Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12715 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#12717 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1781 |
| commutator | C22×C4 | C22×C4×C79 |
| Frattini, cosocle | C2×#41 | C23×C79 |
| prime radical | C79×#3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C2×C4×C79 ◁ C2×2−1+4×C79 ◁ G | |
944 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)(3,10)(4,15,18,27)(5,9,20,26)(6,11)(7,13,22,19)(8,17,24,12)(14,25)(16,21)(23,28)(29,30)(31,32)(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);(1,3,6,14)(2,7,11,22)(4,12,18,17)(5,19,20,13)(8,21,24,32)(9,25,26,10)(15,29,27,23)(16,30,31,28);(1,4)(2,8)(3,12)(5,16)(6,18)(7,21)(9,23)(10,27)(11,24)(13,28)(14,17)(15,25)(19,30)(20,31)(22,32)(26,29)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5705);