Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#17409 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C2×#121×C79 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C4×Q8×C79 ◁ G | |
912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18,19,23,21,22,25,30)(20,27,26,32,29,24,28,31)(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)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(18,22)(19,25)(24,27)(26,28);(3,9)(6,12)(10,15)(13,16)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31);(2,7)(6,12)(8,14)(13,16)(17,20,21,29)(18,24,22,27)(19,26,25,28)(23,31,30,32)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 10397);