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D28644 #72292263

order57288
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD28644

Splittings

FittingC3×C4×C7×C11×C31 ⋊ C2
HallC3×C7×C11×C31 ⋊ D4
HallC7×C11×C31 ⋊ D12
HallC3×C11×C31 ⋊ D28
HallC3×C7×C31 ⋊ D44
HallC11×C31 ⋊ D84
HallC3×C7×C11 ⋊ D124
HallC7×C31 ⋊ D132
HallC3×C31 ⋊ D308
HallC7×C11 ⋊ D372
HallC3×C11 ⋊ D868
SylowC31 ⋊ D924
HallC3×C7 ⋊ D1364
SylowC11 ⋊ D2604
SylowC7 ⋊ D4092
SylowC3 ⋊ D9548

Subgroups

subgroupG/H
center, FrattiniC2C2×D7161
commutator, socle, cosocleC2×C3×C7×C11×C31C22
FittingC3×C4×C7×C11×C31C2
hypercenterC4D7161
hypocenterC3×C7×C11×C31D4
LCSG ▷ C2×C3×C7×C11×C31 ▷ C3×C7×C11×C31
UCSC1 ◁ C2 ◁ C4

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 56
transitive degreeμt(G) = 28644

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);(2,4)(6,7)(9,14)(10,13)(11,12)(16,25)(17,24)(18,23)(19,22)(20,21)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)

Polycyclic presentation:

2^3,3,7,11,31|b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^10,g^a=g^30

Conjugacy classes & character table

14325 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, 31]);;
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.7^30);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);