Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C26 |
| direct product | C2×C79×#2053 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C22×C79 | #394 |
| prime radical | C2×#575×C79 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C23×C4×C79 ◁ G | |
1366 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)(5,14)(6,19)(7,24)(8,12)(9,13)(10,16)(11,21)(15,25)(17,27)(18,26)(20,28)(22,29)(23,32)(30,33)(31,34);(3,5)(4,10)(6,20)(7,17)(8,16)(9,18)(11,28)(12,26)(13,27)(14,24)(15,22)(19,25)(21,31)(23,30)(29,33)(32,34);(3,6,7,21,9,23,8,22)(4,11,12,19,13,29,24,32)(5,15,16,30,18,31,17,20)(10,25,14,28,27,34,26,33)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 16522);