Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#2132 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #99 |
| commutator | C2×C4 | C22×C4×C17 |
| Frattini, cosocle | C2×Q8 | C23×C17 |
| socle | C2×C17 | C2×#99 |
| prime radical | C17×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C2×C4×C17 ◁ C2×#41×C17 ◁ G | |
250 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 250 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) = 33 |
| transitive degree | μt(G) = 272 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,27,23,28,20,25)(21,26,29,33,30,24,31,32);(19,24)(21,30)(25,32)(26,28)(27,33)(29,31);(18,20,23,22)(19,25,28,27)(21,29,30,31)(24,32,26,33)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 864);