Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#1813 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #117 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| socle | C2×C17 | #380 |
| prime radical | C2×#41×C17 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C2×M16×C17 ◁ G | |
252 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 116 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)(18,19,23,27,24,28,22,37)(20,29,32,40,33,44,31,25)(21,34,35,41,36,26,38,42)(30,45,46,49,47,43,48,39);(18,20,22,31,24,33,23,32)(19,25,27,29,28,40,37,44)(21,30,35,46,36,47,38,48)(26,39,41,43,34,49,42,45);(18,21)(19,26)(20,30)(22,35)(23,38)(24,36)(25,39)(27,41)(28,34)(29,43)(31,46)(32,48)(33,47)(37,42)(40,49)(44,45)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 545);