Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#33569 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C79 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C23×C79 ◁ C22×Q8×C79 ◁ G | |
1082 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)(3,12)(4,17)(5,20)(6,11)(7,15)(8,18)(9,19)(10,23)(13,25)(14,28)(16,24)(21,29)(22,27)(26,30)(31,32)(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)(5,21)(6,14)(8,24)(9,28)(10,26)(11,25)(12,17)(15,19)(16,23)(18,31)(20,32)(22,30)(27,29);(1,4,6,19)(2,8,11,27)(3,13,14,15)(5,22,23,18)(7,24,25,29)(9,20,17,10)(12,26,28,32)(16,31,21,30);(3,14)(7,25)(12,28)(13,15)(16,21)(24,29)(26,32)(30,31)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26557);