Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C83 · Schur multiplier (H2) = C23×C4 |
| direct product | C83×#21140 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C83 | C22×D4 |
| commutator | C2×C4 | C23×C4×C83 |
| Frattini, cosocle | C22×C4 | C24×C83 |
| socle | C22×C83 | C2×C4×D4 |
| prime radical | C2×#135×C83 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C83 ◁ C2×#97×C83 ◁ G | |
764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 4 generators on 123 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,19,15,18,13,27)(11,16,23,30,21,31,22,37)(12,24,25,17,26,33,28,32)(20,34,36,29,35,39,38,40)(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);(2,5)(6,8)(9,11,14,23,15,21,13,22)(10,16,19,30,18,31,27,37)(12,20,25,36,26,35,28,38)(17,29,33,39,32,40,24,34);(9,12)(10,17)(11,20)(13,28)(14,25)(15,26)(16,29)(18,32)(19,33)(21,35)(22,38)(23,36)(24,27)(30,39)(31,40)(34,37);(1,3,4,7)(2,6,5,8)(10,18)(11,21)(17,32)(19,27)(20,35)(22,23)(24,33)(36,38)5312 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 14128);