Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#33494 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C79 | C24 |
| commutator | C22 | C23×C8×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C22×C79 | C23×C8 |
| prime radical | #532×C79 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C79 ◁ G | |
556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,17,13,18,22,30,14,15,23,26,24,27,34,37)(11,16,20,28,21,29,33,38,19,25,31,35,32,36,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(1,3)(2,5)(4,8)(6,7)(11,19)(16,25)(20,31)(21,32)(28,35)(29,36)(33,39)(38,40);(3,8)(5,7)(10,15)(11,19)(17,26)(18,27)(20,31)(21,32)(30,37)(33,39);(2,6)(5,7)(9,11)(10,16)(12,20)(13,21)(14,19)(15,25)(17,28)(18,29)(22,33)(23,31)(24,32)(26,35)(27,36)(30,38)(34,39)(37,40)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26482);