Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7217 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#7217 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #340 |
| commutator | C2×C8 | C42×C79 |
| Frattini, cosocle | C2×C4×C8 | C22×C79 |
| prime radical | C23×C79 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 2 generators on 111 points:
(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(18,23)(19,27)(20,29)(21,28)(24,25)(26,32)(30,31)(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);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,18,20,24,21,25,19,23,22,26,27,31,28,30,29,32)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 205);