Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#2143 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | C2×D8 |
| commutator | C8 | C22×C4×C17 |
| Frattini, cosocle | C2×C8 | C23×C17 |
| socle | C2×C17 | C2×#102 |
| prime radical | C2×C17×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C22×C8×C17 ◁ G | |
220 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 100 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)(18,19,22,27,24,28,23,38)(20,29,32,46,30,45,39,43)(21,33,35,44,37,48,36,26)(25,34,41,47,40,49,42,31);(18,20)(19,25)(21,34)(22,32)(23,39)(24,30)(26,43)(27,41)(28,40)(29,33)(31,36)(35,47)(37,49)(38,42)(44,46)(45,48);(20,30)(25,40)(29,45)(31,47)(32,39)(34,49)(41,42)(43,46)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 875);