Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C73 · Schur multiplier (H2) = C22×C4 |
| direct product | C73×#13196 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C73 | C2×D4 |
| commutator | C2×C4 | C22×C8×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C22×C73 | C8×D4 |
| prime radical | C2×#135×C73 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C73 ◁ C22×C16×C73 ◁ G | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 109 points:
(2,4)(5,6,11,17,12,16,27,33,9,15,19,14,10,28,8,25)(7,18,23,32,24,34,20,35,21,13,29,30,22,31,26,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(1,2,3,4)(5,7,19,29,12,24,8,26,9,21,11,23,10,22,27,20)(6,13,14,32,16,31,25,35,15,18,17,30,28,34,33,36);(5,8,10,19,9,27,12,11)(6,14,16,25,15,17,28,33)(7,20,22,23,21,26,24,29)(13,30,31,36,18,32,34,35)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6184);