Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#17992 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C22×D4 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C42 | C24×C79 |
| socle | C22×C79 | C2×C4×D4 |
| prime radical | #466×C79 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ #431×C79 ◁ G | |
576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,5,8,6,9,14,3)(4,11,13,10,12,7,15,16)(17,18)(19,26)(20,22)(21,25)(23,27)(24,29)(28,31)(30,32)(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);(4,12)(7,11)(10,16)(13,15)(18,22)(20,29)(23,30)(24,25)(26,31)(28,32);(1,3,5,2,6,8,14,9)(4,10,13,7,12,16,15,11)(19,27)(23,26)(28,32)(30,31);(1,4)(2,7)(3,10)(5,13)(6,12)(8,16)(9,11)(14,15)(17,19,21,27)(18,23,25,26)(20,28,29,32)(22,30,24,31)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 10980);