Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C73×#2242 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C73 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C73 |
| prime radical | C22×D8×C73 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C73 ◁ C2×C42×C73 ◁ C2×C4×C8×C73 ◁ G | |
610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(3,4)(5,8)(6,7)(9,10)(11,12)(13,25)(14,19)(15,33)(16,36)(17,37)(18,24)(20,29)(21,26)(22,35)(23,34)(27,39)(28,38)(30,42)(31,41)(32,40);(4,7)(8,10)(11,13,26,31,17,28,35,32,18,29,16,27,34,30,15,14)(12,19,33,42,23,39,36,20,24,40,22,38,37,41,21,25);(3,5,6,9)(4,8,7,10)(11,14,15,30,34,27,16,29,18,32,35,28,17,31,26,13)(12,20,21,39,37,42,22,19,24,25,36,41,23,38,33,40)5548 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 26684);