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) = C22×C4 |
| direct product | C17×#2129 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | C2×#99 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C2×D4 | 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 | |
238 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 238 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)(20,28,31,43)(21,25)(22,35,37,26)(23,27)(24,30)(29,44,32,46)(33,48,49,41)(34,42)(36,39)(38,45)(40,47);(18,20,23,31)(19,24,27,39)(21,32,34,29)(22,36,37,30)(25,40,42,38)(26,43,35,28)(33,45,49,47)(41,44,48,46);(18,21,23,34)(19,25,27,42)(20,29,31,32)(22,33,37,49)(24,38,39,40)(26,41,35,48)(28,44,43,46)(30,45,36,47)391 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 861);