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Dic12420 #72285752

order49680
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic12420

Splittings

HallC5×C23×C27 ⋊ Q16
HallC23×C27 ⋊ Dic20
HallC5×C27 ⋊ Dic92
HallC5×C23 ⋊ Dic108
SylowC27 ⋊ Dic460
SylowC23 ⋊ Dic540
SylowC5 ⋊ Dic2484

Subgroups

subgroupG/H
centerC2D12420
commutator, cosocleC4×C5×C23×C27C22
FrattiniC4×C9C2×D345
FittingC5×C8×C23×C27C2
socle, prime radicalC2×C3×C5×C23D36
hypercenterC8D3105
hypocenterC5×C23×C27Q16
LCSG ▷ C4×C5×C23×C27 ▷ C2×C5×C23×C27 ▷ C5×C23×C27
UCSC1 ◁ C2 ◁ C4 ◁ C8

144 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 71
transitive degreeμt(G) = 49680

A minimal faithful permutation representation — 2 generators on 71 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(45,48)(46,47)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)

Polycyclic presentation:

2^4,3,5,23,3^2|a^2=d,b^2=cd,c^2=d,e^3=h,h^3=i,b^a=bc,c^a=cd,e^a=e^2h^2i^2,f^a=f^4,g^a=g^22,h^a=h^2i^2,i^a=i^2

Conjugacy classes & character table

12423 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, 23, 3, 3]);;
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, 8, 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*F.9^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^22);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);