Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#12887 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#106 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C23×C79 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,21,25,36)(12,26,28,37)(13,29)(14,31)(15,20)(16,24,34,39)(17,27,35,40)(18,30)(19,32)(22,38)(23,33)(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);(1,3,4,7)(2,5,6,8)(9,11,22,39,14,12,23,40,15,25,13,24,32,28,30,27)(10,16,33,36,19,35,38,26,20,34,18,21,31,17,29,37);(2,6)(5,8)(9,12,13,27,14,25,30,39,15,28,22,40,32,11,23,24)(10,17,18,26,19,16,29,21,20,35,33,37,31,34,38,36)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5875);