Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7316 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#7317 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1304 |
| commutator | C24 | C42×C79 |
| Frattini, cosocle | C23×Q8 | C22×C79 |
| prime radical | C24×C79 | C42 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C23×C79 ◁ C23×Q8×C79 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 2 generators on 111 points:
(1,2,4,8,6,10,17,25)(3,11,12,23,14,7,15,21)(5,18,19,24,20,9,16,26)(13,22,29,32,30,27,28,31)(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);(1,3,5,13,6,14,20,30)(2,7,9,22,10,11,18,27)(4,15,16,29,17,12,19,28)(8,23,24,32,25,21,26,31)2212 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 305);