Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C17 · Schur multiplier (H2) = C1 |
| direct product | C17×SD128 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | D32 |
| commutator, Frattini, cosocle | C32 | C22×C17 |
| prime radical | C17×D32 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C4×C17 ◁ C8×C17 ◁ C16×C17 ◁ C17×C32 ◁ G | |
38 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)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(19,49)(20,80)(21,47)(22,78)(23,45)(24,76)(25,43)(26,74)(27,41)(28,72)(29,39)(30,70)(31,37)(32,68)(33,35)(34,66)(36,64)(38,62)(40,60)(42,58)(44,56)(46,54)(48,52)(51,81)(53,79)(55,77)(57,75)(59,73)(61,71)(63,69)(65,67)595 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 162);