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C3×D3451 #35208478

order20706
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D3451
unambiguous semidirect productC17⋊(C3×D203)
unambiguous semidirect productC29⋊(C3×D119)

Splittings

Fitting, socle, HallC3×C7×C17×C29 ⋊ C2
derived, HallC7×C17×C29 ⋊ C2×C3
HallC3×C17×C29 ⋊ D7
HallC3×C7×C29 ⋊ D17
HallC17×C29 ⋊ C3×D7
HallC3×C7×C17 ⋊ D29
HallC7×C29 ⋊ C3×D17
HallC7×C17 ⋊ C3×D29
HallC3×C29 ⋊ D119
HallC3×C17 ⋊ D203
SylowC29 ⋊ C3×D119
HallC3×C7 ⋊ D493
SylowC17 ⋊ C3×D203
SylowC7 ⋊ C3×D493

Subgroups

subgroupG/H
center, hypercenterC3D3451
commutator, cosocle, hypocenterC7×C17×C29C2×C3
FrattiniC1G
Fitting, socleC3×C7×C17×C29C2
LCSG ▷ C7×C17×C29
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 56
transitive degreeμt(G) = 10353

A minimal faithful permutation representation — 2 generators on 56 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);(5,10)(6,9)(7,8)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)

Conjugacy classes & character table

5181 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20706, 10);