Finite Group DatabasebetaFinite Group Databasebeta| order | 13356 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C3⋊(C7⋊#6157)) |
| unambiguous semidirect product | C7⋊(C3⋊(C3⋊#6157)) |
| Fitting, socle, derived, Hall | C32×C7×C53 ⋊ C4 |
| Hall | C32×C53 ⋊ Dic7 |
| Hall | C7×C53 ⋊ C3⋊Dic3 |
| Hall | C32×C7 ⋊ #6157 |
| Sylow | C53 ⋊ C3⋊Dic21 |
| Sylow | C32 ⋊ C7⋊#6157 |
| Sylow | C7 ⋊ C3⋊(C3⋊#6157) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C7×C53 | C4 |
| prime radical, cosocle | C32×C7×D53 | C2 |
| LCS | G ▷ C32×C7×C53 | |
| UCS | C1 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 3339 |
A minimal faithful permutation representation — 3 generators on 66 points:
(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,43)(23,34,35,40)(27,32,38,37)(29,41,42,33)(36,44,45,50)(39,46,47,49)(48,51,52,53)(55,56)(57,58)(59,60)(62,63)(65,66);(1,2,5,10,19,32,21,8,16,27,39,48,51,44,30,42,28,40,49,45,35,22,13,6,11,20,33,41,25,12,3,7,14,23,36,46,34,17,29,43,50,53,52,47,38,26,15,24,37,31,18,9,4)(54,55,57,59,60,58,56)(64,65,66);(61,62,63)Polycyclic presentation:
2^2,3^2,7,53|a^2=b,c^a=c^2,d^a=d^2,e^a=e^6,f^a=f^30,f^b=f^52885 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13356, 94);