Finite Group Databasebeta

← Prev (#72289616) Next (#72289618) →

Dic13530 #72289617

order54120
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic13530

Splittings

HallC3×C5×C11×C41 ⋊ Q8
HallC5×C11×C41 ⋊ Dic6
HallC3×C11×C41 ⋊ Dic10
HallC3×C5×C41 ⋊ Dic22
HallC11×C41 ⋊ Dic30
HallC5×C41 ⋊ Dic66
HallC3×C5×C11 ⋊ Dic82
HallC3×C41 ⋊ Dic110
HallC5×C11 ⋊ Dic246
SylowC41 ⋊ Dic330
HallC3×C11 ⋊ Dic410
HallC3×C5 ⋊ Dic902
SylowC11 ⋊ Dic1230
SylowC5 ⋊ Dic2706
SylowC3 ⋊ Dic4510

Subgroups

subgroupG/H
center, FrattiniC2C2×D6765
commutator, socle, prime radical, cosocleC2×C3×C5×C11×C41C22
FittingC3×C4×C5×C11×C41C2
hypercenterC4D6765
hypocenterC3×C5×C11×C41Q8
LCSG ▷ C2×C3×C5×C11×C41 ▷ C3×C5×C11×C41
UCSC1 ◁ C2 ◁ C4

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 68
transitive degreeμt(G) = 54120

A minimal faithful permutation representation — 2 generators on 68 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)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68);(1,5,3,7)(2,6,4,8)(10,11)(13,16)(14,15)(18,27)(19,26)(20,25)(21,24)(22,23)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2^3,3,5,11,41|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^10,g^a=g^40

Conjugacy classes & character table

13533 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, 11, 41]);;
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.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);