Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#1395 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #99 |
| commutator | C2×C4 | C42×C17 |
| Frattini, cosocle | C2×#41 | C22×C17 |
| socle | C2×C17 | #342 |
| prime radical | C2×#41×C17 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C2×C4×C17 ◁ C2×#41×C17 ◁ G | |
162 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 2 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,28,30,40)(21,32,33,25)(22,35)(23,27)(24,29)(26,36)(31,45,44,41)(34,39,47,46)(37,49)(38,43)(42,48);(18,20,22,31,23,30,36,44)(19,24,26,37,27,38,35,48)(21,29,34,42,33,43,47,49)(25,28,39,41,32,40,46,45)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 127);