes & character table

570 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3×#14583752⋊C3, C3×#14583753⋊C3, C3×D13⋊C3×D53, …
maximal normal subgroup ofC3×#14583752⋊C3, C3×#14583753⋊C3, C3×D13⋊C3×D53, …
maximal quotient ofC3×Dic689⋊C3, C2×C3×(C13⋊C3)⋊D53

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12402, 25);