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Dic13398 #72289206

order53592
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic13398

Splittings

HallC3×C7×C11×C29 ⋊ Q8
HallC7×C11×C29 ⋊ Dic6
HallC3×C11×C29 ⋊ Dic14
HallC3×C7×C29 ⋊ Dic22
HallC11×C29 ⋊ Dic42
HallC3×C7×C11 ⋊ Dic58
HallC7×C29 ⋊ Dic66
HallC3×C29 ⋊ Dic154
HallC7×C11 ⋊ Dic174
HallC3×C11 ⋊ Dic406
SylowC29 ⋊ Dic462
HallC3×C7 ⋊ Dic638
SylowC11 ⋊ Dic1218
SylowC7 ⋊ Dic1914
SylowC3 ⋊ Dic4466

Subgroups

subgroupG/H
center, FrattiniC2C2×D6699
commutator, socle, prime radical, cosocleC2×C3×C7×C11×C29C22
FittingC3×C4×C7×C11×C29C2
hypercenterC4D6699
hypocenterC3×C7×C11×C29Q8
LCSG ▷ C2×C3×C7×C11×C29 ▷ C3×C7×C11×C29
UCSC1 ◁ C2 ◁ C4

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 58
transitive degreeμt(G) = 53592

A minimal faithful permutation representation — 2 generators on 58 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)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58);(1,5,3,7)(2,6,4,8)(10,11)(13,18)(14,17)(15,16)(20,29)(21,28)(22,27)(23,26)(24,25)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)

Polycyclic presentation:

2^3,3,7,11,29|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^10,g^a=g^28

Conjugacy classes & character table

13401 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 7, 11, 29]);;
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.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);