metabelian, supersoluble, monomial
Aliases: C6.2Dic6, C62.5C22, C6.4(C4×S3), C22.7S32, (C3×C6).3Q8, C32⋊5(C4⋊C4), C3⋊Dic3⋊2C4, (C3×C6).16D4, (C2×C6).12D6, C3⋊2(Dic3⋊C4), C6.11(C3⋊D4), (C2×Dic3).2S3, (C6×Dic3).1C2, C2.2(D6⋊S3), C2.2(C32⋊2Q8), C2.5(C6.D6), (C3×C6).16(C2×C4), (C2×C3⋊Dic3).3C2, SmallGroup(144,67)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.C22
G = < a,b,c,d | a6=b6=1, c2=d2=a3, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=b3c >
Subgroups: 160 in 60 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C3×Dic3, C3⋊Dic3, C62, Dic3⋊C4, C6×Dic3, C2×C3⋊Dic3, C62.C22
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, Dic6, C4×S3, C3⋊D4, S32, Dic3⋊C4, C6.D6, D6⋊S3, C32⋊2Q8, C62.C22
Character table of C62.C22
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 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 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | i | -i | -i | i | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | i | -i | -i | i | i | i | -i | -i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | i | i | -i | -i | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -i | -i | i | i | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -i | i | i | -i | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -i | i | i | -i | -i | -i | i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | 0 | 2 | 0 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | 0 | -2 | 0 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | -1 | -1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ15 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -2 | -2 | 1 | 1 | -1 | 1 | 1 | 0 | -√3 | √3 | 0 | √3 | -√3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 1 | 1 | -2 | -2 | -1 | 1 | 1 | -√3 | 0 | 0 | √3 | 0 | 0 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -2 | -2 | 1 | 1 | -1 | 1 | 1 | 0 | √3 | -√3 | 0 | -√3 | √3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 2 | -2 | -2 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 1 | 1 | -2 | -2 | -1 | 1 | 1 | √3 | 0 | 0 | -√3 | 0 | 0 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ19 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | 2 | -1 | 1 | 1 | 1 | -1 | 0 | -√-3 | √-3 | 0 | -√-3 | √-3 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | 2 | -1 | 1 | 1 | 1 | -1 | 0 | √-3 | -√-3 | 0 | √-3 | -√-3 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | -2 | -2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -1 | 2 | -2 | 1 | 1 | -1 | -√-3 | 0 | 0 | √-3 | 0 | 0 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | -2 | -2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -1 | 2 | -2 | 1 | 1 | -1 | √-3 | 0 | 0 | -√-3 | 0 | 0 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ23 | 2 | -2 | 2 | -2 | -1 | 2 | -1 | 0 | -2i | 0 | 2i | 0 | 0 | 1 | -2 | -1 | 1 | -2 | 2 | 1 | -1 | 1 | -i | 0 | 0 | -i | 0 | 0 | i | i | complex lifted from C4×S3 |
| ρ24 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | -2i | 0 | 2i | 0 | 0 | 0 | -2 | 1 | 2 | -2 | 1 | -1 | 1 | -1 | 1 | 0 | -i | -i | 0 | i | i | 0 | 0 | complex lifted from C4×S3 |
| ρ25 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | 2i | 0 | -2i | 0 | 0 | 0 | -2 | 1 | 2 | -2 | 1 | -1 | 1 | -1 | 1 | 0 | i | i | 0 | -i | -i | 0 | 0 | complex lifted from C4×S3 |
| ρ26 | 2 | -2 | 2 | -2 | -1 | 2 | -1 | 0 | 2i | 0 | -2i | 0 | 0 | 1 | -2 | -1 | 1 | -2 | 2 | 1 | -1 | 1 | i | 0 | 0 | i | 0 | 0 | -i | -i | complex lifted from C4×S3 |
| ρ27 | 4 | 4 | 4 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ28 | 4 | -4 | 4 | -4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | 2 | -2 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C6.D6 |
| ρ29 | 4 | 4 | -4 | -4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D6⋊S3, Schur index 2 |
| ρ30 | 4 | -4 | -4 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 2 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊2Q8, Schur index 2 |
►On 48 points
(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)
(1 17 5 15 3 13)(2 18 6 16 4 14)(7 47 11 45 9 43)(8 48 12 46 10 44)(19 29 21 25 23 27)(20 30 22 26 24 28)(31 38 33 40 35 42)(32 39 34 41 36 37)
(1 33 4 36)(2 34 5 31)(3 35 6 32)(7 27 10 30)(8 28 11 25)(9 29 12 26)(13 40 16 37)(14 41 17 38)(15 42 18 39)(19 46 22 43)(20 47 23 44)(21 48 24 45)
(1 21 4 24)(2 20 5 23)(3 19 6 22)(7 33 10 36)(8 32 11 35)(9 31 12 34)(13 29 16 26)(14 28 17 25)(15 27 18 30)(37 47 40 44)(38 46 41 43)(39 45 42 48)
G:=sub<Sym(48)| (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), (1,17,5,15,3,13)(2,18,6,16,4,14)(7,47,11,45,9,43)(8,48,12,46,10,44)(19,29,21,25,23,27)(20,30,22,26,24,28)(31,38,33,40,35,42)(32,39,34,41,36,37), (1,33,4,36)(2,34,5,31)(3,35,6,32)(7,27,10,30)(8,28,11,25)(9,29,12,26)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45), (1,21,4,24)(2,20,5,23)(3,19,6,22)(7,33,10,36)(8,32,11,35)(9,31,12,34)(13,29,16,26)(14,28,17,25)(15,27,18,30)(37,47,40,44)(38,46,41,43)(39,45,42,48)>;
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), (1,17,5,15,3,13)(2,18,6,16,4,14)(7,47,11,45,9,43)(8,48,12,46,10,44)(19,29,21,25,23,27)(20,30,22,26,24,28)(31,38,33,40,35,42)(32,39,34,41,36,37), (1,33,4,36)(2,34,5,31)(3,35,6,32)(7,27,10,30)(8,28,11,25)(9,29,12,26)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45), (1,21,4,24)(2,20,5,23)(3,19,6,22)(7,33,10,36)(8,32,11,35)(9,31,12,34)(13,29,16,26)(14,28,17,25)(15,27,18,30)(37,47,40,44)(38,46,41,43)(39,45,42,48) );
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)], [(1,17,5,15,3,13),(2,18,6,16,4,14),(7,47,11,45,9,43),(8,48,12,46,10,44),(19,29,21,25,23,27),(20,30,22,26,24,28),(31,38,33,40,35,42),(32,39,34,41,36,37)], [(1,33,4,36),(2,34,5,31),(3,35,6,32),(7,27,10,30),(8,28,11,25),(9,29,12,26),(13,40,16,37),(14,41,17,38),(15,42,18,39),(19,46,22,43),(20,47,23,44),(21,48,24,45)], [(1,21,4,24),(2,20,5,23),(3,19,6,22),(7,33,10,36),(8,32,11,35),(9,31,12,34),(13,29,16,26),(14,28,17,25),(15,27,18,30),(37,47,40,44),(38,46,41,43),(39,45,42,48)]])
C62.C22 is a maximal subgroup of
C62.3D4 C62.4D4 C62.6D4 C62.7D4 C62.8C23 C62.9C23 Dic3⋊6Dic6 Dic3.Dic6 C62.17C23 C62.20C23 C62.24C23 D6⋊6Dic6 C62.29C23 C62.31C23 C62.35C23 C62.37C23 C62.38C23 C62.39C23 C62.40C23 C62.43C23 C62.44C23 S3×Dic3⋊C4 C62.47C23 C62.58C23 D6⋊3Dic6 D6⋊4Dic6 C62.70C23 C4×D6⋊S3 C62.72C23 C62.82C23 C4×C32⋊2Q8 C12⋊Dic6 C62.94C23 C62.98C23 C62.99C23 C62.56D4 C62⋊3Q8 C62.111C23 C62.113C23 C62⋊7D4 C62⋊4Q8 C18.Dic6 C62.D6 C62.81D6 C62.85D6
C62.C22 is a maximal quotient of
C12.15Dic6 C12.6Dic6 C12.8Dic6 C62.5Q8 C62.6Q8 C18.Dic6 C62.3D6 C62.81D6 C62.85D6
Matrix representation of C62.C22 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 7 | 3 | 0 | 0 | 0 | 0 |
| 10 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,10,0,0,0,0,3,6,0,0,0,0,0,0,8,5,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C62.C22 in GAP, Magma, Sage, TeX
C_6^2.C_2^2
% in TeX
G:=Group("C6^2.C2^2"); // GroupNames label
G:=SmallGroup(144,67);
// by ID
G=gap.SmallGroup(144,67);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,31,490,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^2=d^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations
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