Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#2135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | C2×#99 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C2×Q8 | C23×C17 |
| prime radical | C2×#41×C17 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×C4×C17 ◁ C2×#41×C17 ◁ G | |
230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,23,28,24,29,21,26)(20,30,33,48,34,45,31,47)(22,36,35,46,38,27,39,44)(25,40,41,37,42,49,43,32);(18,20,24,34)(19,25,29,42)(21,33,23,31)(22,37,38,32)(26,43,28,41)(27,45,36,30)(35,40,39,49)(44,47,46,48);(18,21,24,23)(19,26,29,28)(20,31,34,33)(22,35,38,39)(25,41,42,43)(27,44,36,46)(30,47,45,48)(32,40,37,49)391 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 867);