Finite Group DatabasebetaFinite Group Databasebeta| order | 1224 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 129024 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C3 |
| abelian | C23×C32×C17 |
| multiplicative | (ℤ/2163)ˣ, (ℤ/2884)ˣ, (ℤ/3708)ˣ, (ℤ/4326)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1224 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(7,8,9)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29);(3,4)(10,11,12);(5,6)1224 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1224, 164);