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D561 #11765494

order1122
isoclinismisoclinic class of D561
automorphismAut(G) order 179520
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD561
unambiguous semidirect productC3⋊D187
unambiguous semidirect productC11⋊D51
unambiguous semidirect productC17⋊D33

Splittings

Fitting, socle, derived, HallC3×C11×C17 ⋊ C2
HallC11×C17 ⋊ D3
HallC3×C17 ⋊ D11
HallC3×C11 ⋊ D17
SylowC17 ⋊ D33
SylowC11 ⋊ D51
SylowC3 ⋊ D187

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C11×C17C2
LCSG ▷ C3×C11×C17
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 561

A minimal faithful permutation representation — 2 generators on 31 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);(2,3)(5,14)(6,13)(7,12)(8,11)(9,10)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)

Polycyclic presentation:

2,3,11,17|b^a=b^2,c^a=c^10,d^a=d^16

Conjugacy classes & character table

282 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD17×D33, D11×D51, D3×D187, …
maximal normal subgroup ofD17×D33, D11×D51, D3×D187, …
maximal quotient ofDic561, C2×D561, D1683, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1122, 7);