metacyclic, supersoluble, monomial, Z-group
Aliases: C67⋊C6, D67⋊C3, C67⋊C3⋊C2, SmallGroup(402,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C67 — C67⋊C6 |
Generators and relations for C67⋊C6
G = < a,b | a67=b6=1, bab-1=a30 >
Character table of C67⋊C6
| class | 1 | 2 | 3A | 3B | 6A | 6B | 67A | 67B | 67C | 67D | 67E | 67F | 67G | 67H | 67I | 67J | 67K | |
| size | 1 | 67 | 67 | 67 | 67 | 67 | 6 | 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 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6766+ζ6738+ζ6737+ζ6730+ζ6729+ζ67 | ζ6761+ζ6746+ζ6740+ζ6727+ζ6721+ζ676 | ζ6750+ζ6743+ζ6741+ζ6726+ζ6724+ζ6717 | ζ6765+ζ6760+ζ6758+ζ679+ζ677+ζ672 | ζ6759+ζ6739+ζ6736+ζ6731+ζ6728+ζ678 | ζ6763+ζ6753+ζ6749+ζ6718+ζ6714+ζ674 | ζ6762+ζ6756+ζ6751+ζ6716+ζ6711+ζ675 | ζ6757+ζ6745+ζ6735+ζ6732+ζ6722+ζ6710 | ζ6752+ζ6748+ζ6734+ζ6733+ζ6719+ζ6715 | ζ6764+ζ6747+ζ6744+ζ6723+ζ6720+ζ673 | ζ6755+ζ6754+ζ6742+ζ6725+ζ6713+ζ6712 | orthogonal faithful |
| ρ8 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6757+ζ6745+ζ6735+ζ6732+ζ6722+ζ6710 | ζ6765+ζ6760+ζ6758+ζ679+ζ677+ζ672 | ζ6759+ζ6739+ζ6736+ζ6731+ζ6728+ζ678 | ζ6764+ζ6747+ζ6744+ζ6723+ζ6720+ζ673 | ζ6755+ζ6754+ζ6742+ζ6725+ζ6713+ζ6712 | ζ6761+ζ6746+ζ6740+ζ6727+ζ6721+ζ676 | ζ6750+ζ6743+ζ6741+ζ6726+ζ6724+ζ6717 | ζ6752+ζ6748+ζ6734+ζ6733+ζ6719+ζ6715 | ζ6762+ζ6756+ζ6751+ζ6716+ζ6711+ζ675 | ζ6766+ζ6738+ζ6737+ζ6730+ζ6729+ζ67 | ζ6763+ζ6753+ζ6749+ζ6718+ζ6714+ζ674 | orthogonal faithful |
| ρ9 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6761+ζ6746+ζ6740+ζ6727+ζ6721+ζ676 | ζ6759+ζ6739+ζ6736+ζ6731+ζ6728+ζ678 | ζ6757+ζ6745+ζ6735+ζ6732+ζ6722+ζ6710 | ζ6755+ζ6754+ζ6742+ζ6725+ζ6713+ζ6712 | ζ6752+ζ6748+ζ6734+ζ6733+ζ6719+ζ6715 | ζ6750+ζ6743+ζ6741+ζ6726+ζ6724+ζ6717 | ζ6766+ζ6738+ζ6737+ζ6730+ζ6729+ζ67 | ζ6765+ζ6760+ζ6758+ζ679+ζ677+ζ672 | ζ6764+ζ6747+ζ6744+ζ6723+ζ6720+ζ673 | ζ6763+ζ6753+ζ6749+ζ6718+ζ6714+ζ674 | ζ6762+ζ6756+ζ6751+ζ6716+ζ6711+ζ675 | orthogonal faithful |
| ρ10 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6759+ζ6739+ζ6736+ζ6731+ζ6728+ζ678 | ζ6752+ζ6748+ζ6734+ζ6733+ζ6719+ζ6715 | ζ6765+ζ6760+ζ6758+ζ679+ζ677+ζ672 | ζ6762+ζ6756+ζ6751+ζ6716+ζ6711+ζ675 | ζ6764+ζ6747+ζ6744+ζ6723+ζ6720+ζ673 | ζ6757+ζ6745+ζ6735+ζ6732+ζ6722+ζ6710 | ζ6761+ζ6746+ζ6740+ζ6727+ζ6721+ζ676 | ζ6755+ζ6754+ζ6742+ζ6725+ζ6713+ζ6712 | ζ6763+ζ6753+ζ6749+ζ6718+ζ6714+ζ674 | ζ6750+ζ6743+ζ6741+ζ6726+ζ6724+ζ6717 | ζ6766+ζ6738+ζ6737+ζ6730+ζ6729+ζ67 | orthogonal faithful |
| ρ11 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6762+ζ6756+ζ6751+ζ6716+ζ6711+ζ675 | ζ6766+ζ6738+ζ6737+ζ6730+ζ6729+ζ67 | ζ6763+ζ6753+ζ6749+ζ6718+ζ6714+ζ674 | ζ6757+ζ6745+ζ6735+ζ6732+ζ6722+ζ6710 | ζ6761+ζ6746+ζ6740+ζ6727+ζ6721+ζ676 | ζ6764+ζ6747+ζ6744+ζ6723+ζ6720+ζ673 | ζ6755+ζ6754+ζ6742+ζ6725+ζ6713+ζ6712 | ζ6750+ζ6743+ζ6741+ζ6726+ζ6724+ζ6717 | ζ6759+ζ6739+ζ6736+ζ6731+ζ6728+ζ678 | ζ6752+ζ6748+ζ6734+ζ6733+ζ6719+ζ6715 | ζ6765+ζ6760+ζ6758+ζ679+ζ677+ζ672 | orthogonal faithful |
| ρ12 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6755+ζ6754+ζ6742+ζ6725+ζ6713+ζ6712 | ζ6762+ζ6756+ζ6751+ζ6716+ζ6711+ζ675 | ζ6764+ζ6747+ζ6744+ζ6723+ζ6720+ζ673 | ζ6750+ζ6743+ζ6741+ζ6726+ζ6724+ζ6717 | ζ6766+ζ6738+ζ6737+ζ6730+ζ6729+ζ67 | ζ6752+ζ6748+ζ6734+ζ6733+ζ6719+ζ6715 | ζ6765+ζ6760+ζ6758+ζ679+ζ677+ζ672 | ζ6763+ζ6753+ζ6749+ζ6718+ζ6714+ζ674 | ζ6761+ζ6746+ζ6740+ζ6727+ζ6721+ζ676 | ζ6759+ζ6739+ζ6736+ζ6731+ζ6728+ζ678 | ζ6757+ζ6745+ζ6735+ζ6732+ζ6722+ζ6710 | orthogonal faithful |
| ρ13 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6763+ζ6753+ζ6749+ζ6718+ζ6714+ζ674 | ζ6750+ζ6743+ζ6741+ζ6726+ζ6724+ζ6717 | ζ6766+ζ6738+ζ6737+ζ6730+ζ6729+ζ67 | ζ6759+ζ6739+ζ6736+ζ6731+ζ6728+ζ678 | ζ6757+ζ6745+ζ6735+ζ6732+ζ6722+ζ6710 | ζ6762+ζ6756+ζ6751+ζ6716+ζ6711+ζ675 | ζ6764+ζ6747+ζ6744+ζ6723+ζ6720+ζ673 | ζ6761+ζ6746+ζ6740+ζ6727+ζ6721+ζ676 | ζ6765+ζ6760+ζ6758+ζ679+ζ677+ζ672 | ζ6755+ζ6754+ζ6742+ζ6725+ζ6713+ζ6712 | ζ6752+ζ6748+ζ6734+ζ6733+ζ6719+ζ6715 | orthogonal faithful |
| ρ14 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6752+ζ6748+ζ6734+ζ6733+ζ6719+ζ6715 | ζ6764+ζ6747+ζ6744+ζ6723+ζ6720+ζ673 | ζ6755+ζ6754+ζ6742+ζ6725+ζ6713+ζ6712 | ζ6766+ζ6738+ζ6737+ζ6730+ζ6729+ζ67 | ζ6763+ζ6753+ζ6749+ζ6718+ζ6714+ζ674 | ζ6765+ζ6760+ζ6758+ζ679+ζ677+ζ672 | ζ6759+ζ6739+ζ6736+ζ6731+ζ6728+ζ678 | ζ6762+ζ6756+ζ6751+ζ6716+ζ6711+ζ675 | ζ6750+ζ6743+ζ6741+ζ6726+ζ6724+ζ6717 | ζ6757+ζ6745+ζ6735+ζ6732+ζ6722+ζ6710 | ζ6761+ζ6746+ζ6740+ζ6727+ζ6721+ζ676 | orthogonal faithful |
| ρ15 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6764+ζ6747+ζ6744+ζ6723+ζ6720+ζ673 | ζ6763+ζ6753+ζ6749+ζ6718+ζ6714+ζ674 | ζ6762+ζ6756+ζ6751+ζ6716+ζ6711+ζ675 | ζ6761+ζ6746+ζ6740+ζ6727+ζ6721+ζ676 | ζ6750+ζ6743+ζ6741+ζ6726+ζ6724+ζ6717 | ζ6755+ζ6754+ζ6742+ζ6725+ζ6713+ζ6712 | ζ6752+ζ6748+ζ6734+ζ6733+ζ6719+ζ6715 | ζ6766+ζ6738+ζ6737+ζ6730+ζ6729+ζ67 | ζ6757+ζ6745+ζ6735+ζ6732+ζ6722+ζ6710 | ζ6765+ζ6760+ζ6758+ζ679+ζ677+ζ672 | ζ6759+ζ6739+ζ6736+ζ6731+ζ6728+ζ678 | orthogonal faithful |
| ρ16 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6750+ζ6743+ζ6741+ζ6726+ζ6724+ζ6717 | ζ6757+ζ6745+ζ6735+ζ6732+ζ6722+ζ6710 | ζ6761+ζ6746+ζ6740+ζ6727+ζ6721+ζ676 | ζ6752+ζ6748+ζ6734+ζ6733+ζ6719+ζ6715 | ζ6765+ζ6760+ζ6758+ζ679+ζ677+ζ672 | ζ6766+ζ6738+ζ6737+ζ6730+ζ6729+ζ67 | ζ6763+ζ6753+ζ6749+ζ6718+ζ6714+ζ674 | ζ6759+ζ6739+ζ6736+ζ6731+ζ6728+ζ678 | ζ6755+ζ6754+ζ6742+ζ6725+ζ6713+ζ6712 | ζ6762+ζ6756+ζ6751+ζ6716+ζ6711+ζ675 | ζ6764+ζ6747+ζ6744+ζ6723+ζ6720+ζ673 | orthogonal faithful |
| ρ17 | 6 | 0 | 0 | 0 | 0 | 0 | ζ6765+ζ6760+ζ6758+ζ679+ζ677+ζ672 | ζ6755+ζ6754+ζ6742+ζ6725+ζ6713+ζ6712 | ζ6752+ζ6748+ζ6734+ζ6733+ζ6719+ζ6715 | ζ6763+ζ6753+ζ6749+ζ6718+ζ6714+ζ674 | ζ6762+ζ6756+ζ6751+ζ6716+ζ6711+ζ675 | ζ6759+ζ6739+ζ6736+ζ6731+ζ6728+ζ678 | ζ6757+ζ6745+ζ6735+ζ6732+ζ6722+ζ6710 | ζ6764+ζ6747+ζ6744+ζ6723+ζ6720+ζ673 | ζ6766+ζ6738+ζ6737+ζ6730+ζ6729+ζ67 | ζ6761+ζ6746+ζ6740+ζ6727+ζ6721+ζ676 | ζ6750+ζ6743+ζ6741+ζ6726+ζ6724+ζ6717 | orthogonal faithful |
►On 67 points: primitive
Generators in S67
(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 62 63 64 65 66 67)
(2 39 38 67 30 31)(3 10 8 66 59 61)(4 48 45 65 21 24)(5 19 15 64 50 54)(6 57 52 63 12 17)(7 28 22 62 41 47)(9 37 29 60 32 40)(11 46 36 58 23 33)(13 55 43 56 14 26)(16 35 20 53 34 49)(18 44 27 51 25 42)
G:=sub<Sym(67)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67), (2,39,38,67,30,31)(3,10,8,66,59,61)(4,48,45,65,21,24)(5,19,15,64,50,54)(6,57,52,63,12,17)(7,28,22,62,41,47)(9,37,29,60,32,40)(11,46,36,58,23,33)(13,55,43,56,14,26)(16,35,20,53,34,49)(18,44,27,51,25,42)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67), (2,39,38,67,30,31)(3,10,8,66,59,61)(4,48,45,65,21,24)(5,19,15,64,50,54)(6,57,52,63,12,17)(7,28,22,62,41,47)(9,37,29,60,32,40)(11,46,36,58,23,33)(13,55,43,56,14,26)(16,35,20,53,34,49)(18,44,27,51,25,42) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67)], [(2,39,38,67,30,31),(3,10,8,66,59,61),(4,48,45,65,21,24),(5,19,15,64,50,54),(6,57,52,63,12,17),(7,28,22,62,41,47),(9,37,29,60,32,40),(11,46,36,58,23,33),(13,55,43,56,14,26),(16,35,20,53,34,49),(18,44,27,51,25,42)]])
Matrix representation of C67⋊C6 ►in GL6(𝔽1609)
| 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 |
| 1608 | 208 | 802 | 234 | 802 | 208 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1105 | 595 | 828 | 792 | 74 | 1587 |
| 577 | 1073 | 814 | 31 | 993 | 1608 |
| 1176 | 1410 | 176 | 113 | 605 | 457 |
| 1249 | 1304 | 1477 | 1208 | 1283 | 1036 |
| 24 | 433 | 764 | 964 | 25 | 412 |
G:=sub<GL(6,GF(1609))| [0,0,0,0,0,1608,1,0,0,0,0,208,0,1,0,0,0,802,0,0,1,0,0,234,0,0,0,1,0,802,0,0,0,0,1,208],[1,1105,577,1176,1249,24,0,595,1073,1410,1304,433,0,828,814,176,1477,764,0,792,31,113,1208,964,0,74,993,605,1283,25,0,1587,1608,457,1036,412] >;
C67⋊C6 in GAP, Magma, Sage, TeX
C_{67}\rtimes C_6 % in TeX
G:=Group("C67:C6"); // GroupNames label
G:=SmallGroup(402,1);
// by ID
G=gap.SmallGroup(402,1);
# by ID
G:=PCGroup([3,-2,-3,-67,3566,1004]);
// Polycyclic
G:=Group<a,b|a^67=b^6=1,b*a*b^-1=a^30>;
// generators/relations
Export
Subgroup lattice of C67⋊C6 in TeX
Character table of C67⋊C6 in TeX