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C3×D5369 #44567258

order32214
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D5369
unambiguous semidirect productC59⋊(C3×D91)

Splittings

Fitting, socle, HallC3×C7×C13×C59 ⋊ C2
derived, HallC7×C13×C59 ⋊ C2×C3
HallC3×C13×C59 ⋊ D7
HallC3×C7×C59 ⋊ D13
HallC13×C59 ⋊ C3×D7
HallC7×C59 ⋊ C3×D13
HallC3×C7×C13 ⋊ D59
HallC3×C59 ⋊ D91
HallC7×C13 ⋊ C3×D59
SylowC59 ⋊ C3×D91
HallC3×C13 ⋊ D413
HallC3×C7 ⋊ D767
SylowC13 ⋊ C3×D413
SylowC7 ⋊ C3×D767

Subgroups

subgroupG/H
center, hypercenterC3D5369
commutator, cosocle, hypocenterC7×C13×C59C2×C3
FrattiniC1G
Fitting, socleC3×C7×C13×C59C2
LCSG ▷ C7×C13×C59
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 82
transitive degreeμt(G) = 16107

A minimal faithful permutation representation — 2 generators on 82 points:

(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21,22,23)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82);(5,10)(6,9)(7,8)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Conjugacy classes & character table

8058 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32214, 12);