Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22×C4 |
| direct product | C17×#1976 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #121 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| socle | C2×C17 | #390 |
| prime radical | C2×C17×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ G | |
284 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 284 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)(20,24,29,38)(21,31)(22,34)(23,26)(25,32)(27,35)(28,40,41,49)(30,44,43,37)(33,45)(36,47,48,42)(39,46);(18,20)(19,24,26,38)(21,28)(22,30)(23,29)(25,36,39,48)(27,37,34,44)(31,40,45,49)(32,42)(33,41)(35,43)(46,47);(18,21,22,32,23,33,35,46)(19,25,27,31,26,39,34,45)(20,28,30,42,29,41,43,47)(24,36,37,40,38,48,44,49)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 708);