Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C22×C4 |
| direct product | C79×#13247 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1972 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C2×#509×C79 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C22×C4×C79 ◁ #405×C79 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,7,3)(4,18)(5,16,25,11)(6,10,13,15)(8,17,24,12)(9,26)(14,21)(19,20)(22,30)(23,31)(27,29)(28,32)(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);(1,3,13,16,8,17,5,15,7,2,6,11,24,12,25,10)(4,19,21,28,22,9,27,31,23,29,26,30,32,14,20,18);(1,4,6,21,8,22,25,27,7,23,13,26,24,32,5,20)(2,9,11,18,12,29,10,28,3,14,16,31,17,19,15,30)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6235);