Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12952 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#12953 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1864 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C22×D8×C79 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C22×C4×C79 ◁ #405×C79 ◁ G | |
416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,6,13,8,14,7,25)(3,15,17,28,19,31,18,29)(4,10,21,12,5,23,22,11)(9,20,26,16,27,24,30,32)(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);(1,3)(2,9,14,27)(4,20)(5,24)(6,18)(7,17)(8,19)(10,15,23,31)(11,28,12,29)(13,30,25,26)(16,22)(21,32);(2,10)(4,21)(5,22)(6,7)(9,15)(11,13)(12,25)(14,23)(16,24)(17,18)(20,32)(26,29)(27,31)(28,30)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5941);