Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C83 · Schur multiplier (H2) = C22×C42 |
| direct product | C83×#10866 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C83 | #117 |
| commutator | C2×C4 | C2×C42×C83 |
| Frattini, cosocle | C23×C4 | C23×C83 |
| socle | C22×C83 | #380 |
| prime radical | C24×C83 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C83 ◁ C22×M16×C83 ◁ G | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 3 generators on 119 points:
(5,6)(7,15,18,11)(8,19)(9,22)(10,13)(12,20)(14,23)(16,29,31,33)(17,25,32,28)(21,34)(24,30,27,36)(26,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,24)(13,15)(14,25)(19,33)(20,30)(21,31)(22,28)(23,32)(26,27)(29,34)(35,36);(5,8,9,20,10,21,23,35)(6,12,14,19,13,26,22,34)(7,16,17,30,18,31,32,36)(11,24,25,33,15,27,28,29)5312 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 3854);