Finite Group DatabasebetaFinite Group Databasebeta| order | 12393 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C17 · Schur multiplier (H2) = C32 |
| direct product | C17×#10624485 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C17 | #6857 |
| commutator, Frattini, cosocle | C32×C9 | C32×C17 |
| prime radical | C3×C17×#732 | C3 |
| LCS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C17 ◁ C33×C17 ◁ C32×C9×C17 ◁ G | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 98 |
| transitive degree | μt(G) = 1377 |
A minimal faithful permutation representation — 2 generators on 98 points:
(1,2,7)(3,12,34,16,44,58,15,30,64)(4,17,46)(5,10,24)(6,11,25)(8,26,50,33,62,42,28,59,48)(9,29,21)(13,37,70,39,71,57,45,77,78)(14,40,56,18,36,54,43,67,22)(19,49,63)(20,32,23)(27,60,74,61,81,38,68,80,41)(31,66,51)(35,69,75,65,79,72,76,53,73)(47,55,52)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98);(1,3,13)(2,8,27)(4,18,38)(5,16,45)(6,15,39)(7,22,53)(9,30,65)(10,33,68)(11,28,61)(12,35,17)(14,41,21)(19,50,72)(20,43,74)(23,26,57)(24,56,79)(25,54,69)(29,62,70)(31,67,77)(32,44,76)(34,60,66)(36,37,49)(40,71,47)(42,75,51)(46,59,78)(48,73,52)(55,58,80)(63,64,81)697 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12393, 45);