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Dic10395 #72274433

order41580
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic10395

Splittings

derived, HallC5×C7×C11×C27 ⋊ C4
HallC7×C11×C27 ⋊ Dic5
HallC5×C11×C27 ⋊ Dic7
HallC5×C7×C27 ⋊ Dic11
HallC5×C7×C11 ⋊ Dic27
HallC11×C27 ⋊ Dic35
HallC7×C27 ⋊ Dic55
HallC5×C27 ⋊ Dic77
HallC7×C11 ⋊ Dic135
HallC5×C11 ⋊ Dic189
HallC5×C7 ⋊ Dic297
SylowC27 ⋊ Dic385
SylowC11 ⋊ Dic945
SylowC7 ⋊ Dic1485
SylowC5 ⋊ Dic2079

Subgroups

subgroupG/H
center, hypercenterC2D10395
commutator, hypocenterC5×C7×C11×C27C4
FrattiniC2×C9D1155
Fitting, cosocleC2×C5×C7×C11×C27C2
socle, prime radicalC2×C3×C5×C7×C11D9
LCSG ▷ C5×C7×C11×C27
UCSC1 ◁ C2

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 41580

A minimal faithful permutation representation — 2 generators on 54 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)(32,33,34,35,36)(37,38,39,40,41,42,43)(44,45,46,47,48,49,50,51,52,53,54);(1,3,2,4)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(33,36)(34,35)(38,43)(39,42)(40,41)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2^2,3,5,7,11,3^2|a^2=b,c^3=g,g^3=h,c^a=c^2g^2h^2,d^a=d^4,e^a=e^6,f^a=f^10,g^a=g^2h^2,h^a=h^2

Conjugacy classes & character table

10398 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");;
gap> coll := SingleCollector(F, [2, 2, 3, 5, 7, 11, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);