Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#2180 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #442 |
| commutator | C8 | C22×C4×C17 |
| Frattini, cosocle | C2×C8 | C23×C17 |
| prime radical | C17×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×C4×C17 ◁ C17×#97 ◁ G | |
142 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 72 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)(20,30,32,41)(21,34)(22,26)(23,38)(24,29)(25,31,42,48)(27,35)(28,40)(33,45,39,47)(36,46,49,43)(37,44);(18,20)(19,25)(21,31)(22,36)(23,39)(24,32)(26,41)(27,45)(28,46)(29,42)(30,44)(33,40)(34,47)(35,48)(37,49)(38,43);(18,21,23,22,24,35,40,37)(19,26,28,34,29,44,38,27)(20,31,39,36,32,48,33,49)(25,41,46,47,42,30,43,45)391 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 912);