Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C22 |
| direct product | C79×#13269 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #391 |
| commutator | C22×C4 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1985 |
| prime radical | C23×C79 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×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:
(1,2,6,12,7,11,25,17,8,13,14,16,5,10,24,3)(4,18,22,30,20,27,32,31,23,28,29,15,21,9,26,19)(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);(1,3,14,2,7,12,24,11,8,17,6,13,5,16,25,10)(4,19,22,28,20,30,32,9,23,31,29,18,21,15,26,27);(1,4,5,21,8,23,7,20)(2,9,11,18,13,27,10,28)(3,15,12,19,17,30,16,31)(6,22,24,26,14,29,25,32)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6257);