Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#443 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | D4 |
| commutator | C4 | C22×C4×C19 |
| Frattini, cosocle | C2×C4 | C23×C19 |
| socle | C2×C19 | C4×D4 |
| prime radical | C19×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C2×C8×C19 ◁ G | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 304 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,28,25,22,29,27)(23,26,31,34,32,30,35,33);(20,22)(21,25)(23,30)(24,27)(26,32)(28,29)(31,33)(34,35);(20,23,24,31,25,32,29,35)(21,26,28,34,22,30,27,33)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 124);