Finite Group DatabasebetaFinite Group Databasebeta| order | 12393 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C17 · Schur multiplier (H2) = C32×C9 |
| direct product | C17×#10624552 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C33×C17 | C33 |
| commutator | C32 | C32×C9×C17 |
| Frattini, cosocle | C33 | C33×C17 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33×C17 ◁ G | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 12393 |
A minimal faithful permutation representation — 3 generators on 80 points:
(1,2,4,3,5,7,6,8,9)(10,11,14)(12,15,21)(13,17,22)(16,26,20)(18,23,29)(19,25,30)(24,33,28)(27,32,35)(31,36,34)(37,38,41)(39,43,49)(40,44,48)(42,50,47)(45,52,57)(46,53,56)(51,58,55)(54,60,62)(59,63,61)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80);(10,12,18,13,19,27,20,28,34)(11,15,23,17,25,32,16,24,31)(14,21,29,22,30,35,26,33,36)(38,42,44)(41,48,50)(43,51,53)(49,56,58)(52,59,60)(57,62,63);(11,16,17)(14,22,26)(15,24,25)(21,30,33)(23,31,32)(29,35,36)(37,39,45,47,55,61,40,46,54)(38,43,52,42,51,59,44,53,60)(41,49,57,50,58,63,48,56,62)2601 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12393, 112);