Finite Group DatabasebetaFinite Group Databasebeta| order | 12393 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C17×C27 · Schur multiplier (H2) = C32 |
| direct product | C17×#10624445 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C17 | H(3,𝔽3) |
| commutator | C32 | C3×C17×C27 |
| Frattini, cosocle | C32×C9 | C32×C17 |
| socle | C32×C17 | C32⋊C9 |
| prime radical | C33×C17 | C3×C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C17 ◁ C32×C9×C17 ◁ G | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 4131 |
A minimal faithful permutation representation — 2 generators on 71 points:
(1,2,5,3,6,11,8,13,18,4,7,12,9,14,19,16,21,24,10,15,20,17,22,25,23,26,27)(28,29,32)(30,36,40)(31,35,41)(33,42,38)(34,45,39)(37,47,50)(43,53,54)(44,52,49)(46,51,48)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71);(28,30,37,31,38,48,39,49,54)(29,33,43,35,44,47,34,36,46)(32,40,50,41,42,51,45,52,53)2601 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12393, 5);