Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C83 · Schur multiplier (H2) = C24 |
| direct product | C83×#8415 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C83 | #379 |
| commutator | C42 | C22×C4×C83 |
| Frattini, cosocle | C2×C42 | C23×C83 |
| prime radical | C83×#3515 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G | |
704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 3 generators on 123 points:
(2,6,7,8)(9,10)(11,19,22,15)(12,23,25,32)(13,18)(14,17)(16,24,30,26)(20,35)(21,34,39,29)(27,31)(28,36)(33,38)(37,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,28)(17,19)(18,29)(23,40)(24,36)(25,37)(26,38)(27,39)(30,33)(31,34)(32,35);(1,3,5,4)(2,6,7,8)(9,12,14,25)(10,16,17,30)(11,20,22,37)(13,26,27,24)(15,28,19,33)(18,32,31,23)(21,38,39,36)(29,35,34,40)3569 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 1403);