Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C83 · Schur multiplier (H2) = C22 |
| direct product | C83×#12063 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C83 | C2×D4 |
| commutator | C2×C4 | C22×C8×C83 |
| Frattini, cosocle | C4×C8 | C23×C83 |
| socle | C22×C83 | C8×D4 |
| prime radical | C2×D8×C83 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C83 ◁ C4×C16×C83 ◁ G | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 10624 |
A minimal faithful permutation representation — 3 generators on 107 points:
(1,2,4,6)(3,5,8,7)(9,10,12,16,13,11,18,19,14,17,21,24,22,20,15,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(2,5)(3,8)(6,7);(2,6)(5,7)(9,11,12,19,13,17,18,24,14,20,21,23,22,10,15,16)6308 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 5051);