Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C83 · Schur multiplier (H2) = C1 |
| direct product | C83×#7547 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C83 | D32 |
| commutator | C32 | C2×C4×C83 |
| Frattini, cosocle | C2×C32 | C22×C83 |
| socle | C2×C83 | #1423 |
| prime radical | C22×C83 | Q64 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C83 ◁ C2×C4×C83 ◁ C2×C8×C83 ◁ C2×C16×C83 ◁ C2×C32×C83 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 211 |
| transitive degree | μt(G) = 10624 |
A minimal faithful permutation representation — 2 generators on 211 points:
(1,2,5,12,9,15,8,38)(3,16,19,53,23,56,22,74)(4,24,26,75,29,78,28,84)(6,32,30,85,35,87,34,88)(7,36,31,51,37,90,33,14)(10,39,40,65,43,64,91,18)(11,44,45,61,47,60,92,17)(13,49,48,80,50,79,86,25)(20,68,66,117,71,119,70,120)(21,72,67,107,73,121,69,55)(27,82,81,124,83,125,58,77)(41,97,96,127,98,128,103,109)(42,63,89,95,99,116,126,108)(46,59,76,100,101,113,123,112)(52,94,102,111,104,110,122,57)(54,93,105,115,106,114,118,62)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211);(1,3,17,57,6,20,58,108,33,69,112,128,79,114,28,64,8,22,60,110,34,70,83,116,37,73,113,127,80,115,29,65,9,23,61,111,35,71,81,95,31,67,100,97,49,93,26,39,5,19,44,94,30,66,27,63,7,21,59,109,25,62,4,18)(2,10,24,54,13,41,76,72,36,89,82,117,85,102,45,53,12,40,75,105,48,96,101,107,51,99,124,119,87,104,47,56,15,43,78,106,50,98,123,121,90,126,125,120,88,122,92,74,38,91,84,118,86,103,46,55,14,42,77,68,32,52,11,16)5810 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 535);