Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C23 |
| direct product | C73×#9177 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | #386 |
| commutator | C42 | C22×C4×C73 |
| Frattini, cosocle | C2×C42 | C23×C73 |
| prime radical | C22×D4×C73 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G | |
494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2,5,8,6,10,4,9)(3,11,13,15,14,16,12,7)(17,18)(19,27,20,23)(21,26)(22,25)(24,29,31,28)(30,32)(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);(1,3)(2,7,10,15)(4,12)(5,13)(6,14)(8,11,9,16)(17,19,21,20)(18,23)(22,29,30,28)(24,25)(26,27)(31,32);(2,8)(4,5)(7,11)(9,10)(12,13)(15,16)(17,20,21,19)(18,24,26,31)(22,28,30,29)(23,25,27,32)3139 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 2165);