Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C73 · Schur multiplier (H2) = C2×C4 |
| direct product | C4×#368×C73 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C73 | D8 |
| commutator | C8 | C2×C42×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C22×C73 | C2×#107 |
| prime radical | C23×C73 | C2×Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C73 ◁ C2×C42×C73 ◁ C2×C4×C8×C73 ◁ G | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2,3,4)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(5,6,9,14,11,15,10,25)(7,16,19,31,21,32,20,35)(8,22,23,29,24,36,17,13)(12,26,27,34,28,33,30,18);(5,7,17,33,10,20,24,34,11,21,23,26,9,19,8,18)(6,12,22,31,14,27,29,32,15,28,36,35,25,30,13,16)6424 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5663);