Finite Group DatabasebetaFinite Group Databasebeta| order | 13356 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D63⋊D53 |
| unambiguous semidirect product | D371⋊D9 |
| unambiguous semidirect product | D477⋊D7 |
| Fitting, derived, Hall | C7×C9×C53 ⋊ C22 |
| Hall | C9×C53 ⋊ C2×D7 |
| Hall | C7×C53 ⋊ C2×D9 |
| Hall | C7×C9 ⋊ C2×D53 |
| Sylow | C53 ⋊ D7×D9 |
| Sylow | C9 ⋊ D7×D53 |
| Sylow | C7 ⋊ D9×D53 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×C9×C53 | C22 |
| Frattini | C3 | D21⋊D53 |
| socle | C3×C7×C53 | C2×D3 |
| LCS | G ▷ C7×C9×C53 | |
| UCS | C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 3339 |
A minimal faithful permutation representation — 2 generators on 69 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)(62,64)(63,67)(65,66)(68,69);(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)(61,69,64,67,66,65,63,62,68)Polycyclic presentation:
2^2,3,7,53,3|c^3=f^2,d^a=d^6,e^a=e^52,c^b=c^2f,d^b=d^6,f^b=f^2888 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13356, 30);