Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C23 |
| direct product | C73×#12830 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C73 | #418 |
| commutator | C2×C8 | C22×C4×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C2×C73 | #1804 |
| prime radical | C2×#135×C73 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C73 ◁ C22×C4×C73 ◁ #405×C73 ◁ G | |
378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 137 points:
(1,2,6,13,8,11,7,30)(3,14,17,35,18,9,29,34)(4,19,22,51,20,52,31,41)(5,23,26,39,28,55,27,12)(10,25,36,53,38,59,37,21)(15,43,46,64,44,60,47,62)(16,48,50,40,24,56,58,33)(32,49,61,45,42,57,63,54)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,3,7,17,8,18,6,29)(2,9,30,34,11,14,13,35)(4,15,31,46,20,44,22,47)(5,24,27,58,28,16,26,50)(10,32,37,61,38,42,36,63)(12,40,55,56,39,33,23,48)(19,43,41,64,52,60,51,62)(21,54,59,49,53,45,25,57);(1,4)(2,10)(3,15)(5,25)(6,22)(7,31)(8,20)(9,32)(11,38)(12,41)(13,36)(14,42)(16,49)(17,46)(18,44)(19,23)(21,27)(24,57)(26,53)(28,59)(29,47)(30,37)(33,62)(34,61)(35,63)(39,51)(40,64)(43,48)(45,50)(52,55)(54,58)(56,60)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5818);