Finite Group DatabasebetaFinite Group Databasebeta| order | 17152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C67 · Schur multiplier (H2) = C25 |
| direct product | C67×#33608 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C67 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C67 |
| prime radical | C2×#41×C67 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C67 ◁ C2×#41×C67 ◁ C2×#131×C67 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 8576 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2)(3,4)(5,6,11,16)(7,17,22,42)(8,13,23,34)(9,26,28,54)(10,29,27,15)(12,30,33,20)(14,37,38,18)(19,39,45,61)(21,49,48,41)(24,52,50,67)(25,53,51,36)(31,55,57,46)(32,43,40,56)(35,60,59,44)(47,66,65,63)(58,64,62,68)(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);(5,7,18,43,10,21,28,30,11,22,37,56,27,48,9,20)(6,12,26,41,15,32,38,42,16,33,54,49,29,40,14,17)(8,19,44,64,25,47,50,55,23,45,60,68,51,65,24,46)(13,31,52,63,36,58,59,61,34,57,67,66,53,62,35,39);(2,4)(5,8)(6,13)(7,19)(9,24)(10,25)(11,23)(12,31)(14,35)(15,36)(16,34)(17,39)(18,44)(20,46)(21,47)(22,45)(26,52)(27,51)(28,50)(29,53)(30,55)(32,58)(33,57)(37,60)(38,59)(40,62)(41,63)(42,61)(43,64)(48,65)(49,66)(54,67)(56,68);(8,23)(13,34)(19,45)(24,50)(25,51)(31,57)(35,59)(36,53)(39,61)(44,60)(46,55)(47,65)(52,67)(58,62)(63,66)(64,68)3484 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17152, 26596);