Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C83 · Schur multiplier (H2) = C24 |
| direct product | C83×#11337 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C83 | #127 |
| commutator | C22×C4 | C22×C4×C83 |
| Frattini, cosocle | C23×C4 | C23×C83 |
| socle | C22×C83 | #392 |
| prime radical | C24×C83 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C83 ◁ C23×C4×C83 ◁ G | |
686 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 686 conjugacy classes of subgroups — too many to draw. Only its 254 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2,3,4)(5,6)(7,16)(8,21)(9,26)(10,14)(11,15)(12,18)(13,24)(17,27)(19,29)(20,28)(22,30)(23,32)(25,31)(33,35)(34,36)(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);(5,7,9,18,11,20,10,19)(6,12,14,16,15,29,26,28)(8,22,23,33,25,34,24,17)(13,30,31,27,32,36,21,35);(5,8,9,24,11,25,10,23)(6,13,14,21,15,32,26,31)(7,17,19,22,20,33,18,34)(12,27,28,36,29,35,16,30)4316 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 4325);