Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#22461 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | D42 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C2×C79 | D4×C22⋊C4 |
| prime radical | C79×#3584 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ G | |
1316 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)(3,12)(4,18,21,26)(5,10)(6,11)(7,15)(8,20,25,19)(9,22)(13,23,30,32)(14,17)(16,24)(27,29,28,31)(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)(2,7)(4,13)(5,16)(6,17)(8,23)(9,14)(10,12)(11,24)(15,22)(18,28)(19,31)(20,29)(21,30)(25,32)(26,27);(1,4)(2,8)(3,13)(5,19)(6,21)(7,23)(9,18)(10,26)(11,25)(12,27)(14,28)(15,29)(16,31)(17,30)(20,22)(24,32);(1,2)(3,14)(4,8)(5,10)(6,11)(7,16)(9,22)(12,17)(13,28)(15,24)(18,20)(19,26)(21,25)(23,31)(27,30)(29,32)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 15449);