Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C22 |
| direct product | C4×SD64×C79 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | D16 |
| commutator | C16 | C22×C4×C79 |
| Frattini, cosocle | C2×C16 | C23×C79 |
| socle | C22×C79 | C2×D16 |
| prime radical | C2×D16×C79 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C42×C79 ◁ C4×C8×C79 ◁ C4×C16×C79 ◁ G | |
202 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 10112 |
A minimal faithful permutation representation — 3 generators on 115 points:
(1,2,3,4)(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);(5,6,10,15)(7,11,19,26)(8,20,22,36)(9,23,21,14)(12,27,29,16)(13,30,31,17)(18,32,33,25)(24,35,28,34);(6,11)(7,16)(8,17)(9,21)(12,13)(14,32)(15,26)(18,35)(19,27)(20,24)(22,30)(23,25)(28,36)(29,31)(33,34)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6638);