Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C22 |
| direct product | C79×#13180 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #459 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C23×C79 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ C2×C4×C8×C79 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 points:
(1,2,4,3)(5,6)(7,18)(8,23)(9,16)(10,15)(11,17)(12,20)(13,24)(14,25)(19,33)(21,28)(22,34)(26,31)(27,32)(29,36)(30,35)(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);(1,3,4,2)(5,7)(6,12)(8,19)(9,21)(10,27)(11,22)(13,28)(14,32)(15,29)(16,33)(17,30)(18,23)(20,25)(24,35)(26,36)(31,34);(5,8,9,24,11,26,10,25)(6,13,15,23,17,14,16,31)(7,19,21,35,22,36,27,20)(12,28,29,18,30,32,33,34)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6168);