Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C22×C4 |
| direct product | C79×#13259 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #459 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1982 |
| prime radical | C79×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C2×C4×C8×C79 ◁ G | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,4,12,6,13,21,9,7,14,22,28,24,11,5,10)(3,15,18,27,19,29,16,30,20,31,23,8,17,25,32,26)(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);(1,3,6,17,7,20,24,19)(2,8,13,30,14,27,11,26)(4,16,21,18,22,32,5,23)(9,25,28,31,10,29,12,15);(1,4,6,21,7,22,24,5)(2,9,11,12,14,10,13,28)(3,16,17,18,20,32,19,23)(8,25,26,15,27,29,30,31)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6247);