Finite Group DatabasebetaFinite Group Databasebeta| order | 31213 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C7×C13×C49 · Schur multiplier (H2) = C7 |
| direct product | C13×C49⋊C49 |
| unambiguous semidirect product | C49⋊(C13×C49) |
| unambiguous semidirect product | (C13×C49)⋊C49 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C72×C13 | C72 |
| commutator | C7 | C7×C13×C49 |
| Frattini, cosocle | C72 | C72×C13 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C72×C13 ◁ G | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 31213 |
A minimal faithful permutation representation — 2 generators on 111 points:
(2,4,7,12,20,32,42)(5,9,15,24,8,13,21)(10,17,27,16,25,14,22)(18,29,23,33,26,35,28)(30,39,37,34,43,38,36)(40,49,48,47,46,45,44)(50,51,53,56,60,65,71,52,54,57,61,66,72,78,55,58,62,67,73,79,84,59,63,68,74,80,85,89,64,69,75,81,86,90,93,70,76,82,87,91,94,96,77,83,88,92,95,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111);(1,2,5,10,18,30,40,3,4,8,14,23,34,44,6,7,9,16,26,36,45,11,12,13,17,28,37,46,19,20,15,22,29,38,47,31,32,21,25,33,39,48,41,42,24,27,35,43,49)5005 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31213, 4);