Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C17 · Schur multiplier (H2) = C24×C4 |
| direct product | C17×#2873 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C24 |
| commutator | C2 | C22×C42×C17 |
| Frattini, cosocle | C2×C4 | C24×C17 |
| socle | C22×C17 | C23×C4 |
| prime radical | C24×C17 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ G | |
564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(22,23,25,29,26,28,32,35)(24,27,31,34,30,33,36,37);(18,19,20,21)(22,24)(23,27)(25,31)(26,30)(28,33)(29,34)(32,36)(35,37);(24,30)(27,33)(31,36)(34,37);(23,28)(27,33)(29,35)(34,37)1156 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 1605);