Finite Group DatabasebetaFinite Group Databasebeta| order | 1458 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C27 · Schur multiplier (H2) = C32 |
| direct product | C2×C3×C81⋊C3 |
| unambiguous semidirect product | (C2×C81)⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C27 | C32 |
| commutator | C3 | C2×C32×C27 |
| Frattini, cosocle | C27 | C2×C33 |
| socle | C2×C32 | C3×C27 |
| prime radical | C2×C33 | C27 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C27 ◁ G | |
164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 86 |
| transitive degree | μt(G) = 486 |
A minimal faithful permutation representation — 3 generators on 86 points:
(1,2)(3,4,5);(6,7,11,8,13,23,16,28,42,9,14,24,17,29,43,33,50,64,19,31,44,35,52,65,55,71,79,10,15,22,18,30,40,34,51,61,20,32,41,36,53,62,56,72,77,38,54,63,58,73,78,74,83,85,21,12,25,37,26,45,57,47,66,39,27,46,59,48,67,75,69,80,60,49,68,76,70,81,84,82,86);(7,12,15)(11,22,25)(13,26,30)(14,27,32)(23,40,45)(24,41,46)(28,47,51)(29,48,53)(31,49,54)(42,61,66)(43,62,67)(44,63,68)(50,69,72)(52,70,73)(64,77,80)(65,78,81)(71,82,83)(79,85,86)594 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 392);