Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#1515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| prime radical | C23×C17 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C2×C42×C17 ◁ G | |
208 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 88 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)(19,24,26,38)(20,29)(21,33)(25,40,41,49)(27,39,28,34)(31,46)(32,45)(35,36)(37,42,48,43)(44,47);(18,19,22,27)(20,25,32,42)(21,34,33,38)(23,28,30,26)(24,36,39,35)(29,43,45,41)(31,40,47,48)(37,46,49,44);(18,20,30,45)(19,25,26,41)(21,31,36,46)(22,32,23,29)(24,37,38,48)(27,42,28,43)(33,47,35,44)(34,40,39,49)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 247);