Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#7928 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C79 |
| Frattini, cosocle | C23×C4 | C23×C79 |
| socle | C22×C79 | C2×#95 |
| prime radical | C24×C79 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C22×C42×C79 ◁ G | |
594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,20,25,22,24,21,26)(19,23,27,31,29,30,28,32)(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)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,21,28,22,29,20,27)(18,23,26,32,24,30,25,31);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,24)(23,30)(25,26)(31,32)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 916);