Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C79 · Schur multiplier (H2) = C214 |
| direct product | C79×#63090 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | C23×D4 |
| commutator, Frattini, cosocle | C4 | C26×C79 |
| socle | C2×C79 | C24×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×#41×C79 ◁ G | |
10836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 6 generators on 111 points:
(1,2)(3,12)(4,8)(5,10)(6,9)(7,15)(11,19)(13,26)(14,21)(16,25)(17,22)(18,23)(20,29)(24,30)(27,32)(28,31)(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);(1,3)(2,7)(4,13)(5,14)(6,16)(8,20)(9,21)(10,12)(11,25)(15,19)(17,27)(18,28)(22,31)(23,26)(24,32)(29,30);(1,4)(2,8)(3,13)(5,18)(6,17)(7,20)(9,22)(10,23)(11,24)(12,26)(14,28)(15,29)(16,27)(19,30)(21,31)(25,32);(4,17)(8,22)(13,27)(18,30)(20,31)(23,24)(26,32)(28,29);(2,9)(7,21)(8,22)(10,11)(12,25)(20,31)(23,24)(26,32);(1,5,6,19)(2,10,9,11)(3,14,16,15)(4,18,17,30)(7,12,21,25)(8,23,22,24)(13,28,27,29)(20,26,31,32)6478 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 56078);