Finite Group DatabasebetaFinite Group Databasebeta| order | 17152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22284 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C67 · Schur multiplier (H2) = C24×C4 |
| direct product | C67×#22285 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C67 | #2328 |
| commutator | C2×C4 | C23×C4×C67 |
| Frattini, cosocle | C22×C4 | C24×C67 |
| prime radical | C2×#135×C67 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C67 ◁ C22×C4×C67 ◁ G | |
772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 4288 |
A minimal faithful permutation representation — 4 generators on 131 points:
(1,2)(3,14)(4,20,24,34)(5,25)(6,12)(7,13)(8,17)(9,23,35,49)(10,27)(11,26)(15,40,43,29)(16,44)(18,31)(19,38)(21,32,42,57)(22,33,52,58)(28,36)(30,47)(37,54)(39,45)(41,61,63,56)(46,59)(48,64,53,51)(50,60,62,55)(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);(1,3,6,17,7,19,26,18)(2,8,11,14,13,31,12,38)(4,21,49,15,24,42,23,43)(5,16,27,45,28,47,54,46)(9,32,20,29,35,57,34,40)(10,30,25,39,37,44,36,59)(22,50,64,41,52,62,51,63)(33,56,48,55,58,61,53,60);(1,4,7,24)(2,9)(3,15)(5,22,28,52)(6,23,26,49)(8,29,31,40)(10,33)(11,34)(12,20)(13,35)(14,32,38,57)(16,41)(17,21)(18,42)(19,43)(25,53)(27,51,54,64)(30,55,44,60)(36,48)(37,58)(39,56,59,61)(45,50)(46,62)(47,63);(1,5,26,54,7,28,6,27)(2,10,12,36,13,37,11,25)(3,16,18,46,19,47,17,45)(4,22,49,64,24,52,23,51)(8,30,38,59,31,44,14,39)(9,33,20,48,35,58,34,53)(15,41,42,62,43,63,21,50)(29,55,57,61,40,60,32,56)3082 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17152, 15273);