Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C73 · Schur multiplier (H2) = C2×C4 |
| direct product | C73×#12881 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | #117 |
| commutator | C2×C4 | C22×C8×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C22×C73 | #432 |
| prime radical | C23×C73 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C2×#97×C73 ◁ G | |
298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 109 points:
(2,3)(5,6,9,16,10,13,25,29,11,14,24,30,8,15,22,31)(7,17,19,32,20,27,34,36,21,12,23,28,18,26,33,35)(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);(1,2,4,3)(5,7,8,18,11,21,10,20)(6,12,15,27,14,17,13,26)(9,23,22,34,24,19,25,33)(16,32,31,35,30,28,29,36);(1,3,4,2)(6,13,14,15)(9,24)(12,26,17,27)(16,31,30,29)(19,23)(22,25)(28,36,32,35)(33,34)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5869);