Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C83 · Schur multiplier (H2) = C25 |
| direct product | C83×#17525 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C83 | C22×D4 |
| commutator | C2×C4 | C23×C4×C83 |
| Frattini, cosocle | C42 | C24×C83 |
| socle | C22×C83 | C22×C22⋊C4 |
| prime radical | C83×#3482 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C83 ◁ C4×#41×C83 ◁ G | |
934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 10624 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2,6,9)(3,10,12,15)(4,8,5,13)(7,14,16,11)(17,18,21,26,22,24,20,25)(19,27,29,23,30,32,28,31)(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);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19,22,30)(18,23,24,31)(20,28,21,29)(25,27,26,32);(1,2)(3,7)(4,13)(5,8)(6,9)(10,11)(12,16)(14,15)(18,24)(23,31)(25,26)(27,32);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,21)(18,25,24,26)(19,28,30,29)(23,27,31,32)4814 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 10513);