Finite Group DatabasebetaFinite Group Databasebeta| order | 13356 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | D371⋊(C3⋊D3) |
| Fitting, socle, derived, Hall | C32×C7×C53 ⋊ C22 |
| Hall | C32×C53 ⋊ C2×D7 |
| Hall | C7×C53 ⋊ C2×C3⋊D3 |
| Hall | C32×C7 ⋊ C2×D53 |
| Sylow | C53 ⋊ D7×C3⋊D3 |
| Sylow | C32 ⋊ D7×D53 |
| Sylow | C7 ⋊ C3⋊D3×D53 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C7×C53 | C22 |
| LCS | G ▷ C32×C7×C53 | |
| UCS | C1 | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 3339 |
A minimal faithful permutation representation — 3 generators on 66 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,55)(56,57)(58,59)(62,63)(65,66);(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)(55,56)(57,58)(59,60)(64,65,66);(61,62,63)Polycyclic presentation:
2^2,3^2,7,53|e^a=e^6,f^a=f^52,c^b=c^2,d^b=d^2,e^b=e^6888 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13356, 99);