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Dic13090 #72288679

order52360
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic13090

Splittings

HallC5×C7×C11×C17 ⋊ Q8
HallC7×C11×C17 ⋊ Dic10
HallC5×C11×C17 ⋊ Dic14
HallC5×C7×C17 ⋊ Dic22
HallC5×C7×C11 ⋊ Dic34
HallC11×C17 ⋊ Dic70
HallC7×C17 ⋊ Dic110
HallC5×C17 ⋊ Dic154
HallC7×C11 ⋊ Dic170
HallC5×C11 ⋊ Dic238
HallC5×C7 ⋊ Dic374
SylowC17 ⋊ Dic770
SylowC11 ⋊ Dic1190
SylowC7 ⋊ Dic1870
SylowC5 ⋊ Dic2618

Subgroups

subgroupG/H
center, FrattiniC2C2×D6545
commutator, socle, prime radical, cosocleC2×C5×C7×C11×C17C22
FittingC4×C5×C7×C11×C17C2
hypercenterC4D6545
hypocenterC5×C7×C11×C17Q8
LCSG ▷ C2×C5×C7×C11×C17 ▷ C5×C7×C11×C17
UCSC1 ◁ C2 ◁ C4

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 48
transitive degreeμt(G) = 52360

A minimal faithful permutation representation — 2 generators on 48 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);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,20)(16,19)(17,18)(22,31)(23,30)(24,29)(25,28)(26,27)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)

Polycyclic presentation:

2^3,5,7,11,17|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^6,f^a=f^10,g^a=g^16

Conjugacy classes & character table

13093 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, 5, 7, 11, 17]);;
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.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);