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×#13191 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C73 | C2×D4 |
| commutator | C2×C4 | C22×C8×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C22×C73 | C2×C22⋊C8 |
| prime radical | #532×C73 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C73 ◁ C2×C4×C8×C73 ◁ G | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 3 generators on 109 points:
(2,4)(5,6,9,14,10,13,23,26,11,15,24,29,8,22,17,32)(7,16,19,30,20,28,33,35,21,31,34,36,18,12,25,27)(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);(5,7,17,25,8,18,24,34,11,21,23,33,10,20,9,19)(6,12,26,30,22,31,14,27,15,28,32,36,13,16,29,35);(1,2,3,4)6424 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6179);