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Dic10296 #72274352

order41184
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic10296

Splittings

HallC9×C11×C13 ⋊ Q32
HallC11×C13 ⋊ Dic72
HallC9×C13 ⋊ Dic88
HallC9×C11 ⋊ Dic104
SylowC13 ⋊ Dic792
SylowC11 ⋊ Dic936
SylowC9 ⋊ Dic1144

Subgroups

subgroupG/H
centerC2D10296
commutator, cosocleC8×C9×C11×C13C22
FrattiniC3×C8C2×D429
FittingC9×C11×C13×C16C2
socle, prime radicalC2×C3×C11×C13D24
hypercenterC16D1287
hypocenterC9×C11×C13Q32
LCSG ▷ C8×C9×C11×C13 ▷ C4×C9×C11×C13 ▷ C2×C9×C11×C13 ▷ C9×C11×C13
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16

144 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 63 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) = 65
transitive degreeμt(G) = 41184

A minimal faithful permutation representation — 2 generators on 65 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,41)(35,40)(36,39)(37,38)(43,52)(44,51)(45,50)(46,49)(47,48)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Polycyclic presentation:

2^5,3,11,13,3|a^2=e,b^2=cd,c^2=de,d^2=e,f^3=i,b^a=bc,c^a=cd,d^a=de,f^a=f^2i^2,g^a=g^10,h^a=h^12,i^a=i^2

Conjugacy classes & character table

10299 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, 2, 3, 11, 13, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> SetConjugate(coll, 8, 1, F.8^12);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);