Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16×C83 · Schur multiplier (H2) = C2 |
| direct product | C83×#7336 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C83 | #99 |
| commutator | C23 | C2×C16×C83 |
| Frattini, cosocle | C2×M32 | C22×C83 |
| socle | C2×C83 | #1314 |
| prime radical | C24×C83 | C16 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C83 ◁ C2×C8×C83 ◁ C2×M32×C83 ◁ G | |
170 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 2656 |
A minimal faithful permutation representation — 2 generators on 115 points:
(1,2,3,8,4,9,12,20,5,10,13,21,14,22,25,30,6,7,11,17,15,18,23,27,16,19,24,28,26,29,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(2,7)(9,18)(10,19)(22,29)4316 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 324);