Finite Group DatabasebetaFinite Group Databasebeta| order | 12810 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D6405 |
| unambiguous semidirect product | C3⋊D2135 |
| unambiguous semidirect product | C5⋊D1281 |
| unambiguous semidirect product | C7⋊D915 |
| unambiguous semidirect product | C61⋊D105 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C5×C7×C61 | C2 |
| LCS | G ▷ C3×C5×C7×C61 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 76 |
| transitive degree | μt(G) = 6405 |
A minimal faithful permutation representation — 2 generators on 76 points:
(1,2,3)(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);(2,3)(5,8)(6,7)(10,15)(11,14)(12,13)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)Polycyclic presentation:
2,3,5,7,61|b^a=b^2,c^a=c^4,d^a=d^6,e^a=e^603204 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D61×D105, D7×D915, D15×D427, … |
| maximal normal subgroup of | D61×D105, D7×D915, D15×D427, … |
| maximal quotient of | Dic6405, C2×D6405, D19215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12810, 25);