Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C83 · Schur multiplier (H2) = C2 |
| direct product | C83×#7361 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C83 | #354 |
| commutator | C42 | C2×C8×C83 |
| Frattini, cosocle | #423 | C22×C83 |
| socle | C2×C83 | #1319 |
| prime radical | C23×C83 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C83 ◁ C2×C4×C83 ◁ C22×C4×C83 ◁ #423×C83 ◁ G | |
184 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 1328 |
A minimal faithful permutation representation — 2 generators on 99 points:
(1,2,4,8,6,10,13,15,7,11,14,16,5,9,3,12)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99);(1,3)(2,8,11,16)(4,6)(5,14)(7,13)(9,12,10,15)2822 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 349);