Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C73 · Schur multiplier (H2) = C24×C4 |
| direct product | C73×#22281 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | C22×D4 |
| commutator | C2×C4 | C23×C4×C73 |
| Frattini, cosocle | C22×C4 | C24×C73 |
| socle | C22×C73 | C2×C4×D4 |
| prime radical | C73×#3150 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C2×C4×C8×C73 ◁ G | |
818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 113 points:
(1,2)(3,6)(4,7)(5,8)(9,10,14,18,15,19,13,28)(11,16,22,30,23,31,21,37)(12,24,27,32,25,38,26,17)(20,33,36,39,34,40,35,29)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(1,3,6,2)(4,7,5,8)(9,11,14,22,15,23,13,21)(10,16,18,30,19,31,28,37)(12,20,27,36,25,34,26,35)(17,29,24,33,32,39,38,40);(1,4,6,5)(2,7,3,8)(9,12,13,26,15,25,14,27)(10,17,18,24,19,32,28,38)(11,20,21,35,23,34,22,36)(16,29,30,33,31,39,37,40);(1,5,6,4)(2,7,3,8)(12,25)(17,32)(20,34)(24,38)(26,27)(29,39)(33,40)(35,36)3796 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 15269);