metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C61⋊C4, D61.C2, SmallGroup(244,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C61 — C61⋊C4 |
Generators and relations for C61⋊C4
G = < a,b | a61=b4=1, bab-1=a50 >
Character table of C61⋊C4
| class | 1 | 2 | 4A | 4B | 61A | 61B | 61C | 61D | 61E | 61F | 61G | 61H | 61I | 61J | 61K | 61L | 61M | 61N | 61O | |
| size | 1 | 61 | 61 | 61 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 4 | 0 | 0 | 0 | ζ6151+ζ6149+ζ6112+ζ6110 | ζ6146+ζ6143+ζ6118+ζ6115 | ζ6148+ζ6140+ζ6121+ζ6113 | ζ6141+ζ6137+ζ6124+ζ6120 | ζ6153+ζ6134+ζ6127+ζ618 | ζ6136+ζ6131+ζ6130+ζ6125 | ζ6159+ζ6139+ζ6122+ζ612 | ζ6142+ζ6135+ζ6126+ζ6119 | ζ6154+ζ6145+ζ6116+ζ617 | ζ6157+ζ6144+ζ6117+ζ614 | ζ6160+ζ6150+ζ6111+ζ61 | ζ6147+ζ6132+ζ6129+ζ6114 | ζ6158+ζ6133+ζ6128+ζ613 | ζ6156+ζ6155+ζ616+ζ615 | ζ6152+ζ6138+ζ6123+ζ619 | orthogonal faithful |
| ρ6 | 4 | 0 | 0 | 0 | ζ6156+ζ6155+ζ616+ζ615 | ζ6152+ζ6138+ζ6123+ζ619 | ζ6141+ζ6137+ζ6124+ζ6120 | ζ6151+ζ6149+ζ6112+ζ6110 | ζ6157+ζ6144+ζ6117+ζ614 | ζ6146+ζ6143+ζ6118+ζ6115 | ζ6160+ζ6150+ζ6111+ζ61 | ζ6148+ζ6140+ζ6121+ζ6113 | ζ6153+ζ6134+ζ6127+ζ618 | ζ6159+ζ6139+ζ6122+ζ612 | ζ6136+ζ6131+ζ6130+ζ6125 | ζ6154+ζ6145+ζ6116+ζ617 | ζ6147+ζ6132+ζ6129+ζ6114 | ζ6158+ζ6133+ζ6128+ζ613 | ζ6142+ζ6135+ζ6126+ζ6119 | orthogonal faithful |
| ρ7 | 4 | 0 | 0 | 0 | ζ6158+ζ6133+ζ6128+ζ613 | ζ6142+ζ6135+ζ6126+ζ6119 | ζ6151+ζ6149+ζ6112+ζ6110 | ζ6156+ζ6155+ζ616+ζ615 | ζ6159+ζ6139+ζ6122+ζ612 | ζ6152+ζ6138+ζ6123+ζ619 | ζ6136+ζ6131+ζ6130+ζ6125 | ζ6141+ζ6137+ζ6124+ζ6120 | ζ6157+ζ6144+ζ6117+ζ614 | ζ6160+ζ6150+ζ6111+ζ61 | ζ6146+ζ6143+ζ6118+ζ6115 | ζ6153+ζ6134+ζ6127+ζ618 | ζ6154+ζ6145+ζ6116+ζ617 | ζ6147+ζ6132+ζ6129+ζ6114 | ζ6148+ζ6140+ζ6121+ζ6113 | orthogonal faithful |
| ρ8 | 4 | 0 | 0 | 0 | ζ6153+ζ6134+ζ6127+ζ618 | ζ6151+ζ6149+ζ6112+ζ6110 | ζ6147+ζ6132+ζ6129+ζ6114 | ζ6154+ζ6145+ζ6116+ζ617 | ζ6146+ζ6143+ζ6118+ζ6115 | ζ6141+ζ6137+ζ6124+ζ6120 | ζ6142+ζ6135+ζ6126+ζ6119 | ζ6158+ζ6133+ζ6128+ζ613 | ζ6136+ζ6131+ζ6130+ζ6125 | ζ6152+ζ6138+ζ6123+ζ619 | ζ6148+ζ6140+ζ6121+ζ6113 | ζ6160+ζ6150+ζ6111+ζ61 | ζ6159+ζ6139+ζ6122+ζ612 | ζ6157+ζ6144+ζ6117+ζ614 | ζ6156+ζ6155+ζ616+ζ615 | orthogonal faithful |
| ρ9 | 4 | 0 | 0 | 0 | ζ6141+ζ6137+ζ6124+ζ6120 | ζ6136+ζ6131+ζ6130+ζ6125 | ζ6142+ζ6135+ζ6126+ζ6119 | ζ6148+ζ6140+ζ6121+ζ6113 | ζ6154+ζ6145+ζ6116+ζ617 | ζ6160+ζ6150+ζ6111+ζ61 | ζ6157+ζ6144+ζ6117+ζ614 | ζ6152+ζ6138+ζ6123+ζ619 | ζ6147+ζ6132+ζ6129+ζ6114 | ζ6153+ζ6134+ζ6127+ζ618 | ζ6159+ζ6139+ζ6122+ζ612 | ζ6158+ζ6133+ζ6128+ζ613 | ζ6156+ζ6155+ζ616+ζ615 | ζ6151+ζ6149+ζ6112+ζ6110 | ζ6146+ζ6143+ζ6118+ζ6115 | orthogonal faithful |
| ρ10 | 4 | 0 | 0 | 0 | ζ6154+ζ6145+ζ6116+ζ617 | ζ6141+ζ6137+ζ6124+ζ6120 | ζ6158+ζ6133+ζ6128+ζ613 | ζ6147+ζ6132+ζ6129+ζ6114 | ζ6136+ζ6131+ζ6130+ζ6125 | ζ6148+ζ6140+ζ6121+ζ6113 | ζ6152+ζ6138+ζ6123+ζ619 | ζ6156+ζ6155+ζ616+ζ615 | ζ6160+ζ6150+ζ6111+ζ61 | ζ6146+ζ6143+ζ6118+ζ6115 | ζ6142+ζ6135+ζ6126+ζ6119 | ζ6159+ζ6139+ζ6122+ζ612 | ζ6157+ζ6144+ζ6117+ζ614 | ζ6153+ζ6134+ζ6127+ζ618 | ζ6151+ζ6149+ζ6112+ζ6110 | orthogonal faithful |
| ρ11 | 4 | 0 | 0 | 0 | ζ6136+ζ6131+ζ6130+ζ6125 | ζ6154+ζ6145+ζ6116+ζ617 | ζ6159+ζ6139+ζ6122+ζ612 | ζ6160+ζ6150+ζ6111+ζ61 | ζ6141+ζ6137+ζ6124+ζ6120 | ζ6147+ζ6132+ζ6129+ζ6114 | ζ6156+ζ6155+ζ616+ζ615 | ζ6157+ζ6144+ζ6117+ζ614 | ζ6148+ζ6140+ζ6121+ζ6113 | ζ6151+ζ6149+ζ6112+ζ6110 | ζ6158+ζ6133+ζ6128+ζ613 | ζ6142+ζ6135+ζ6126+ζ6119 | ζ6152+ζ6138+ζ6123+ζ619 | ζ6146+ζ6143+ζ6118+ζ6115 | ζ6153+ζ6134+ζ6127+ζ618 | orthogonal faithful |
| ρ12 | 4 | 0 | 0 | 0 | ζ6147+ζ6132+ζ6129+ζ6114 | ζ6148+ζ6140+ζ6121+ζ6113 | ζ6156+ζ6155+ζ616+ζ615 | ζ6158+ζ6133+ζ6128+ζ613 | ζ6160+ζ6150+ζ6111+ζ61 | ζ6142+ζ6135+ζ6126+ζ6119 | ζ6146+ζ6143+ζ6118+ζ6115 | ζ6151+ζ6149+ζ6112+ζ6110 | ζ6159+ζ6139+ζ6122+ζ612 | ζ6136+ζ6131+ζ6130+ζ6125 | ζ6152+ζ6138+ζ6123+ζ619 | ζ6157+ζ6144+ζ6117+ζ614 | ζ6153+ζ6134+ζ6127+ζ618 | ζ6154+ζ6145+ζ6116+ζ617 | ζ6141+ζ6137+ζ6124+ζ6120 | orthogonal faithful |
| ρ13 | 4 | 0 | 0 | 0 | ζ6142+ζ6135+ζ6126+ζ6119 | ζ6159+ζ6139+ζ6122+ζ612 | ζ6146+ζ6143+ζ6118+ζ6115 | ζ6152+ζ6138+ζ6123+ζ619 | ζ6158+ζ6133+ζ6128+ζ613 | ζ6157+ζ6144+ζ6117+ζ614 | ζ6154+ζ6145+ζ6116+ζ617 | ζ6136+ζ6131+ζ6130+ζ6125 | ζ6156+ζ6155+ζ616+ζ615 | ζ6147+ζ6132+ζ6129+ζ6114 | ζ6153+ζ6134+ζ6127+ζ618 | ζ6151+ζ6149+ζ6112+ζ6110 | ζ6141+ζ6137+ζ6124+ζ6120 | ζ6148+ζ6140+ζ6121+ζ6113 | ζ6160+ζ6150+ζ6111+ζ61 | orthogonal faithful |
| ρ14 | 4 | 0 | 0 | 0 | ζ6152+ζ6138+ζ6123+ζ619 | ζ6157+ζ6144+ζ6117+ζ614 | ζ6136+ζ6131+ζ6130+ζ6125 | ζ6146+ζ6143+ζ6118+ζ6115 | ζ6156+ζ6155+ζ616+ζ615 | ζ6153+ζ6134+ζ6127+ζ618 | ζ6147+ζ6132+ζ6129+ζ6114 | ζ6160+ζ6150+ζ6111+ζ61 | ζ6151+ζ6149+ζ6112+ζ6110 | ζ6158+ζ6133+ζ6128+ζ613 | ζ6154+ζ6145+ζ6116+ζ617 | ζ6141+ζ6137+ζ6124+ζ6120 | ζ6148+ζ6140+ζ6121+ζ6113 | ζ6142+ζ6135+ζ6126+ζ6119 | ζ6159+ζ6139+ζ6122+ζ612 | orthogonal faithful |
| ρ15 | 4 | 0 | 0 | 0 | ζ6146+ζ6143+ζ6118+ζ6115 | ζ6153+ζ6134+ζ6127+ζ618 | ζ6160+ζ6150+ζ6111+ζ61 | ζ6136+ζ6131+ζ6130+ζ6125 | ζ6151+ζ6149+ζ6112+ζ6110 | ζ6154+ζ6145+ζ6116+ζ617 | ζ6158+ζ6133+ζ6128+ζ613 | ζ6159+ζ6139+ζ6122+ζ612 | ζ6141+ζ6137+ζ6124+ζ6120 | ζ6156+ζ6155+ζ616+ζ615 | ζ6147+ζ6132+ζ6129+ζ6114 | ζ6148+ζ6140+ζ6121+ζ6113 | ζ6142+ζ6135+ζ6126+ζ6119 | ζ6152+ζ6138+ζ6123+ζ619 | ζ6157+ζ6144+ζ6117+ζ614 | orthogonal faithful |
| ρ16 | 4 | 0 | 0 | 0 | ζ6157+ζ6144+ζ6117+ζ614 | ζ6156+ζ6155+ζ616+ζ615 | ζ6154+ζ6145+ζ6116+ζ617 | ζ6153+ζ6134+ζ6127+ζ618 | ζ6152+ζ6138+ζ6123+ζ619 | ζ6151+ζ6149+ζ6112+ζ6110 | ζ6148+ζ6140+ζ6121+ζ6113 | ζ6147+ζ6132+ζ6129+ζ6114 | ζ6146+ζ6143+ζ6118+ζ6115 | ζ6142+ζ6135+ζ6126+ζ6119 | ζ6141+ζ6137+ζ6124+ζ6120 | ζ6136+ζ6131+ζ6130+ζ6125 | ζ6160+ζ6150+ζ6111+ζ61 | ζ6159+ζ6139+ζ6122+ζ612 | ζ6158+ζ6133+ζ6128+ζ613 | orthogonal faithful |
| ρ17 | 4 | 0 | 0 | 0 | ζ6160+ζ6150+ζ6111+ζ61 | ζ6147+ζ6132+ζ6129+ζ6114 | ζ6157+ζ6144+ζ6117+ζ614 | ζ6159+ζ6139+ζ6122+ζ612 | ζ6148+ζ6140+ζ6121+ζ6113 | ζ6158+ζ6133+ζ6128+ζ613 | ζ6151+ζ6149+ζ6112+ζ6110 | ζ6153+ζ6134+ζ6127+ζ618 | ζ6142+ζ6135+ζ6126+ζ6119 | ζ6141+ζ6137+ζ6124+ζ6120 | ζ6156+ζ6155+ζ616+ζ615 | ζ6152+ζ6138+ζ6123+ζ619 | ζ6146+ζ6143+ζ6118+ζ6115 | ζ6136+ζ6131+ζ6130+ζ6125 | ζ6154+ζ6145+ζ6116+ζ617 | orthogonal faithful |
| ρ18 | 4 | 0 | 0 | 0 | ζ6159+ζ6139+ζ6122+ζ612 | ζ6158+ζ6133+ζ6128+ζ613 | ζ6153+ζ6134+ζ6127+ζ618 | ζ6157+ζ6144+ζ6117+ζ614 | ζ6142+ζ6135+ζ6126+ζ6119 | ζ6156+ζ6155+ζ616+ζ615 | ζ6141+ζ6137+ζ6124+ζ6120 | ζ6154+ζ6145+ζ6116+ζ617 | ζ6152+ζ6138+ζ6123+ζ619 | ζ6148+ζ6140+ζ6121+ζ6113 | ζ6151+ζ6149+ζ6112+ζ6110 | ζ6146+ζ6143+ζ6118+ζ6115 | ζ6136+ζ6131+ζ6130+ζ6125 | ζ6160+ζ6150+ζ6111+ζ61 | ζ6147+ζ6132+ζ6129+ζ6114 | orthogonal faithful |
| ρ19 | 4 | 0 | 0 | 0 | ζ6148+ζ6140+ζ6121+ζ6113 | ζ6160+ζ6150+ζ6111+ζ61 | ζ6152+ζ6138+ζ6123+ζ619 | ζ6142+ζ6135+ζ6126+ζ6119 | ζ6147+ζ6132+ζ6129+ζ6114 | ζ6159+ζ6139+ζ6122+ζ612 | ζ6153+ζ6134+ζ6127+ζ618 | ζ6146+ζ6143+ζ6118+ζ6115 | ζ6158+ζ6133+ζ6128+ζ613 | ζ6154+ζ6145+ζ6116+ζ617 | ζ6157+ζ6144+ζ6117+ζ614 | ζ6156+ζ6155+ζ616+ζ615 | ζ6151+ζ6149+ζ6112+ζ6110 | ζ6141+ζ6137+ζ6124+ζ6120 | ζ6136+ζ6131+ζ6130+ζ6125 | 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 12 61 51)(3 23 60 40)(4 34 59 29)(5 45 58 18)(6 56 57 7)(8 17 55 46)(9 28 54 35)(10 39 53 24)(11 50 52 13)(14 22 49 41)(15 33 48 30)(16 44 47 19)(20 27 43 36)(21 38 42 25)(26 32 37 31)
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), (2,12,61,51)(3,23,60,40)(4,34,59,29)(5,45,58,18)(6,56,57,7)(8,17,55,46)(9,28,54,35)(10,39,53,24)(11,50,52,13)(14,22,49,41)(15,33,48,30)(16,44,47,19)(20,27,43,36)(21,38,42,25)(26,32,37,31)>;
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), (2,12,61,51)(3,23,60,40)(4,34,59,29)(5,45,58,18)(6,56,57,7)(8,17,55,46)(9,28,54,35)(10,39,53,24)(11,50,52,13)(14,22,49,41)(15,33,48,30)(16,44,47,19)(20,27,43,36)(21,38,42,25)(26,32,37,31) );
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)], [(2,12,61,51),(3,23,60,40),(4,34,59,29),(5,45,58,18),(6,56,57,7),(8,17,55,46),(9,28,54,35),(10,39,53,24),(11,50,52,13),(14,22,49,41),(15,33,48,30),(16,44,47,19),(20,27,43,36),(21,38,42,25),(26,32,37,31)]])
C61⋊C4 is a maximal quotient of C61⋊C8
Matrix representation of C61⋊C4 ►in GL4(𝔽733) generated by
| 92 | 1 | 0 | 0 |
| 667 | 0 | 1 | 0 |
| 558 | 0 | 0 | 1 |
| 72 | 375 | 713 | 376 |
| 577 | 501 | 583 | 75 |
| 450 | 429 | 45 | 172 |
| 430 | 427 | 567 | 305 |
| 653 | 52 | 388 | 626 |
G:=sub<GL(4,GF(733))| [92,667,558,72,1,0,0,375,0,1,0,713,0,0,1,376],[577,450,430,653,501,429,427,52,583,45,567,388,75,172,305,626] >;
C61⋊C4 in GAP, Magma, Sage, TeX
C_{61}\rtimes C_4 % in TeX
G:=Group("C61:C4"); // GroupNames label
G:=SmallGroup(244,3);
// by ID
G=gap.SmallGroup(244,3);
# by ID
G:=PCGroup([3,-2,-2,-61,6,398,1085]);
// Polycyclic
G:=Group<a,b|a^61=b^4=1,b*a*b^-1=a^50>;
// generators/relations
Export
Subgroup lattice of C61⋊C4 in TeX
Character table of C61⋊C4 in TeX