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Dic6270 #72236062

order25080
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic6270

Splittings

HallC3×C5×C11×C19 ⋊ Q8
HallC5×C11×C19 ⋊ Dic6
HallC3×C11×C19 ⋊ Dic10
HallC3×C5×C19 ⋊ Dic22
HallC11×C19 ⋊ Dic30
HallC3×C5×C11 ⋊ Dic38
HallC5×C19 ⋊ Dic66
HallC3×C19 ⋊ Dic110
HallC5×C11 ⋊ Dic114
HallC3×C11 ⋊ Dic190
SylowC19 ⋊ Dic330
HallC3×C5 ⋊ Dic418
SylowC11 ⋊ Dic570
SylowC5 ⋊ Dic1254
SylowC3 ⋊ Dic2090

Subgroups

subgroupG/H
center, FrattiniC2C2×D3135
commutator, socle, prime radical, cosocleC2×C3×C5×C11×C19C22
FittingC3×C4×C5×C11×C19C2
hypercenterC4D3135
hypocenterC3×C5×C11×C19Q8
LCSG ▷ C2×C3×C5×C11×C19 ▷ C3×C5×C11×C19
UCSC1 ◁ C2 ◁ C4

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 46
transitive degreeμt(G) = 25080

A minimal faithful permutation representation — 2 generators on 46 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);(1,5,3,7)(2,6,4,8)(10,11)(13,16)(14,15)(18,27)(19,26)(20,25)(21,24)(22,23)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)

Polycyclic presentation:

2^3,3,5,11,19|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^10,g^a=g^18

Conjugacy classes & character table

6273 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofDic12540
maximal normal subgroup ofDic12540

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, 19]);;
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.7^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);