Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C2×C4 |
| direct product | C73×#13281 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C73 | C2×D8 |
| commutator | C2×C8 | C22×C4×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C73 ◁ C2×C42×C73 ◁ C2×C4×C8×C73 ◁ G | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 18688 |
A minimal faithful permutation representation — 3 generators on 117 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,25)(16,22)(17,32)(18,34)(19,24)(20,27)(21,30)(23,33)(26,39)(28,35)(29,41)(31,38)(36,43)(37,42)(40,44)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(2,4)(6,9)(10,11)(13,15,16,26,18,28,30,42,19,29,31,43,33,44,17,27)(14,20,32,40,23,36,38,41,24,37,21,35,34,39,22,25);(5,7,8,12)(6,10,9,11)(13,16,18,30,19,31,33,17)(14,21,23,22,24,32,34,38)(15,26,28,42,29,43,44,27)(20,35,36,25,37,40,39,41)5548 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6269);