Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C73 · Schur multiplier (H2) = C2×C4 |
| direct product | C73×#8246 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C73 |
| Frattini, cosocle | C2×C42 | C23×C73 |
| socle | C22×C73 | C4×C22⋊C4 |
| prime radical | C23×C73 | #117 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C2×C4×C8×C73 ◁ G | |
376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2,5,9,6,10,4,7)(3,8,12,15,13,16,11,14)(17,18,20,23)(19,25,26,22)(21,29,28,24)(27,30,32,31)(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);(2,7,10,9)(8,14,16,15)(17,19,21,27,20,26,28,32)(18,22,24,30,23,25,29,31);(1,3,5,12,6,13,4,11)(2,8,9,15,10,16,7,14)(19,26)(22,25)(27,32)(30,31)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 1234);