Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C24 |
| direct product | C2×C79×#1395 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #99 |
| commutator | C2×C4 | C2×C42×C79 |
| Frattini, cosocle | C2×#41 | C23×C79 |
| socle | C22×C79 | #342 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C22×C4×C79 ◁ C22×#41×C79 ◁ G | |
766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 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);(3,4,7,12)(5,15,17,31)(6,13,20,29)(8,11,22,28)(9,14,23,30)(10,24,25,16)(18,27,32,34)(19,26,21,33);(3,5,8,18,9,19,6,16)(4,10,13,26,14,27,11,15)(7,21,22,24,23,17,20,32)(12,25,29,33,30,34,28,31)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5576);