Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C9 · Schur multiplier (H2) = C3 |
| direct product | C4×#6873 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | H(3,𝔽3) |
| commutator | C32 | C3×C4×C9 |
| Frattini, cosocle | C2×C3×C9 | C2×C32 |
| socle | C2×C3 | C2×C32⋊C9 |
| prime radical | C2×H(3,𝔽3) | C2×C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C3×C4×C9 ◁ G | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 324 |
A minimal faithful permutation representation — 2 generators on 85 points:
(1,2,3,4)(5,6,10,8,13,25,20,31,47,22,12,28,43,30,52,66,55,73,9,14,24,21,32,46,42,56,69)(7,15,33,18,35,58,40,60,77,17,11,23,39,27,45,64,50,68,19,29,51,41,54,72,65,75,83)(16,34,44,37,59,67,62,78,81,38,26,48,63,49,70,80,71,82,36,53,57,61,74,76,79,84,85);(5,7,16)(6,11,26)(8,18,37)(9,19,36)(10,23,44)(12,29,53)(13,27,49)(14,15,34)(17,38,22)(20,40,62)(21,41,61)(24,33,57)(25,45,67)(28,51,48)(30,54,74)(31,50,71)(32,35,59)(39,63,43)(42,65,79)(46,58,76)(47,68,81)(52,72,70)(55,75,84)(56,60,78)(64,80,66)(69,77,85)(73,83,82)204 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 83);