Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C83 · Schur multiplier (H2) = C2×C4 |
| direct product | C83×#12904 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C83 | C2×#106 |
| commutator | C2×C8 | C22×C4×C83 |
| Frattini, cosocle | C22×C8 | C23×C83 |
| prime radical | C23×C83 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C83 ◁ C2×C42×C83 ◁ C2×C4×C8×C83 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 3 generators on 119 points:
(2,4)(5,6)(7,15,17,27)(8,19)(9,12)(10,23)(11,16,24,29)(13,21)(14,22)(18,32,31,28)(20,33)(25,36,35,30)(26,34)(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);(5,7,8,16,10,18,21,30,9,17,20,29,22,31,34,36)(6,11,13,15,14,25,19,28,12,24,26,27,23,35,33,32);(1,2,3,4)(6,12)(11,24)(13,26)(14,23)(15,27)(19,33)(25,35)(28,32)4316 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 5892);