Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#17965 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C79 | C2×D4 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C42 | C24×C79 |
| socle | C22×C79 | C2×C4×D4 |
| prime radical | #121×C79 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C79 ◁ C2×C4×C8×C79 ◁ G | |
594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,4,7,5,8,12,15)(3,9,11,16,10,6,13,14)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31)(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);(3,10)(6,9)(11,13)(14,16)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,2,4,7,5,8,12,15)(3,6,11,14,10,9,13,16);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,13)(15,16)(18,23)(22,31)(24,28)(25,32)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 10953);