Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#1912 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #386 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| prime radical | C2×#41×C17 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C4×C17 ◁ G | |
314 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 314 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)(18,19,23,27,24,28,22,38)(20,29,33,40,32,44,31,25)(21,34,36,42,37,26,35,41)(30,43,47,39,46,49,45,48);(18,20)(19,25,28,40)(21,30)(22,31)(23,33)(24,32)(26,39,34,48)(27,29,38,44)(35,45)(36,47)(37,46)(41,49,42,43);(18,21,24,37)(19,26,28,34)(20,30,32,46)(22,36,23,35)(25,39,40,48)(27,42,38,41)(29,43,44,49)(31,47,33,45)493 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 644);