Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#2123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | C2×#99 |
| commutator | C23 | C22×C4×C17 |
| Frattini, cosocle | C2×D4 | C23×C17 |
| prime radical | C2×#41×C17 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×C4×C17 ◁ C2×#41×C17 ◁ G | |
234 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 234 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)(20,28)(21,25)(22,35,36,26)(23,27)(24,30,38,45)(29,42)(31,43)(32,46)(33,48,49,40)(34,41)(37,44,39,47);(18,20,23,31)(19,24,27,38)(21,32,34,29)(22,30,36,45)(25,39,41,37)(26,28,35,43)(33,47,49,44)(40,46,48,42);(18,21,23,34)(19,25,27,41)(20,29,31,32)(22,33,36,49)(24,37,38,39)(26,40,35,48)(28,42,43,46)(30,44,45,47)391 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 855);