Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C83 · Schur multiplier (H2) = C1 |
| direct product | C83×#7544 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C83 | D4 |
| commutator | C4 | C2×C32×C83 |
| Frattini, cosocle | C2×C32 | C22×C83 |
| socle | C2×C83 | C4⋊C32 |
| prime radical | C22×C83 | M64 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C83 ◁ C2×C32×C83 ◁ G | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 2 generators on 147 points:
(1,2,5,12,6,13,24,37,7,14,25,38,28,40,51,59,8,11,26,34,29,35,52,56,20,36,46,57,47,58,62,64,9,15,27,39,30,41,53,60,17,10,31,32,44,33,54,55,4,21,22,48,23,49,50,63,3,16,18,42,19,43,45,61)(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);(1,3,4,17,9,20,8,7)(2,10,11,16,15,14,21,36)(5,18,22,31,27,46,26,25)(6,19,23,44,30,47,29,28)(12,32,34,42,39,38,48,57)(13,33,35,43,41,40,49,58)(24,45,50,54,53,62,52,51)(37,55,56,61,60,59,63,64)9296 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 532);