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D1001 #14523009

order2002
isoclinismisoclinic class of D1001
automorphismAut(G) order 720720
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD1001
unambiguous semidirect productC7⋊D143
unambiguous semidirect productC11⋊D91
unambiguous semidirect productC13⋊D77

Splittings

Fitting, socle, derived, HallC7×C11×C13 ⋊ C2
HallC11×C13 ⋊ D7
HallC7×C13 ⋊ D11
HallC7×C11 ⋊ D13
SylowC13 ⋊ D77
SylowC11 ⋊ D91
SylowC7 ⋊ D143

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C11×C13C2
LCSG ▷ C7×C11×C13
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 1001

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);(2,7)(3,6)(4,5)(9,18)(10,17)(11,16)(12,15)(13,14)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)

Polycyclic presentation:

2,7,11,13|b^a=b^6,c^a=c^10,d^a=d^12

Conjugacy classes & character table

502 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD13×D77, D11×D91, D7×D143, …
maximal normal subgroup ofD13×D77, D11×D91, D7×D143, …
maximal quotient ofDic1001, C2×D1001, D7007, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2002, 8);