Finite Group Databasebeta

← Prev (#72290985) Next (#72290987) →

Dic14355 #72290986

order57420
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic14355

Splittings

derived, HallC5×C9×C11×C29 ⋊ C4
HallC9×C11×C29 ⋊ Dic5
HallC5×C11×C29 ⋊ Dic9
HallC5×C9×C29 ⋊ Dic11
HallC5×C9×C11 ⋊ Dic29
HallC11×C29 ⋊ Dic45
HallC9×C29 ⋊ Dic55
HallC5×C29 ⋊ Dic99
HallC9×C11 ⋊ Dic145
HallC5×C11 ⋊ Dic261
HallC5×C9 ⋊ Dic319
SylowC29 ⋊ Dic495
SylowC11 ⋊ Dic1305
SylowC9 ⋊ Dic1595
SylowC5 ⋊ Dic2871

Subgroups

subgroupG/H
center, hypercenterC2D14355
commutator, hypocenterC5×C9×C11×C29C4
FrattiniC2×C3D4785
Fitting, cosocleC2×C5×C9×C11×C29C2
socle, prime radicalC2×C3×C5×C11×C29D3
LCSG ▷ C5×C9×C11×C29
UCSC1 ◁ C2

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 58
transitive degreeμt(G) = 57420

A minimal faithful permutation representation — 2 generators on 58 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)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,18)(16,17)(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:

2^2,3,5,11,29,3|a^2=b,c^3=g,c^a=c^2g^2,d^a=d^4,e^a=e^10,f^a=f^28,g^a=g^2

Conjugacy classes & character table

14358 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, 3, 5, 11, 29, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.7);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.7^2);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);