Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C79 · Schur multiplier (H2) = C42 |
| direct product | C79×#12541 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C79 | C42 |
| commutator | C4 | C43×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C2×#97 |
| prime radical | C23×C79 | C2×M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C79 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 points:
(5,6,8,12,9,14,20,28,10,13,21,27,22,29,35,36)(7,15,17,31,18,24,33,26,16,30,19,34,32,11,23,25)(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);(5,7)(6,11,13,24)(8,19)(9,18)(10,16)(12,26,27,25)(14,15,29,30)(17,21)(20,23)(22,32)(28,34,36,31)(33,35);(1,2,3,4)(7,16)(11,24)(15,30)(17,19)(18,32)(23,33)(25,26)(31,34)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5529);