Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#1917 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C17 | C23 |
| commutator | C23 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| socle, prime radical | C23×C17 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C17 ◁ G | |
228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,21)(22,23)(24,27)(25,26)(28,29)(30,31,35,32,34,36,33,37);(23,26)(27,29)(30,32,35,36,34,37,33,31);(19,21)(22,24,25,28)(23,27,26,29)(30,33,34,35)(31,32,36,37)748 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 649);