metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary
Aliases: C61⋊C5, SmallGroup(305,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C61 — C61⋊C5 |
Generators and relations for C61⋊C5
G = < a,b | a61=b5=1, bab-1=a34 >
Character table of C61⋊C5
| class | 1 | 5A | 5B | 5C | 5D | 61A | 61B | 61C | 61D | 61E | 61F | 61G | 61H | 61I | 61J | 61K | 61L | |
| size | 1 | 61 | 61 | 61 | 61 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | ζ5 | ζ53 | ζ52 | ζ54 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ3 | 1 | ζ54 | ζ52 | ζ53 | ζ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ4 | 1 | ζ52 | ζ5 | ζ54 | ζ53 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ5 | 1 | ζ53 | ζ54 | ζ5 | ζ52 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 5 |
| ρ6 | 5 | 0 | 0 | 0 | 0 | ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 | ζ6149+ζ6136+ζ6119+ζ6114+ζ614 | ζ6148+ζ6146+ζ6145+ζ6139+ζ615 | ζ6160+ζ6152+ζ6141+ζ6127+ζ613 | ζ6138+ζ6137+ζ6128+ζ6111+ζ618 | ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 | ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 | ζ6158+ζ6134+ζ6120+ζ619+ζ61 | ζ6159+ζ6154+ζ6143+ζ6121+ζ616 | ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 | ζ6155+ζ6140+ζ6118+ζ617+ζ612 | ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 | complex faithful |
| ρ7 | 5 | 0 | 0 | 0 | 0 | ζ6149+ζ6136+ζ6119+ζ6114+ζ614 | ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 | ζ6159+ζ6154+ζ6143+ζ6121+ζ616 | ζ6138+ζ6137+ζ6128+ζ6111+ζ618 | ζ6158+ζ6134+ζ6120+ζ619+ζ61 | ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 | ζ6155+ζ6140+ζ6118+ζ617+ζ612 | ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 | ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 | ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 | ζ6148+ζ6146+ζ6145+ζ6139+ζ615 | ζ6160+ζ6152+ζ6141+ζ6127+ζ613 | complex faithful |
| ρ8 | 5 | 0 | 0 | 0 | 0 | ζ6158+ζ6134+ζ6120+ζ619+ζ61 | ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 | ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 | ζ6155+ζ6140+ζ6118+ζ617+ζ612 | ζ6148+ζ6146+ζ6145+ζ6139+ζ615 | ζ6160+ζ6152+ζ6141+ζ6127+ζ613 | ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 | ζ6159+ζ6154+ζ6143+ζ6121+ζ616 | ζ6149+ζ6136+ζ6119+ζ6114+ζ614 | ζ6138+ζ6137+ζ6128+ζ6111+ζ618 | ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 | ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 | complex faithful |
| ρ9 | 5 | 0 | 0 | 0 | 0 | ζ6160+ζ6152+ζ6141+ζ6127+ζ613 | ζ6138+ζ6137+ζ6128+ζ6111+ζ618 | ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 | ζ6159+ζ6154+ζ6143+ζ6121+ζ616 | ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 | ζ6158+ζ6134+ζ6120+ζ619+ζ61 | ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 | ζ6155+ζ6140+ζ6118+ζ617+ζ612 | ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 | ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 | ζ6149+ζ6136+ζ6119+ζ6114+ζ614 | ζ6148+ζ6146+ζ6145+ζ6139+ζ615 | complex faithful |
| ρ10 | 5 | 0 | 0 | 0 | 0 | ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 | ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 | ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 | ζ6158+ζ6134+ζ6120+ζ619+ζ61 | ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 | ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 | ζ6148+ζ6146+ζ6145+ζ6139+ζ615 | ζ6160+ζ6152+ζ6141+ζ6127+ζ613 | ζ6155+ζ6140+ζ6118+ζ617+ζ612 | ζ6149+ζ6136+ζ6119+ζ6114+ζ614 | ζ6159+ζ6154+ζ6143+ζ6121+ζ616 | ζ6138+ζ6137+ζ6128+ζ6111+ζ618 | complex faithful |
| ρ11 | 5 | 0 | 0 | 0 | 0 | ζ6138+ζ6137+ζ6128+ζ6111+ζ618 | ζ6158+ζ6134+ζ6120+ζ619+ζ61 | ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 | ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 | ζ6155+ζ6140+ζ6118+ζ617+ζ612 | ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 | ζ6149+ζ6136+ζ6119+ζ6114+ζ614 | ζ6148+ζ6146+ζ6145+ζ6139+ζ615 | ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 | ζ6160+ζ6152+ζ6141+ζ6127+ζ613 | ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 | ζ6159+ζ6154+ζ6143+ζ6121+ζ616 | complex faithful |
| ρ12 | 5 | 0 | 0 | 0 | 0 | ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 | ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 | ζ6155+ζ6140+ζ6118+ζ617+ζ612 | ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 | ζ6160+ζ6152+ζ6141+ζ6127+ζ613 | ζ6149+ζ6136+ζ6119+ζ6114+ζ614 | ζ6159+ζ6154+ζ6143+ζ6121+ζ616 | ζ6138+ζ6137+ζ6128+ζ6111+ζ618 | ζ6148+ζ6146+ζ6145+ζ6139+ζ615 | ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 | ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 | ζ6158+ζ6134+ζ6120+ζ619+ζ61 | complex faithful |
| ρ13 | 5 | 0 | 0 | 0 | 0 | ζ6148+ζ6146+ζ6145+ζ6139+ζ615 | ζ6159+ζ6154+ζ6143+ζ6121+ζ616 | ζ6138+ζ6137+ζ6128+ζ6111+ζ618 | ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 | ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 | ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 | ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 | ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 | ζ6158+ζ6134+ζ6120+ζ619+ζ61 | ζ6155+ζ6140+ζ6118+ζ617+ζ612 | ζ6160+ζ6152+ζ6141+ζ6127+ζ613 | ζ6149+ζ6136+ζ6119+ζ6114+ζ614 | complex faithful |
| ρ14 | 5 | 0 | 0 | 0 | 0 | ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 | ζ6155+ζ6140+ζ6118+ζ617+ζ612 | ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 | ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 | ζ6149+ζ6136+ζ6119+ζ6114+ζ614 | ζ6148+ζ6146+ζ6145+ζ6139+ζ615 | ζ6138+ζ6137+ζ6128+ζ6111+ζ618 | ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 | ζ6160+ζ6152+ζ6141+ζ6127+ζ613 | ζ6159+ζ6154+ζ6143+ζ6121+ζ616 | ζ6158+ζ6134+ζ6120+ζ619+ζ61 | ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 | complex faithful |
| ρ15 | 5 | 0 | 0 | 0 | 0 | ζ6159+ζ6154+ζ6143+ζ6121+ζ616 | ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 | ζ6158+ζ6134+ζ6120+ζ619+ζ61 | ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 | ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 | ζ6155+ζ6140+ζ6118+ζ617+ζ612 | ζ6160+ζ6152+ζ6141+ζ6127+ζ613 | ζ6149+ζ6136+ζ6119+ζ6114+ζ614 | ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 | ζ6148+ζ6146+ζ6145+ζ6139+ζ615 | ζ6138+ζ6137+ζ6128+ζ6111+ζ618 | ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 | complex faithful |
| ρ16 | 5 | 0 | 0 | 0 | 0 | ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 | ζ6160+ζ6152+ζ6141+ζ6127+ζ613 | ζ6149+ζ6136+ζ6119+ζ6114+ζ614 | ζ6148+ζ6146+ζ6145+ζ6139+ζ615 | ζ6159+ζ6154+ζ6143+ζ6121+ζ616 | ζ6138+ζ6137+ζ6128+ζ6111+ζ618 | ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 | ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 | ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 | ζ6158+ζ6134+ζ6120+ζ619+ζ61 | ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 | ζ6155+ζ6140+ζ6118+ζ617+ζ612 | complex faithful |
| ρ17 | 5 | 0 | 0 | 0 | 0 | ζ6155+ζ6140+ζ6118+ζ617+ζ612 | ζ6148+ζ6146+ζ6145+ζ6139+ζ615 | ζ6160+ζ6152+ζ6141+ζ6127+ζ613 | ζ6149+ζ6136+ζ6119+ζ6114+ζ614 | ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 | ζ6159+ζ6154+ζ6143+ζ6121+ζ616 | ζ6158+ζ6134+ζ6120+ζ619+ζ61 | ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 | ζ6138+ζ6137+ζ6128+ζ6111+ζ618 | ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 | ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 | ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 | complex 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 10 21 59 35)(3 19 41 56 8)(4 28 61 53 42)(5 37 20 50 15)(6 46 40 47 49)(7 55 60 44 22)(9 12 39 38 29)(11 30 18 32 36)(13 48 58 26 43)(14 57 17 23 16)(24 25 34 54 51)(27 52 33 45 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), (2,10,21,59,35)(3,19,41,56,8)(4,28,61,53,42)(5,37,20,50,15)(6,46,40,47,49)(7,55,60,44,22)(9,12,39,38,29)(11,30,18,32,36)(13,48,58,26,43)(14,57,17,23,16)(24,25,34,54,51)(27,52,33,45,31)>;
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), (2,10,21,59,35)(3,19,41,56,8)(4,28,61,53,42)(5,37,20,50,15)(6,46,40,47,49)(7,55,60,44,22)(9,12,39,38,29)(11,30,18,32,36)(13,48,58,26,43)(14,57,17,23,16)(24,25,34,54,51)(27,52,33,45,31) );
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)], [(2,10,21,59,35),(3,19,41,56,8),(4,28,61,53,42),(5,37,20,50,15),(6,46,40,47,49),(7,55,60,44,22),(9,12,39,38,29),(11,30,18,32,36),(13,48,58,26,43),(14,57,17,23,16),(24,25,34,54,51),(27,52,33,45,31)]])
Matrix representation of C61⋊C5 ►in GL5(𝔽1831)
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 1110 | 1287 | 1215 | 848 |
| 1 | 0 | 0 | 0 | 0 |
| 1340 | 1602 | 1813 | 393 | 151 |
| 736 | 1182 | 752 | 91 | 511 |
| 975 | 607 | 1232 | 675 | 601 |
| 114 | 1164 | 133 | 1323 | 632 |
G:=sub<GL(5,GF(1831))| [0,0,0,0,1,1,0,0,0,1110,0,1,0,0,1287,0,0,1,0,1215,0,0,0,1,848],[1,1340,736,975,114,0,1602,1182,607,1164,0,1813,752,1232,133,0,393,91,675,1323,0,151,511,601,632] >;
C61⋊C5 in GAP, Magma, Sage, TeX
C_{61}\rtimes C_5 % in TeX
G:=Group("C61:C5"); // GroupNames label
G:=SmallGroup(305,1);
// by ID
G=gap.SmallGroup(305,1);
# by ID
G:=PCGroup([2,-5,-61,181]);
// Polycyclic
G:=Group<a,b|a^61=b^5=1,b*a*b^-1=a^34>;
// generators/relations
Export
Subgroup lattice of C61⋊C5 in TeX
Character table of C61⋊C5 in TeX