Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C26 |
| direct product | C73×#11029 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C73 | #121 |
| commutator | C22×C4 | C22×C4×C73 |
| Frattini, cosocle | C23×C4 | C23×C73 |
| prime radical | C24×C73 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C73 ◁ C23×C4×C73 ◁ G | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 3 generators on 113 points:
(1,2)(3,4)(9,10,14,20,15,18,13,19)(11,21,24,32,25,16,23,33)(12,26,29,35,30,17,28,36)(22,31,38,40,39,37,27,34)(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);(2,4)(5,6)(7,8)(9,11,15,25)(10,16,18,21)(12,27,30,38)(13,24,14,23)(17,34,26,40)(19,33,20,32)(22,29,39,28)(31,36,37,35);(6,8)(9,12,15,30)(10,17,18,26)(11,22,25,39)(13,29,14,28)(16,31,21,37)(19,36,20,35)(23,38,24,27)(32,40,33,34)3796 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 4017);