Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C83 · Schur multiplier (H2) = C1 |
| direct product | C83×#7548 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C83 | #1424 |
| commutator | C32 | C2×C4×C83 |
| Frattini, cosocle | C2×C32 | C22×C83 |
| prime radical | C22×C83 | SD64 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C83 ◁ C2×C4×C83 ◁ C2×C8×C83 ◁ C2×C16×C83 ◁ C2×C32×C83 ◁ G | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 2 generators on 147 points:
(2,8,10,26)(3,14)(4,19)(5,20)(6,23)(9,30)(11,39,36,53)(12,43,37,63)(13,44,38,29)(15,22)(16,33)(17,35)(18,34)(21,48)(24,51)(25,52,41,27)(28,56,55,40)(31,58,59,54)(32,61)(42,64,62,60)(45,46)(47,57)(49,50)(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);(1,2,9,31,4,11,32,55,20,40,47,53,50,58,18,26,6,13,34,52,21,42,57,63,24,37,61,64,15,25,30,44,7,10,35,59,22,36,16,28,51,56,46,39,48,54,3,8,23,38,14,27,49,62,45,43,5,12,33,60,19,41,17,29)3818 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 536);