Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C23×C4 |
| direct product | C19×#555 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C19 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ G | |
236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 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)(22,26)(23,27)(24,25)(29,30)(33,34);(21,25)(23,26)(28,29,32,30)(31,34,35,33);(20,22,24,27)(21,26,25,23)(28,30,32,29)(31,34,35,33);(20,23,24,26)(21,27,25,22)(28,31,32,35)(29,33,30,34)475 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 236);