Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#33656 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C2×C79 | C2×#481 |
| prime radical | C79×#3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C22×M16×C79 ◁ G | |
556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2)(3,12)(4,8)(5,19)(6,11)(7,15)(9,17)(10,18)(13,29)(14,28)(16,26)(20,24)(21,22)(23,30)(25,32)(27,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(5,20)(9,25)(12,28)(15,30)(17,32)(19,24)(21,31)(26,29);(1,3,4,13,6,14,18,16)(2,7,10,27,11,23,8,22)(5,21,17,30,20,31,32,15)(9,26,19,28,25,29,24,12);(1,4,6,18)(2,8,11,10)(3,13,14,16)(5,17,20,32)(7,22,23,27)(9,24,25,19)(12,29,28,26)(15,21,30,31)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26644);