Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#473 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C23×C19 |
| prime radical | #41×C19 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ G | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,26,31)(22,32,36,42)(23,37,39,45)(24,30,25,29)(27,35,43,34)(28,40,44,38)(33,47,49,41)(46,50,51,48);(20,22,25,35,26,36,24,34)(21,27,30,42,31,43,29,32)(23,33,40,50,39,49,38,48)(28,41,45,51,44,47,37,46);(20,23)(21,28)(22,33)(24,38)(25,40)(26,39)(27,41)(29,37)(30,45)(31,44)(32,46)(34,48)(35,50)(36,49)(42,51)(43,47)304 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 154);