Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#9380 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #386 |
| commutator | C42 | C22×C4×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| prime radical | C23×C79 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
424 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)(3,11,4,7)(5,10)(6,9)(8,13,15,12)(14,16)(17,18,21,26,22,27,20,25)(19,28,30,23,31,32,29,24)(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);(1,3,5,4)(2,7)(6,13,14,12)(8,9)(10,11)(15,16)(17,19)(18,23)(20,30)(21,29)(22,31)(24,27)(25,32)(26,28);(1,4,5,3)(2,8,10,15)(6,12,14,13)(7,9,11,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,27)(25,32)(26,28)2923 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 2368);