Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#1915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #386 |
| commutator | C23 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| prime radical | C2×#41×C17 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C4×C17 ◁ G | |
308 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 308 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)(18,19,22,27,23,26,28,41)(20,24,31,38,30,39,35,42)(21,32,34,43,33,25,36,40)(29,37,45,48,44,47,46,49);(18,20,28,31,23,30,22,35)(19,24,27,42,26,39,41,38)(21,29,34,46,33,44,36,45)(25,37,43,48,32,47,40,49);(18,21)(19,25)(20,29)(22,36)(23,33)(24,37)(26,32)(27,43)(28,34)(30,44)(31,46)(35,45)(38,49)(39,47)(40,41)(42,48)493 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 647);