Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#1535 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C42 | C23×C17 |
| prime radical | C17×#121 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C2×C42×C17 ◁ G | |
200 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 1088 |
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)(18,19,22,27,23,26,21,25)(20,28,29,33,30,24,31,32)(34,35)(36,42,44,49)(37,41)(38,40)(39,43,47,45)(46,48);(18,20)(19,24)(21,29)(22,31)(23,30)(25,32)(26,28)(27,33)(34,36)(35,39)(37,45)(38,44)(40,47)(41,49)(42,48)(43,46);(18,21,23,22)(19,25,26,27)(20,29,30,31)(24,32,28,33)(35,40)(39,47)(41,48)(42,49)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 267);