Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#1909 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #386 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| prime radical | C2×#41×C17 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C4×C17 ◁ G | |
312 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 312 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)(20,28)(21,32,34,25)(22,35)(23,27)(24,30)(26,37)(29,42,45,49)(31,43)(33,40,36,41)(38,44,48,46)(39,47);(18,20,22,30,23,31,37,47)(19,24,26,28,27,39,35,43)(21,29,36,46,34,45,33,44)(25,38,41,42,32,48,40,49);(18,21)(19,25)(20,29)(22,36)(23,34)(24,38)(26,41)(27,32)(28,42)(30,46)(31,45)(33,37)(35,40)(39,48)(43,49)(44,47)493 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 641);