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Dic15810 #72295751

order63240
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic15810

Splittings

HallC3×C5×C17×C31 ⋊ Q8
HallC5×C17×C31 ⋊ Dic6
HallC3×C17×C31 ⋊ Dic10
HallC17×C31 ⋊ Dic30
HallC3×C5×C31 ⋊ Dic34
HallC3×C5×C17 ⋊ Dic62
HallC5×C31 ⋊ Dic102
HallC3×C31 ⋊ Dic170
HallC5×C17 ⋊ Dic186
HallC3×C17 ⋊ Dic310
SylowC31 ⋊ Dic510
SylowC17 ⋊ Dic930
HallC3×C5 ⋊ Dic1054
SylowC5 ⋊ Dic3162
SylowC3 ⋊ Dic5270

Subgroups

subgroupG/H
center, FrattiniC2C2×D7905
commutator, socle, prime radical, cosocleC2×C3×C5×C17×C31C22
FittingC3×C4×C5×C17×C31C2
hypercenterC4D7905
hypocenterC3×C5×C17×C31Q8
LCSG ▷ C2×C3×C5×C17×C31 ▷ C3×C5×C17×C31
UCSC1 ◁ C2 ◁ C4

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 63240

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,11)(13,16)(14,15)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2^3,3,5,17,31|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^16,g^a=g^30

Conjugacy classes & character table

15813 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, 5, 17, 31]);;
gap> SetPower(coll, 1, F.3);;
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.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^30);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);