Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C17 · Schur multiplier (H2) = C1 |
| direct product | C17×#1425 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | D16 |
| commutator | C16 | C2×C4×C17 |
| Frattini, cosocle | C2×C16 | C22×C17 |
| socle | C2×C17 | #366 |
| prime radical | C22×C17 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C2×C4×C17 ◁ C2×C8×C17 ◁ C2×C16×C17 ◁ G | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 2 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,28,25,30,24,46)(20,31,34,56,37,58,36,68)(21,38,40,69,42,71,41,72)(23,44,43,54,45,73,39,29)(26,47,48,64,50,63,74,33)(27,51,52,61,53,60,70,32)(35,66,65,78,67,81,62,57)(49,59,55,75,76,80,77,79);(18,20,32,59,23,35,40,47,22,34,51,75,43,65,42,64,25,37,61,80,45,67,41,63,24,36,60,79,39,62,21,33)(19,26,38,57,29,49,70,68,46,74,72,81,73,77,53,58,30,50,71,78,54,76,52,56,28,48,69,66,44,55,27,31)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 157);