Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C73 · Schur multiplier (H2) = C23×C4 |
| direct product | C73×#20787 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C73 | #117 |
| commutator | C2×C4 | C23×C4×C73 |
| Frattini, cosocle | C22×C4 | C24×C73 |
| socle | C2×C73 | #2283 |
| prime radical | C2×#41×C73 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C73 ◁ #405×C73 ◁ G | |
616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 2336 |
A minimal faithful permutation representation — 4 generators on 105 points:
(1,2)(3,12,15,8)(4,9)(5,10)(6,11)(7,14,20,28)(13,25,27,21)(16,22)(17,23)(18,24)(19,26,30,32)(29,31)(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);(1,3)(2,7,11,20)(4,13)(5,14)(6,15)(8,10,12,24)(9,19,23,30)(16,26)(17,27)(18,28)(21,22,25,31)(29,32);(1,3)(2,8)(4,13)(5,14)(6,15)(7,10)(9,21)(11,12)(16,26)(17,27)(18,28)(19,22)(20,24)(23,25)(29,32)(30,31);(1,4,5,16,6,17,18,29)(2,9,10,22,11,23,24,31)(3,13,14,26,15,27,28,32)(7,19,12,25,20,30,8,21)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 13775);