Finite Group DatabasebetaFinite Group Databasebeta| order | 31213 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C7×C13×C49 · Schur multiplier (H2) = C72 |
| direct product | C13×C72⋊C49 |
| unambiguous semidirect product | C72⋊(C13×C49) |
| unambiguous semidirect product | (C72×C13)⋊C49 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C72×C13 | C72 |
| commutator | C7 | C7×C13×C49 |
| Frattini, cosocle | C72 | C72×C13 |
| prime radical | C73×C13 | C7 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C72×C13 ◁ G | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 4459 |
A minimal faithful permutation representation — 2 generators on 111 points:
(1,2,4,8,15,26,35)(3,6,10,17,28,37,44)(5,11,21,33,42,49,43)(7,13,20,30,39,36,45)(9,18,31,41,48,34,12)(14,24,19,32,27,38,46)(16,29,40,47,25,23,22)(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);(2,5,12,23,24,13,6)(4,9,19,10,11,22,20)(8,16,17,18,30,21,32)(15,27,33,39,31,28,29)(26,36,40,42,37,38,41)(35,44,45,46,47,48,49)5005 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31213, 3);