Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#458 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #120 |
| commutator, Frattini, cosocle | C23 | C23×C19 |
| prime radical | C19×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C23×C19 ◁ G | |
160 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)(20,21,24,27)(22,25,29,32)(23,26,30,33)(28,34,35,31);(20,22)(21,25)(23,28)(24,29)(26,31)(27,32)(30,35)(33,34);(20,23)(21,26)(22,28)(24,30)(25,31)(27,33)(29,35)(32,34)304 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 139);