Finite Group DatabasebetaFinite Group Databasebeta| order | 1312 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C41 · Schur multiplier (H2) = C22 |
| direct product | #102×C41 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C41 | D4 |
| commutator | C4 | C2×C4×C41 |
| Frattini, cosocle | C2×C4 | C22×C41 |
| prime radical | C2×D4×C41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C41 ◁ C2×C4×C41 ◁ G | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 656 |
A minimal faithful permutation representation — 2 generators on 53 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(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);(6,9)(7,12)(10,11)574 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1312, 9);