Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C83 · Schur multiplier (H2) = C22×C4 |
| direct product | C83×#7864 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C83 | C2×C42 |
| commutator | C2×C4 | C2×C42×C83 |
| Frattini, cosocle | C2×C42 | C23×C83 |
| socle | C22×C83 | C2×#95 |
| prime radical | C24×C83 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C83 ◁ G | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 3 generators on 115 points:
(1,2)(3,10,12,16)(4,8)(5,14)(6,9)(7,11,15,13)(17,18,21,25,22,26,20,24)(19,23,27,30,28,31,29,32)(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);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16)(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,30)(26,31);(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,30,31,32)5312 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 852);