Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#21477 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | #120 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C2×C79 | C4×#120 |
| prime radical | C79×#3018 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C22×C8×C79 ◁ G | |
1290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,6,13,7,12,24,11)(3,14,18,27,19,28,17,8)(4,20,22,5,10,9,21,23)(15,29,31,32,26,16,30,25)(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)(2,8)(4,15)(5,16)(6,18)(7,19)(9,25)(10,26)(11,28)(12,27)(13,14)(17,24)(20,32)(21,30)(22,31)(23,29);(1,4)(2,9)(3,15)(5,11)(6,22)(7,10)(8,25)(12,20)(13,23)(14,29)(16,28)(17,30)(18,31)(19,26)(21,24)(27,32);(1,5,6,9,7,23,24,20)(2,10,13,21,12,4,11,22)(3,16,18,25,19,29,17,32)(8,26,14,30,27,15,28,31)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 14465);