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Dic8547 #46803148

order34188
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic8547
unambiguous semidirect productC3⋊Dic2849
unambiguous semidirect productC7⋊Dic1221
unambiguous semidirect productC11⋊Dic777

Splittings

derived, HallC3×C7×C11×C37 ⋊ C4
HallC7×C11×C37 ⋊ Dic3
HallC3×C11×C37 ⋊ Dic7
HallC3×C7×C37 ⋊ Dic11
HallC11×C37 ⋊ Dic21
HallC7×C37 ⋊ Dic33
HallC3×C7×C11 ⋊ Dic37
HallC3×C37 ⋊ Dic77
HallC7×C11 ⋊ Dic111
SylowC37 ⋊ Dic231
HallC3×C11 ⋊ Dic259
HallC3×C7 ⋊ Dic407
SylowC11 ⋊ Dic777
SylowC7 ⋊ Dic1221
SylowC3 ⋊ Dic2849

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D8547
commutator, hypocenterC3×C7×C11×C37C4
Fitting, socle, prime radical, cosocleC2×C3×C7×C11×C37C2
LCSG ▷ C3×C7×C11×C37
UCSC1 ◁ C2

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 62
transitive degreeμt(G) = 34188

A minimal faithful permutation representation — 2 generators on 62 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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62);(1,3,2,4)(6,7)(9,14)(10,13)(11,12)(16,25)(17,24)(18,23)(19,22)(20,21)(27,62)(28,61)(29,60)(30,59)(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:

2,3,7,11,37,2|a^2=f,b^a=b^2,c^a=c^6,d^a=d^10,e^a=e^36

Conjugacy classes & character table

8550 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34188, 20);