Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#1554 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C42 | C23×C17 |
| prime radical | C23×C17 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C2×C42×C17 ◁ G | |
198 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 1088 |
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)(18,19,22,25,23,26,21,30)(20,27,28,32,29,24,31,33)(34,35)(36,42)(37,45)(38,40)(39,43)(41,47)(44,49)(46,48);(18,20,23,29)(19,24,26,27)(21,28,22,31)(25,32,30,33)(34,36,38,44)(35,39,40,48)(37,46,47,43)(41,49,45,42);(18,21,23,22)(19,25,26,30)(20,28,29,31)(24,32,27,33)(35,40)(39,48)(41,45)(42,49)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 286);