Finite Group DatabasebetaFinite Group Databasebeta| order | 30258 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊#23758624 |
| Fitting, socle | C3×C412 ⋊ D3 |
| derived, Hall | C3×C412⋊C3 ⋊ C2 |
| Sylow | C412 ⋊ C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C412⋊C3 | C2 |
| Fitting, socle | C3×C412 | D3 |
| LCS | G ▷ C3×C412⋊C3 | |
| UCS | C1 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 126 |
| transitive degree | μt(G) = 369 |
A minimal faithful permutation representation — 3 generators on 126 points:
(2,3)(4,6,11,23,53,97,77,118,125,60,104,112,31,64,107,58,102,123,117,38,76,110,29,62,69,106,124,108,25,56,100,74,116,126,68,111,114,103,78,119,65)(5,32,54,90,48,40,18,59,86,72,35,85,80,33,61,113,95,87,46,36,16,92,122,42,81,12,24,66,101,120,94,63,45,34,15,43,83,13,28,67,51,47,21,17,9,75,99,115,91,79,41,26,57,93,50,44,20,14,8,70,98,82,88,27,37,109,121,89,73,39,55,71,105,84,96,30,52,49,22,19,10,7);(1,2,3);(4,5,7)(6,10,19)(8,14,31)(9,17,38)(11,22,49)(12,25,24)(13,29,28)(15,34,69)(16,36,74)(18,40,78)(20,44,64)(21,47,76)(23,52,30)(26,58,57)(27,60,37)(32,65,54)(33,68,61)(35,72,114)(39,77,55)(41,79,102)(42,56,81)(43,62,83)(45,63,106)(46,87,116)(48,90,119)(50,93,107)(51,67,110)(53,96,84)(59,103,86)(66,108,101)(70,112,98)(71,97,105)(73,89,118)(75,117,99)(80,85,111)(82,104,88)(91,115,123)(92,100,122)(94,120,124)(95,113,126)(109,125,121)Polycyclic presentation:
2,3^2,41^2|b^a=b^2,c^a=c^2,d^a=de^3,e^a=e^40,d^b=d^28e^32,e^b=d^22e^12906 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30258, 24);