Finite Group DatabasebetaFinite Group Databasebeta| order | 12450 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C83⋊((C5⋊D5)⋊C3) |
| unambiguous semidirect product | (C5⋊D415)⋊C3 |
| Fitting, socle, derived, Hall | C52×C83 ⋊ C2×C3 |
| Hall | C52⋊C3×C83 ⋊ C2 |
| Hall | C5⋊D415 ⋊ C3 |
| Sylow | C83 ⋊ (C5⋊D5)⋊C3 |
| Hall | C52⋊C3 ⋊ D83 |
| Sylow | C52 ⋊ C3×D83 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C52×C83 | C2×C3 |
| LCS | G ▷ C52×C83 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 98 |
| transitive degree | μt(G) = 1245 |
A minimal faithful permutation representation — 2 generators on 98 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85,87)(86,96,92,89,90,98)(88,93,97,94,95,91);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(84,86,91,97,89)(85,88,90,96,94)(87,92,93,95,98)Polycyclic presentation:
2,3,5^2,83|c^a=c^4,d^a=d^4,e^a=e^82,c^b=cd,d^b=c^2d^3461 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12450, 13);