Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#1881 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #120 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| socle | C2×C17 | #386 |
| prime radical | C17×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ G | |
310 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 310 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)(20,29,32,40)(21,33)(22,27)(23,37)(24,28)(25,31,41,36)(26,34)(30,45,48,39)(35,42)(38,43)(44,46,49,47);(18,20,22,36,24,32,23,31)(19,25,27,40,28,41,37,29)(21,30,34,47,35,48,38,46)(26,39,42,49,43,45,33,44);(18,21)(19,26)(20,30)(22,34)(23,38)(24,35)(25,39)(27,42)(28,43)(29,44)(31,46)(32,48)(33,37)(36,47)(40,49)(41,45)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 613);