Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C83 · Schur multiplier (H2) = C1 |
| direct product | C83×#7441 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C83 | #1353 |
| commutator | C2×M16 | C2×C4×C83 |
| Frattini, cosocle | #423 | C22×C83 |
| prime radical | C23×C83 | #104 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C83 ◁ C2×C4×C83 ◁ C22×C4×C83 ◁ #423×C83 ◁ G | |
136 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 2656 |
A minimal faithful permutation representation — 2 generators on 115 points:
(1,2)(3,11,13,23,15,28,14,31)(4,8,17,24)(5,19)(6,9)(7,12,21,29,22,30,27,16)(10,20)(18,26,32,25)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,3)(2,7,9,22)(4,16,17,29)(5,14)(6,15)(8,23)(10,21,19,27)(11,26)(12,18,30,32)(13,20)(24,31)(25,28)2075 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 429);