Finite Group DatabasebetaFinite Group Databasebeta| order | 22698 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#21438969 |
| Fitting, socle, derived | C3×C13×C97 ⋊ C2×C3 |
| Hall | C3×#16838252 ⋊ C2 |
| Hall | C13×C97 ⋊ C3×D3 |
| Sylow | C97 ⋊ C3⋊(D13⋊C3) |
| Sylow | C13 ⋊ C3⋊(D97⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C13×C97 | C2×C3 |
| LCS | G ▷ C3×C13×C97 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 3783 |
A minimal faithful permutation representation — 2 generators on 113 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,87)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,80,78,58,59,57)(51,83,79,63,72,94)(53,90,91,73,71,84)(56,65,89,75,76,85)(60,92,96,74,81,82)(77,88,95,86,93,97)(99,103,104,100,101,105)(102,109,108,106,107,110)(112,113);(1,2,5,11,19,32,51,55,18,29,39,41,61,82,70,89,72,92,90,69,88,97,94,78,45,15,26,14,25,40,60,73,52,33,20,4,8,13,23,37,56,77,76,31,48,59,62,64,84,91,47,66,80,67,46,65,86,75,54,35,22,7,6,12,21,34,53,74,58,36,10,16,24,38,57,79,95,93,50,71,81,83,85,87,96,68,28,43,44,27,42,63,49,30,17,9,3)(98,99,102,101,105,108,107,109,110,104,103,106,100)(111,112,113)Polycyclic presentation:
2,3^2,13,97|c^a=c^2,d^a=d^12,e^a=e^96,d^b=d^3,e^b=e^35639 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22698, 45);