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Dic16275 #72301451

order65100
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic16275

Splittings

derived, HallC3×C7×C25×C31 ⋊ C4
HallC7×C25×C31 ⋊ Dic3
HallC3×C25×C31 ⋊ Dic7
HallC25×C31 ⋊ Dic21
HallC3×C7×C31 ⋊ Dic25
HallC3×C7×C25 ⋊ Dic31
HallC7×C31 ⋊ Dic75
HallC7×C25 ⋊ Dic93
HallC3×C31 ⋊ Dic175
HallC3×C25 ⋊ Dic217
SylowC31 ⋊ Dic525
SylowC25 ⋊ Dic651
HallC3×C7 ⋊ Dic775
SylowC7 ⋊ Dic2325
SylowC3 ⋊ Dic5425

Subgroups

subgroupG/H
center, hypercenterC2D16275
commutator, hypocenterC3×C7×C25×C31C4
FrattiniC2×C5D3255
Fitting, cosocleC2×C3×C7×C25×C31C2
socle, prime radicalC2×C3×C5×C7×C31D5
LCSG ▷ C3×C7×C25×C31
UCSC1 ◁ C2

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 70
transitive degreeμt(G) = 65100

A minimal faithful permutation representation — 2 generators on 70 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);(1,3,2,4)(6,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(34,39)(35,38)(36,37)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2^2,3,5,7,31,5|a^2=b,d^5=g,c^a=c^2,d^a=d^4g^4,e^a=e^6,f^a=f^30,g^a=g^4

Conjugacy classes & character table

16278 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, 3, 5, 7, 31, 5]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.7^4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^30);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);