metacyclic, supersoluble, monomial, Z-group
Aliases: C61⋊C6, D61⋊C3, C61⋊C3⋊C2, SmallGroup(366,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C61 — C61⋊C6 |
Generators and relations for C61⋊C6
G = < a,b | a61=b6=1, bab-1=a14 >
Character table of C61⋊C6
| class | 1 | 2 | 3A | 3B | 6A | 6B | 61A | 61B | 61C | 61D | 61E | 61F | 61G | 61H | 61I | 61J | |
| size | 1 | 61 | 61 | 61 | 61 | 61 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6153+ζ6151+ζ6143+ζ6118+ζ6110+ζ618 | ζ6155+ζ6144+ζ6138+ζ6123+ζ6117+ζ616 | ζ6154+ζ6137+ζ6131+ζ6130+ζ6124+ζ617 | ζ6157+ζ6156+ζ6152+ζ619+ζ615+ζ614 | ζ6149+ζ6146+ζ6134+ζ6127+ζ6115+ζ6112 | ζ6145+ζ6141+ζ6136+ζ6125+ζ6120+ζ6116 | ζ6150+ζ6140+ζ6132+ζ6129+ζ6121+ζ6111 | ζ6160+ζ6148+ζ6147+ζ6114+ζ6113+ζ61 | ζ6158+ζ6142+ζ6139+ζ6122+ζ6119+ζ613 | ζ6159+ζ6135+ζ6133+ζ6128+ζ6126+ζ612 | orthogonal faithful |
| ρ8 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6157+ζ6156+ζ6152+ζ619+ζ615+ζ614 | ζ6158+ζ6142+ζ6139+ζ6122+ζ6119+ζ613 | ζ6149+ζ6146+ζ6134+ζ6127+ζ6115+ζ6112 | ζ6159+ζ6135+ζ6133+ζ6128+ζ6126+ζ612 | ζ6155+ζ6144+ζ6138+ζ6123+ζ6117+ζ616 | ζ6153+ζ6151+ζ6143+ζ6118+ζ6110+ζ618 | ζ6145+ζ6141+ζ6136+ζ6125+ζ6120+ζ6116 | ζ6154+ζ6137+ζ6131+ζ6130+ζ6124+ζ617 | ζ6150+ζ6140+ζ6132+ζ6129+ζ6121+ζ6111 | ζ6160+ζ6148+ζ6147+ζ6114+ζ6113+ζ61 | orthogonal faithful |
| ρ9 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6154+ζ6137+ζ6131+ζ6130+ζ6124+ζ617 | ζ6153+ζ6151+ζ6143+ζ6118+ζ6110+ζ618 | ζ6150+ζ6140+ζ6132+ζ6129+ζ6121+ζ6111 | ζ6149+ζ6146+ζ6134+ζ6127+ζ6115+ζ6112 | ζ6145+ζ6141+ζ6136+ζ6125+ζ6120+ζ6116 | ζ6160+ζ6148+ζ6147+ζ6114+ζ6113+ζ61 | ζ6159+ζ6135+ζ6133+ζ6128+ζ6126+ζ612 | ζ6158+ζ6142+ζ6139+ζ6122+ζ6119+ζ613 | ζ6157+ζ6156+ζ6152+ζ619+ζ615+ζ614 | ζ6155+ζ6144+ζ6138+ζ6123+ζ6117+ζ616 | orthogonal faithful |
| ρ10 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6158+ζ6142+ζ6139+ζ6122+ζ6119+ζ613 | ζ6160+ζ6148+ζ6147+ζ6114+ζ6113+ζ61 | ζ6157+ζ6156+ζ6152+ζ619+ζ615+ζ614 | ζ6150+ζ6140+ζ6132+ζ6129+ζ6121+ζ6111 | ζ6159+ζ6135+ζ6133+ζ6128+ζ6126+ζ612 | ζ6155+ζ6144+ζ6138+ζ6123+ζ6117+ζ616 | ζ6149+ζ6146+ζ6134+ζ6127+ζ6115+ζ6112 | ζ6153+ζ6151+ζ6143+ζ6118+ζ6110+ζ618 | ζ6154+ζ6137+ζ6131+ζ6130+ζ6124+ζ617 | ζ6145+ζ6141+ζ6136+ζ6125+ζ6120+ζ6116 | orthogonal faithful |
| ρ11 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6160+ζ6148+ζ6147+ζ6114+ζ6113+ζ61 | ζ6145+ζ6141+ζ6136+ζ6125+ζ6120+ζ6116 | ζ6158+ζ6142+ζ6139+ζ6122+ζ6119+ζ613 | ζ6154+ζ6137+ζ6131+ζ6130+ζ6124+ζ617 | ζ6150+ζ6140+ζ6132+ζ6129+ζ6121+ζ6111 | ζ6159+ζ6135+ζ6133+ζ6128+ζ6126+ζ612 | ζ6157+ζ6156+ζ6152+ζ619+ζ615+ζ614 | ζ6155+ζ6144+ζ6138+ζ6123+ζ6117+ζ616 | ζ6153+ζ6151+ζ6143+ζ6118+ζ6110+ζ618 | ζ6149+ζ6146+ζ6134+ζ6127+ζ6115+ζ6112 | orthogonal faithful |
| ρ12 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6155+ζ6144+ζ6138+ζ6123+ζ6117+ζ616 | ζ6159+ζ6135+ζ6133+ζ6128+ζ6126+ζ612 | ζ6153+ζ6151+ζ6143+ζ6118+ζ6110+ζ618 | ζ6158+ζ6142+ζ6139+ζ6122+ζ6119+ζ613 | ζ6157+ζ6156+ζ6152+ζ619+ζ615+ζ614 | ζ6149+ζ6146+ζ6134+ζ6127+ζ6115+ζ6112 | ζ6154+ζ6137+ζ6131+ζ6130+ζ6124+ζ617 | ζ6145+ζ6141+ζ6136+ζ6125+ζ6120+ζ6116 | ζ6160+ζ6148+ζ6147+ζ6114+ζ6113+ζ61 | ζ6150+ζ6140+ζ6132+ζ6129+ζ6121+ζ6111 | orthogonal faithful |
| ρ13 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6150+ζ6140+ζ6132+ζ6129+ζ6121+ζ6111 | ζ6154+ζ6137+ζ6131+ζ6130+ζ6124+ζ617 | ζ6159+ζ6135+ζ6133+ζ6128+ζ6126+ζ612 | ζ6145+ζ6141+ζ6136+ζ6125+ζ6120+ζ6116 | ζ6160+ζ6148+ζ6147+ζ6114+ζ6113+ζ61 | ζ6158+ζ6142+ζ6139+ζ6122+ζ6119+ζ613 | ζ6155+ζ6144+ζ6138+ζ6123+ζ6117+ζ616 | ζ6157+ζ6156+ζ6152+ζ619+ζ615+ζ614 | ζ6149+ζ6146+ζ6134+ζ6127+ζ6115+ζ6112 | ζ6153+ζ6151+ζ6143+ζ6118+ζ6110+ζ618 | orthogonal faithful |
| ρ14 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6145+ζ6141+ζ6136+ζ6125+ζ6120+ζ6116 | ζ6149+ζ6146+ζ6134+ζ6127+ζ6115+ζ6112 | ζ6160+ζ6148+ζ6147+ζ6114+ζ6113+ζ61 | ζ6153+ζ6151+ζ6143+ζ6118+ζ6110+ζ618 | ζ6154+ζ6137+ζ6131+ζ6130+ζ6124+ζ617 | ζ6150+ζ6140+ζ6132+ζ6129+ζ6121+ζ6111 | ζ6158+ζ6142+ζ6139+ζ6122+ζ6119+ζ613 | ζ6159+ζ6135+ζ6133+ζ6128+ζ6126+ζ612 | ζ6155+ζ6144+ζ6138+ζ6123+ζ6117+ζ616 | ζ6157+ζ6156+ζ6152+ζ619+ζ615+ζ614 | orthogonal faithful |
| ρ15 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6149+ζ6146+ζ6134+ζ6127+ζ6115+ζ6112 | ζ6157+ζ6156+ζ6152+ζ619+ζ615+ζ614 | ζ6145+ζ6141+ζ6136+ζ6125+ζ6120+ζ6116 | ζ6155+ζ6144+ζ6138+ζ6123+ζ6117+ζ616 | ζ6153+ζ6151+ζ6143+ζ6118+ζ6110+ζ618 | ζ6154+ζ6137+ζ6131+ζ6130+ζ6124+ζ617 | ζ6160+ζ6148+ζ6147+ζ6114+ζ6113+ζ61 | ζ6150+ζ6140+ζ6132+ζ6129+ζ6121+ζ6111 | ζ6159+ζ6135+ζ6133+ζ6128+ζ6126+ζ612 | ζ6158+ζ6142+ζ6139+ζ6122+ζ6119+ζ613 | orthogonal faithful |
| ρ16 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6159+ζ6135+ζ6133+ζ6128+ζ6126+ζ612 | ζ6150+ζ6140+ζ6132+ζ6129+ζ6121+ζ6111 | ζ6155+ζ6144+ζ6138+ζ6123+ζ6117+ζ616 | ζ6160+ζ6148+ζ6147+ζ6114+ζ6113+ζ61 | ζ6158+ζ6142+ζ6139+ζ6122+ζ6119+ζ613 | ζ6157+ζ6156+ζ6152+ζ619+ζ615+ζ614 | ζ6153+ζ6151+ζ6143+ζ6118+ζ6110+ζ618 | ζ6149+ζ6146+ζ6134+ζ6127+ζ6115+ζ6112 | ζ6145+ζ6141+ζ6136+ζ6125+ζ6120+ζ6116 | ζ6154+ζ6137+ζ6131+ζ6130+ζ6124+ζ617 | orthogonal faithful |
►On 61 points: primitive
Generators in S61
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61)
(2 49 48 61 14 15)(3 36 34 60 27 29)(4 23 20 59 40 43)(5 10 6 58 53 57)(7 45 39 56 18 24)(8 32 25 55 31 38)(9 19 11 54 44 52)(12 41 30 51 22 33)(13 28 16 50 35 47)(17 37 21 46 26 42)
G:=sub<Sym(61)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61), (2,49,48,61,14,15)(3,36,34,60,27,29)(4,23,20,59,40,43)(5,10,6,58,53,57)(7,45,39,56,18,24)(8,32,25,55,31,38)(9,19,11,54,44,52)(12,41,30,51,22,33)(13,28,16,50,35,47)(17,37,21,46,26,42)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61), (2,49,48,61,14,15)(3,36,34,60,27,29)(4,23,20,59,40,43)(5,10,6,58,53,57)(7,45,39,56,18,24)(8,32,25,55,31,38)(9,19,11,54,44,52)(12,41,30,51,22,33)(13,28,16,50,35,47)(17,37,21,46,26,42) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61)], [(2,49,48,61,14,15),(3,36,34,60,27,29),(4,23,20,59,40,43),(5,10,6,58,53,57),(7,45,39,56,18,24),(8,32,25,55,31,38),(9,19,11,54,44,52),(12,41,30,51,22,33),(13,28,16,50,35,47),(17,37,21,46,26,42)]])
Matrix representation of C61⋊C6 ►in GL6(𝔽367)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 366 | 279 | 214 | 291 | 214 | 279 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 107 | 61 | 268 | 128 | 116 | 53 |
| 101 | 337 | 313 | 198 | 151 | 92 |
| 113 | 357 | 227 | 272 | 250 | 154 |
| 259 | 301 | 194 | 54 | 355 | 268 |
| 259 | 154 | 11 | 34 | 163 | 99 |
G:=sub<GL(6,GF(367))| [0,0,0,0,0,366,1,0,0,0,0,279,0,1,0,0,0,214,0,0,1,0,0,291,0,0,0,1,0,214,0,0,0,0,1,279],[1,107,101,113,259,259,0,61,337,357,301,154,0,268,313,227,194,11,0,128,198,272,54,34,0,116,151,250,355,163,0,53,92,154,268,99] >;
C61⋊C6 in GAP, Magma, Sage, TeX
C_{61}\rtimes C_6 % in TeX
G:=Group("C61:C6"); // GroupNames label
G:=SmallGroup(366,1);
// by ID
G=gap.SmallGroup(366,1);
# by ID
G:=PCGroup([3,-2,-3,-61,3242,1274]);
// Polycyclic
G:=Group<a,b|a^61=b^6=1,b*a*b^-1=a^14>;
// generators/relations
Export
Subgroup lattice of C61⋊C6 in TeX
Character table of C61⋊C6 in TeX