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Dic13110 #72288705

order52440
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic13110

Splittings

HallC3×C5×C19×C23 ⋊ Q8
HallC5×C19×C23 ⋊ Dic6
HallC3×C19×C23 ⋊ Dic10
HallC19×C23 ⋊ Dic30
HallC3×C5×C23 ⋊ Dic38
HallC3×C5×C19 ⋊ Dic46
HallC5×C23 ⋊ Dic114
HallC5×C19 ⋊ Dic138
HallC3×C23 ⋊ Dic190
HallC3×C19 ⋊ Dic230
SylowC23 ⋊ Dic570
SylowC19 ⋊ Dic690
HallC3×C5 ⋊ Dic874
SylowC5 ⋊ Dic2622
SylowC3 ⋊ Dic4370

Subgroups

subgroupG/H
center, FrattiniC2C2×D6555
commutator, socle, prime radical, cosocleC2×C3×C5×C19×C23C22
FittingC3×C4×C5×C19×C23C2
hypercenterC4D6555
hypocenterC3×C5×C19×C23Q8
LCSG ▷ C2×C3×C5×C19×C23 ▷ C3×C5×C19×C23
UCSC1 ◁ C2 ◁ C4

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 58
transitive degreeμt(G) = 52440

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)(13,16)(14,15)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

2^3,3,5,19,23|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^18,g^a=g^22

Conjugacy classes & character table

13113 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, 5, 19, 23]);;
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.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);