Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#424 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C23 |
| commutator | C22 | C22×C4×C19 |
| Frattini, cosocle | C2×C4 | C23×C19 |
| socle | C22×C19 | C22×C4 |
| prime radical | C23×C19 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ G | |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,23)(24,25,27,30,28,31,34,37)(26,29,33,36,32,35,38,39);(20,21,22,23)(24,26)(25,29)(27,33)(28,32)(30,36)(31,35)(34,38)(37,39);(26,32)(29,35)(33,38)(36,39)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 105);