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C3×D3731 #36375591

order22386
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D3731
unambiguous semidirect productC41⋊(C3×D91)

Splittings

Fitting, socle, HallC3×C7×C13×C41 ⋊ C2
derived, HallC7×C13×C41 ⋊ C2×C3
HallC3×C13×C41 ⋊ D7
HallC3×C7×C41 ⋊ D13
HallC13×C41 ⋊ C3×D7
HallC7×C41 ⋊ C3×D13
HallC3×C7×C13 ⋊ D41
HallC3×C41 ⋊ D91
HallC7×C13 ⋊ C3×D41
SylowC41 ⋊ C3×D91
HallC3×C13 ⋊ D287
HallC3×C7 ⋊ D533
SylowC13 ⋊ C3×D287
SylowC7 ⋊ C3×D533

Subgroups

subgroupG/H
center, hypercenterC3D3731
commutator, cosocle, hypocenterC7×C13×C41C2×C3
FrattiniC1G
Fitting, socleC3×C7×C13×C41C2
LCSG ▷ C7×C13×C41
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 11193

A minimal faithful permutation representation — 2 generators on 64 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);(5,10)(6,9)(7,8)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(25,64)(26,63)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)

Conjugacy classes & character table

5601 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22386, 12);