Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C83 · Schur multiplier (H2) = C23 |
| direct product | C83×#12829 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C83 | #418 |
| commutator | C2×C8 | C22×C4×C83 |
| Frattini, cosocle | C22×C8 | C23×C83 |
| socle | C2×C83 | #1804 |
| prime radical | C2×#135×C83 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C83 ◁ C22×C4×C83 ◁ #405×C83 ◁ G | |
378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 3 generators on 147 points:
(1,2)(3,13,16,8)(4,18)(5,22)(6,28)(7,10)(9,20)(11,26)(12,25)(14,41,42,60)(15,45,23,56)(17,32,48,40)(19,51)(21,54)(24,38)(27,55)(29,37)(30,43,39,53)(31,49,36,58)(33,59)(34,52)(35,46)(44,62,50,64)(47,61,57,63)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(1,3,6,17,7,16,25,48)(2,8,28,40,10,13,12,32)(4,14,29,50,19,42,21,44)(5,23,26,58,27,15,46,49)(9,30,33,61,34,39,38,63)(11,36,55,56,35,31,22,45)(18,41,37,64,51,60,54,62)(20,53,59,47,52,43,24,57);(1,4)(2,9)(3,14)(5,24)(6,29)(7,19)(8,30)(10,34)(11,37)(12,38)(13,39)(15,47)(16,42)(17,50)(18,22)(20,26)(21,25)(23,57)(27,59)(28,33)(31,62)(32,63)(35,54)(36,64)(40,61)(41,45)(43,49)(44,48)(46,52)(51,55)(53,58)(56,60)3818 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 5817);