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Dic8820 #72273133

order35280
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic8820

Splittings

HallC5×C9×C49 ⋊ Q16
HallC9×C49 ⋊ Dic20
HallC5×C49 ⋊ Dic36
SylowC49 ⋊ Dic180
HallC5×C9 ⋊ Dic196
SylowC9 ⋊ Dic980
SylowC5 ⋊ Dic1764

Subgroups

subgroupG/H
centerC2D8820
commutator, cosocleC4×C5×C9×C49C22
FrattiniC3×C4×C7C2×D105
FittingC5×C8×C9×C49C2
socle, prime radicalC2×C3×C5×C7D84
hypercenterC8D2205
hypocenterC5×C9×C49Q16
LCSG ▷ C4×C5×C9×C49 ▷ C2×C5×C9×C49 ▷ C5×C9×C49
UCSC1 ◁ C2 ◁ C4 ◁ C8

162 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 79
transitive degreeμt(G) = 35280

A minimal faithful permutation representation — 2 generators on 79 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,25)(19,24)(20,23)(21,22)(27,30)(28,29)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(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:

2^4,3,5,7,3,7|a^2=d,b^2=cd,c^2=d,e^3=h,g^7=i,b^a=bc,c^a=cd,e^a=e^2h^2,f^a=f^4,g^a=g^6i^6,h^a=h^2,i^a=i^6

Conjugacy classes & character table

8823 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", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 7, 3, 7]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^6*F.9^6);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);