Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21776 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#21781 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #2308 |
| commutator | C23 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C79×#3487 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C22×C4×C79 ◁ G | |
1258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,4,8,6,10,17,25)(3,11,15,23,14,7,12,21)(5,18,19,24,20,9,16,26)(13,22,28,32,29,27,30,31)(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)(2,7)(4,12)(5,13)(6,14)(8,23)(9,22)(10,11)(15,17)(16,28)(18,27)(19,30)(20,29)(21,25)(24,32)(26,31);(1,4,6,17)(2,8,10,25)(3,12,14,15)(5,16,20,19)(7,21,11,23)(9,24,18,26)(13,28,29,30)(22,31,27,32);(1,5,6,20)(2,9,10,18)(3,13,14,29)(4,16,17,19)(7,22,11,27)(8,24,25,26)(12,28,15,30)(21,31,23,32)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 14769);