Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#33371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | C22×#99 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C2×Q8 | C24×C79 |
| prime radical | C79×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C2×Q8×C79 ◁ C2×H(5,𝔽2)×C79 ◁ G | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,5,9,6,11,21,27)(3,12,16,30,17,28,14,29)(4,8,19,26,20,10,18,25)(7,22,23,13,24,31,32,15)(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)(6,17)(8,12)(9,23)(10,28)(11,24)(15,20)(16,21)(18,31)(19,22)(25,29)(26,30)(27,32);(1,4,6,20)(2,8,11,10)(3,13,17,15)(5,19,21,18)(7,12,24,28)(9,26,27,25)(14,22,16,31)(23,30,32,29);(1,5,6,21)(2,9,11,27)(3,14,17,16)(4,18,20,19)(7,23,24,32)(8,25,10,26)(12,29,28,30)(13,31,15,22)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26359);