Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C2×C79×#2135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#99 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C2×Q8 | C24×C79 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ C22×#41×C79 ◁ G | |
1198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 113 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(3,4,8,12,9,13,23,30)(5,14,17,31,18,27,22,33)(6,11,19,28,7,20,21,29)(10,24,25,15,26,32,34,16);(3,5,9,18)(4,10,13,26)(6,16,7,15)(8,22,23,17)(11,27,20,14)(12,25,30,34)(19,32,21,24)(28,33,29,31);(3,6,9,7)(4,11,13,20)(5,15,18,16)(8,19,23,21)(10,14,26,27)(12,28,30,29)(17,32,22,24)(25,31,34,33)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26356);