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D30225 #72295541

order60450
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD30225

Splittings

Fitting, derived, HallC3×C13×C25×C31 ⋊ C2
HallC13×C25×C31 ⋊ D3
HallC3×C25×C31 ⋊ D13
HallC3×C13×C31 ⋊ D25
HallC3×C13×C25 ⋊ D31
HallC25×C31 ⋊ D39
HallC13×C31 ⋊ D75
HallC13×C25 ⋊ D93
HallC3×C31 ⋊ D325
HallC3×C25 ⋊ D403
HallC3×C13 ⋊ D775
SylowC31 ⋊ D975
SylowC25 ⋊ D1209
SylowC13 ⋊ D2325
SylowC3 ⋊ D10075

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC3×C13×C25×C31C2
FrattiniC5D6045
socleC3×C5×C13×C31D5
LCSG ▷ C3×C13×C25×C31
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 72
transitive degreeμt(G) = 30225

A minimal faithful permutation representation — 2 generators on 72 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)(42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72);(2,3)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)

Polycyclic presentation:

2,3,5,13,31,5|c^5=f,b^a=b^2,c^a=c^4f^4,d^a=d^12,e^a=e^30,f^a=f^4

Conjugacy classes & character table

15114 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");;
gap> coll := SingleCollector(F, [2, 3, 5, 13, 31, 5]);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3^4*F.6^4);;
gap> SetConjugate(coll, 4, 1, F.4^12);;
gap> SetConjugate(coll, 5, 1, F.5^30);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);