Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#2170 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C17 | D4 |
| commutator | C4 | C22×C8×C17 |
| Frattini, cosocle | C2×C8 | C23×C17 |
| socle | C2×C17 | C8×D4 |
| prime radical | C17×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C17 ◁ C2×C16×C17 ◁ G | |
146 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 92 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)(18,19,23,27,22,28,37,45,24,29,39,46,38,26,42,44)(20,30,32,35,33,41,47,36,34,43,48,49,31,25,40,21);(18,20)(19,25)(21,27)(22,33)(23,32)(24,34)(26,43)(28,30)(29,41)(31,38)(35,45)(36,46)(37,47)(39,48)(40,42)(44,49);(18,21,23,30,22,35,37,41,24,36,39,43,38,49,42,25)(19,20,27,32,28,33,45,47,29,34,46,48,26,31,44,40)952 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 902);