Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C2 |
| direct product | C73×#8803 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | #383 |
| commutator | C42 | C22×C4×C73 |
| Frattini, cosocle | C2×C42 | C23×C73 |
| prime radical | C23×C73 | #126 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G | |
216 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)(3,11,14,15)(4,7)(5,9)(6,10)(8,13,16,12)(17,18)(19,27,29,32)(20,24,30,23)(21,25)(22,26)(28,31)(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)(8,15,16,11)(17,19,28,30,22,29,21,20)(18,23,25,27,26,24,31,32);(1,3,4,12,6,14,5,13)(2,8,9,11,10,16,7,15)(17,20,21,29,22,30,28,19)(18,24,25,32,26,23,31,27)2920 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 1791);