Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#1858 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #120 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| socle | C2×C17 | #385 |
| prime radical | C2×#41×C17 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ G | |
252 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 108 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)(18,19,23,27,24,28,22,35)(20,29,31,25,32,40,36,39)(21,26,33,42,34,43,37,41)(30,45,48,49,47,44,46,38);(18,20,23,31,24,32,22,36)(19,25,27,40,28,39,35,29)(21,30,33,48,34,47,37,46)(26,38,42,45,43,49,41,44);(18,21)(19,26)(20,30)(22,37)(23,33)(24,34)(25,38)(27,42)(28,43)(29,44)(31,48)(32,47)(35,41)(36,46)(39,49)(40,45)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 590);