Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#12668 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | #106 |
| commutator | C8 | C2×C42×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | C4×#106 |
| prime radical | C23×C79 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C22×C8×C79 ◁ C2×C4×C8×C79 ◁ G | |
308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,4,8,5,10,17,26,6,9,18,25,20,27,31,29)(3,11,14,28,15,21,30,7,16,22,13,24,19,32,12,23)(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);(1,3)(2,7,9,23)(4,12)(5,19)(6,16)(8,21,25,32)(10,28,27,24)(11,29,22,26)(13,17)(14,31)(15,20)(18,30);(1,4,5,17,6,18,20,31)(2,8,10,26,9,25,27,29)(3,12,19,13,16,30,15,14)(7,21,28,11,23,32,24,22)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5656);