Finite Group DatabasebetaFinite Group Databasebeta| order | 19698 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C72×C67 ⋊ C2×C3 |
| Hall | C7×#13064104 ⋊ C2 |
| Hall | C7⋊D469 ⋊ C3 |
| Sylow | C67 ⋊ ΓL(1,𝔽8)⋊D7 |
| Sylow | C72 ⋊ D67⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C67 | C2×C3 |
| LCS | G ▷ C72×C67 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1407 |
A minimal faithful permutation representation — 2 generators on 81 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,59)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,54,55,49,40,64)(34,61,62,50,48,57)(37,42,60,52,53,58)(51,65,66,63,56,67)(69,72,73,70,71,74)(76,77)(78,79)(80,81);(1,2,5,11,19,32,45,15,26,14,25,40,56,66,47,60,64,61,46,53,18,29,39,41,57,63,50,33,20,4,8,13,23,37,52,35,22,7,6,12,21,34,51,62,28,43,44,27,42,31,48,55,58,59,67,65,54,36,10,16,24,38,49,30,17,9,3)(68,69,71,74,73,72,70)(75,76,78,80,81,79,77)Polycyclic presentation:
2,3,7^2,67|c^a=c^6,d^a=d^6,e^a=e^66,c^b=c^2,e^b=e^29561 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19698, 61);