Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#12656 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C79 | C2×D4 |
| commutator | C4 | C2×C4×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C2×C4⋊C8 |
| prime radical | C23×C79 | C2×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C79 ◁ C2×C4×C8×C79 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 points:
(1,2,4,3)(5,6)(7,16,19,27)(8,14)(9,13)(10,15)(11,18,25,20)(12,22)(17,32,34,31)(21,30)(23,29)(24,35,26,33)(28,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(5,7,8,17,9,20,21,35,10,19,22,34,23,18,36,33)(6,11,14,26,13,27,30,31,15,25,12,24,29,16,28,32);(1,3,4,2)(5,8,9,21,10,22,23,36)(6,12,13,28,15,14,29,30)(7,17,20,35,19,34,18,33)(11,24,27,32,25,26,16,31)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5644);