Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61109 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C29 |
| direct product | C79×#61111 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | C2×D42 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C22×C4×C79 ◁ G | |
2600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 5 generators on 111 points:
(2,8)(3,13)(5,9)(6,23)(10,25)(11,12)(14,28)(15,16)(17,27)(18,26)(19,20)(30,31)(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);(1,2)(3,9)(4,17)(5,16)(6,11)(7,12)(8,23)(10,19)(13,24)(14,29)(15,22)(18,31)(20,27)(21,25)(26,28)(30,32);(1,3,7,16)(2,9,12,5)(4,14,21,30)(6,15,23,13)(8,24,11,22)(10,26,27,18)(17,29,25,32)(19,28,20,31);(1,4)(2,10)(3,14)(5,18)(6,20)(7,21)(8,25)(9,26)(11,17)(12,27)(13,28)(15,31)(16,30)(19,23)(22,29)(24,32);(1,5)(2,3)(4,18)(6,22)(7,9)(8,15)(10,14)(11,13)(12,16)(17,28)(19,32)(20,29)(21,26)(23,24)(25,31)(27,30)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54099);