Finite Group Databasebeta

← Prev (#72289438) Next (#72289440) →

D26180 #72289439

order52360
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD26180

Splittings

FittingC4×C5×C7×C11×C17 ⋊ C2
HallC5×C7×C11×C17 ⋊ D4
HallC7×C11×C17 ⋊ D20
HallC5×C11×C17 ⋊ D28
HallC5×C7×C17 ⋊ D44
HallC5×C7×C11 ⋊ D68
HallC11×C17 ⋊ D140
HallC7×C17 ⋊ D220
HallC5×C17 ⋊ D308
HallC7×C11 ⋊ D340
HallC5×C11 ⋊ D476
HallC5×C7 ⋊ D748
SylowC17 ⋊ D1540
SylowC11 ⋊ D2380
SylowC7 ⋊ D3740
SylowC5 ⋊ D5236

Subgroups

subgroupG/H
center, FrattiniC2C2×D6545
commutator, socle, cosocleC2×C5×C7×C11×C17C22
FittingC4×C5×C7×C11×C17C2
hypercenterC4D6545
hypocenterC5×C7×C11×C17D4
LCSG ▷ C2×C5×C7×C11×C17 ▷ C5×C7×C11×C17
UCSC1 ◁ C2 ◁ C4

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 44
transitive degreeμt(G) = 26180

A minimal faithful permutation representation — 2 generators on 44 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);(2,4)(6,9)(7,8)(11,16)(12,15)(13,14)(18,27)(19,26)(20,25)(21,24)(22,23)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)

Polycyclic presentation:

2^3,5,7,11,17|b^2=c,b^a=bc,d^a=d^4,e^a=e^6,f^a=f^10,g^a=g^16

Conjugacy classes & character table

13093 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 7, 11, 17]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);