Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#2148 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #102 |
| commutator | C8 | C22×C4×C17 |
| Frattini, cosocle | C2×C8 | C23×C17 |
| socle | C2×C17 | C2×#102 |
| prime radical | C2×C17×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C22×C8×C17 ◁ G | |
220 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 100 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)(18,19,21,25,23,28,35,27)(20,29,32,43,33,46,30,45)(22,36,34,26,38,42,47,44)(24,37,40,31,41,48,39,49);(18,20)(19,24)(21,30)(22,37)(23,33)(25,39)(26,43)(27,40)(28,41)(29,42)(31,47)(32,35)(34,49)(36,46)(38,48)(44,45);(18,21,23,35)(19,25,28,27)(20,30,33,32)(22,34,38,47)(24,39,41,40)(26,42,44,36)(29,45,46,43)(31,37,49,48)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 880);