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×#2145 |
| 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 | |
224 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 224 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)(20,29,32,46)(21,25)(22,36)(23,28)(24,31,40,48)(26,34)(27,35)(30,45,33,42)(37,44,49,39)(38,41)(43,47);(18,20)(19,24)(21,30)(22,37)(23,32)(25,39)(26,42)(27,44)(28,40)(29,41)(31,47)(33,35)(34,48)(36,46)(38,49)(43,45);(18,21,23,35)(19,25,28,27)(20,30,32,33)(22,34,38,47)(24,39,40,44)(26,41,43,36)(29,45,46,42)(31,37,48,49)544 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 877);