Finite Group DatabasebetaFinite Group Databasebeta| order | 17152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C67 · Schur multiplier (H2) = C25 |
| direct product | C67×#33588 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C67 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C67 |
| prime radical | C2×#135×C67 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C67 ◁ #117×C67 ◁ #405×C67 ◁ G | |
680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 8576 |
A minimal faithful permutation representation — 4 generators on 135 points:
(2,4)(5,6,11,16)(7,12,18,33)(8,21,24,55)(9,25,29,54)(10,30,27,15)(13,36,38,22)(14,39,41,28)(17,43,44,66)(19,49,47,65)(20,52,48,35)(23,56,31,40)(26,42,57,37)(32,58,59,45)(34,62,61,51)(46,67,53,63)(50,64,68,60)(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);(1,2,3,4)(5,7)(6,12)(8,17)(9,19)(10,20)(11,18)(13,32)(14,34)(15,35)(16,33)(21,43)(22,45)(23,46)(24,44)(25,49)(26,50)(27,48)(28,51)(29,47)(30,52)(31,53)(36,58)(37,60)(38,59)(39,62)(40,63)(41,61)(42,64)(54,65)(55,66)(56,67)(57,68);(5,8,11,24)(6,13,16,38)(7,17,18,44)(9,26,29,57)(10,31,27,23)(12,32,33,59)(14,40,41,56)(15,42,30,37)(19,50,47,68)(20,53,48,46)(21,54,55,25)(22,39,36,28)(34,63,61,67)(35,64,52,60)(43,65,66,49)(45,62,58,51);(7,18)(12,33)(17,44)(19,47)(20,48)(32,59)(34,61)(35,52)(43,66)(45,58)(46,53)(49,65)(50,68)(51,62)(60,64)(63,67)3082 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17152, 26576);