Finite Group Databasebeta

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D26796 #72290139

order53592
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD26796

Splittings

FittingC3×C4×C7×C11×C29 ⋊ C2
HallC3×C7×C11×C29 ⋊ D4
HallC7×C11×C29 ⋊ D12
HallC3×C11×C29 ⋊ D28
HallC3×C7×C29 ⋊ D44
HallC11×C29 ⋊ D84
HallC3×C7×C11 ⋊ D116
HallC7×C29 ⋊ D132
HallC3×C29 ⋊ D308
HallC7×C11 ⋊ D348
HallC3×C11 ⋊ D812
SylowC29 ⋊ D924
HallC3×C7 ⋊ D1276
SylowC11 ⋊ D2436
SylowC7 ⋊ D3828
SylowC3 ⋊ D8932

Subgroups

subgroupG/H
center, FrattiniC2C2×D6699
commutator, socle, cosocleC2×C3×C7×C11×C29C22
FittingC3×C4×C7×C11×C29C2
hypercenterC4D6699
hypocenterC3×C7×C11×C29D4
LCSG ▷ C2×C3×C7×C11×C29 ▷ C3×C7×C11×C29
UCSC1 ◁ C2 ◁ C4

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 26796

A minimal faithful permutation representation — 2 generators on 54 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);(2,4)(6,7)(9,14)(10,13)(11,12)(16,25)(17,24)(18,23)(19,22)(20,21)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)

Polycyclic presentation:

2^3,3,7,11,29|b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^10,g^a=g^28

Conjugacy classes & character table

13401 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, 3, 7, 11, 29]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);