Finite Group DatabasebetaFinite Group Databasebeta| order | 30258 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C41 |
| unambiguous semidirect product | C3⋊((C41⋊D41)⋊C3) |
| unambiguous semidirect product | (C3⋊(C41⋊D41))⋊C3 |
| Fitting, socle, derived | C3×C412 ⋊ C2×C3 |
| Hall | C3×C412⋊C3 ⋊ C2 |
| Sylow | C412 ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C412 | C2×C3 |
| LCS | G ▷ C3×C412 | |
| UCS | C1 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 126 |
| transitive degree | μt(G) = 369 |
A minimal faithful permutation representation — 2 generators on 126 points:
(2,3)(4,5,7)(6,22,23,10,11,38)(8,35,37,15,16,67)(9,14,47,19,20,33)(12,60,61,26,27,100)(13,65,55,29,30,24)(17,82,83,40,41,120)(18,76,88,43,44,113)(21,34,63,49,50,28)(25,80,90,58,59,119)(31,110,92,68,69,78)(32,112,45,71,72,57)(36,39,116,46,77,81)(42,85,123,86,87,106)(48,108,98,66,91,93)(51,74,111,56,75,115)(52,101,121,94,84,109)(53,54,114,95,89,73)(62,79,107,102,103,117)(64,96,97,104,105,70)(99,118,125,124,122,126);(1,2,3)(4,6,12,28,64,106,91,116,126,72,113,121,37,65,107,69,111,120,114,47,90,119,33,73,83,115,110,117,30,67,109,88,112,125,81,108,123,104,63,26,10)(5,8,17,42,36,9,18,27,62,78,21,45,58,99,105,51,55,22,52,95,93,98,53,94,11,24,56,96,124,25,57,49,92,102,60,43,19,46,86,40,15)(7,13,31,61,101,82,97,71,20,48,39,80,118,87,54,76,50,74,16,38,79,103,23,35,75,34,44,89,85,122,59,77,66,14,32,70,41,84,100,68,29)Polycyclic presentation:
2,3^2,41^2|c^a=c^2,d^a=d^40,e^a=e^40,d^b=d^28e^32,e^b=d^22e^12849 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30258, 25);