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Dic390 #13095702

order1560
automorphismAut(G) order 149760
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic390

Splittings

HallC3×C5×C13 ⋊ Q8
HallC5×C13 ⋊ Dic6
HallC3×C13 ⋊ Dic10
HallC3×C5 ⋊ Dic26
SylowC13 ⋊ Dic30
SylowC5 ⋊ Dic78
SylowC3 ⋊ Dic130

Subgroups

subgroupG/H
center, FrattiniC2C2×D195
commutator, socle, prime radical, cosocleC2×C3×C5×C13C22
FittingC3×C4×C5×C13C2
hypercenterC4D195
hypocenterC3×C5×C13Q8
LCSG ▷ C2×C3×C5×C13 ▷ C3×C5×C13
UCSC1 ◁ C2 ◁ C4

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 1560

A minimal faithful permutation representation — 2 generators on 29 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);(1,5,3,7)(2,6,4,8)(10,11)(13,16)(14,15)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)

Polycyclic presentation:

2^3,3,5,13|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^12

Conjugacy classes & character table

393 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic780, Dic1170, Dic1950, …
maximal normal subgroup ofDic780
maximal quotient ofDic1170, Dic1950, Dic2730, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1560, 138);