metabelian, soluble, monomial, A-group
Aliases: C34⋊3C4, C33⋊8Dic3, C3⋊(C33⋊C4), C32⋊3(C32⋊C4), C32⋊3(C3⋊Dic3), C3⋊S3.(C3⋊S3), (C3×C3⋊S3).6S3, (C32×C3⋊S3).3C2, SmallGroup(324,163)
Series: Derived ►Chief ►Lower central ►Upper central
| C34 — C34⋊C4 |
Generators and relations for C34⋊C4
G = < a,b,c,d,e | a3=b3=c3=d3=e4=1, ebe-1=ab=ba, ac=ca, ad=da, eae-1=a-1b, bc=cb, bd=db, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 564 in 104 conjugacy classes, 19 normal (7 characteristic)
C1, C2, C3, C3, C4, S3, C6, C32, C32, Dic3, C3×S3, C3⋊S3, C3×C6, C33, C33, C3⋊Dic3, C32⋊C4, S3×C32, C3×C3⋊S3, C34, C33⋊C4, C32×C3⋊S3, C34⋊C4
Quotients: C1, C2, C4, S3, Dic3, C3⋊S3, C3⋊Dic3, C32⋊C4, C33⋊C4, C34⋊C4
Character table of C34⋊C4
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 3K | 3L | 3M | 3N | 3O | 3P | 3Q | 3R | 3S | 3T | 3U | 3V | 4A | 4B | 6A | 6B | 6C | 6D | |
| size | 1 | 9 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 81 | 81 | 18 | 18 | 18 | 18 | |
| 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 | i | -i | -1 | -1 | -1 | -1 | linear of order 4 |
| 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 | -i | i | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | -2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | 1 | 1 | 1 | -2 | symplectic lifted from Dic3, Schur index 2 |
| ρ10 | 2 | -2 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | 1 | -2 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ11 | 2 | -2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | -2 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ12 | 2 | -2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | -2 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ13 | 4 | 0 | 4 | 4 | 4 | 4 | 1 | -2 | -2 | -2 | -2 | -2 | -2 | -2 | 1 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ14 | 4 | 0 | 4 | 4 | 4 | 4 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | 1 | -2 | -2 | -2 | -2 | -2 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ15 | 4 | 0 | -2 | 4 | -2 | -2 | 1 | -1+3√-3/2 | -1+3√-3/2 | 1 | -1-3√-3/2 | -1-3√-3/2 | -1+3√-3/2 | 1 | -2 | 1 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | -1-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ16 | 4 | 0 | -2 | 4 | -2 | -2 | -1+3√-3/2 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | 1 | -2 | -1-3√-3/2 | -1-3√-3/2 | 1 | -1+3√-3/2 | -1+3√-3/2 | -1-3√-3/2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ17 | 4 | 0 | 4 | -2 | -2 | -2 | -1-3√-3/2 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | -1+3√-3/2 | 1 | -1-3√-3/2 | -1+3√-3/2 | 1 | -1-3√-3/2 | -1+3√-3/2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ18 | 4 | 0 | 4 | -2 | -2 | -2 | 1 | -1-3√-3/2 | 1 | -1+3√-3/2 | -1-3√-3/2 | 1 | -1+3√-3/2 | -1-3√-3/2 | -2 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | -1+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ19 | 4 | 0 | -2 | -2 | -2 | 4 | 1 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | 1 | -1-3√-3/2 | -1-3√-3/2 | -1-3√-3/2 | -1+3√-3/2 | -1+3√-3/2 | -1+3√-3/2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ20 | 4 | 0 | -2 | -2 | 4 | -2 | -1-3√-3/2 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | -2 | -1+3√-3/2 | -1-3√-3/2 | -1+3√-3/2 | 1 | -1+3√-3/2 | 1 | -1-3√-3/2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ21 | 4 | 0 | 4 | -2 | -2 | -2 | 1 | -1+3√-3/2 | 1 | -1-3√-3/2 | -1+3√-3/2 | 1 | -1-3√-3/2 | -1+3√-3/2 | -2 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | -1-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ22 | 4 | 0 | -2 | -2 | 4 | -2 | -1+3√-3/2 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | -2 | -1-3√-3/2 | -1+3√-3/2 | -1-3√-3/2 | 1 | -1-3√-3/2 | 1 | -1+3√-3/2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ23 | 4 | 0 | -2 | -2 | 4 | -2 | 1 | -1-3√-3/2 | -1+3√-3/2 | -1-3√-3/2 | 1 | -1-3√-3/2 | 1 | -1+3√-3/2 | -2 | 1 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | -1+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ24 | 4 | 0 | -2 | 4 | -2 | -2 | 1 | -1-3√-3/2 | -1-3√-3/2 | 1 | -1+3√-3/2 | -1+3√-3/2 | -1-3√-3/2 | 1 | -2 | 1 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | -1+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ25 | 4 | 0 | -2 | -2 | 4 | -2 | 1 | -1+3√-3/2 | -1-3√-3/2 | -1+3√-3/2 | 1 | -1+3√-3/2 | 1 | -1-3√-3/2 | -2 | 1 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | -1-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ26 | 4 | 0 | 4 | -2 | -2 | -2 | -1+3√-3/2 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | -1-3√-3/2 | 1 | -1+3√-3/2 | -1-3√-3/2 | 1 | -1+3√-3/2 | -1-3√-3/2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ27 | 4 | 0 | -2 | 4 | -2 | -2 | -1-3√-3/2 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | 1 | -2 | -1+3√-3/2 | -1+3√-3/2 | 1 | -1-3√-3/2 | -1-3√-3/2 | -1+3√-3/2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ28 | 4 | 0 | -2 | -2 | -2 | 4 | -2 | 1 | -1+3√-3/2 | -1+3√-3/2 | -1+3√-3/2 | -1-3√-3/2 | -1-3√-3/2 | -1-3√-3/2 | -2 | 1 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ29 | 4 | 0 | -2 | -2 | -2 | 4 | 1 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | 1 | -1+3√-3/2 | -1+3√-3/2 | -1+3√-3/2 | -1-3√-3/2 | -1-3√-3/2 | -1-3√-3/2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ30 | 4 | 0 | -2 | -2 | -2 | 4 | -2 | 1 | -1-3√-3/2 | -1-3√-3/2 | -1-3√-3/2 | -1+3√-3/2 | -1+3√-3/2 | -1+3√-3/2 | -2 | 1 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
►On 36 points
Generators in S36
(1 7 16)(2 19 34)(3 14 5)(4 36 17)(6 26 32)(8 30 28)(9 18 29)(10 22 13)(11 31 20)(12 15 24)(21 33 27)(23 25 35)
(1 29 21)(2 10 28)(3 23 31)(4 26 12)(5 35 11)(6 24 17)(7 9 33)(8 19 22)(13 30 34)(14 25 20)(15 36 32)(16 18 27)
(1 16 7)(2 8 13)(3 14 5)(4 6 15)(9 29 18)(10 19 30)(11 31 20)(12 17 32)(21 27 33)(22 34 28)(23 25 35)(24 36 26)
(1 21 29)(2 30 22)(3 23 31)(4 32 24)(5 35 11)(6 12 36)(7 33 9)(8 10 34)(13 19 28)(14 25 20)(15 17 26)(16 27 18)
(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)
G:=sub<Sym(36)| (1,7,16)(2,19,34)(3,14,5)(4,36,17)(6,26,32)(8,30,28)(9,18,29)(10,22,13)(11,31,20)(12,15,24)(21,33,27)(23,25,35), (1,29,21)(2,10,28)(3,23,31)(4,26,12)(5,35,11)(6,24,17)(7,9,33)(8,19,22)(13,30,34)(14,25,20)(15,36,32)(16,18,27), (1,16,7)(2,8,13)(3,14,5)(4,6,15)(9,29,18)(10,19,30)(11,31,20)(12,17,32)(21,27,33)(22,34,28)(23,25,35)(24,36,26), (1,21,29)(2,30,22)(3,23,31)(4,32,24)(5,35,11)(6,12,36)(7,33,9)(8,10,34)(13,19,28)(14,25,20)(15,17,26)(16,27,18), (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)>;
G:=Group( (1,7,16)(2,19,34)(3,14,5)(4,36,17)(6,26,32)(8,30,28)(9,18,29)(10,22,13)(11,31,20)(12,15,24)(21,33,27)(23,25,35), (1,29,21)(2,10,28)(3,23,31)(4,26,12)(5,35,11)(6,24,17)(7,9,33)(8,19,22)(13,30,34)(14,25,20)(15,36,32)(16,18,27), (1,16,7)(2,8,13)(3,14,5)(4,6,15)(9,29,18)(10,19,30)(11,31,20)(12,17,32)(21,27,33)(22,34,28)(23,25,35)(24,36,26), (1,21,29)(2,30,22)(3,23,31)(4,32,24)(5,35,11)(6,12,36)(7,33,9)(8,10,34)(13,19,28)(14,25,20)(15,17,26)(16,27,18), (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) );
G=PermutationGroup([[(1,7,16),(2,19,34),(3,14,5),(4,36,17),(6,26,32),(8,30,28),(9,18,29),(10,22,13),(11,31,20),(12,15,24),(21,33,27),(23,25,35)], [(1,29,21),(2,10,28),(3,23,31),(4,26,12),(5,35,11),(6,24,17),(7,9,33),(8,19,22),(13,30,34),(14,25,20),(15,36,32),(16,18,27)], [(1,16,7),(2,8,13),(3,14,5),(4,6,15),(9,29,18),(10,19,30),(11,31,20),(12,17,32),(21,27,33),(22,34,28),(23,25,35),(24,36,26)], [(1,21,29),(2,30,22),(3,23,31),(4,32,24),(5,35,11),(6,12,36),(7,33,9),(8,10,34),(13,19,28),(14,25,20),(15,17,26),(16,27,18)], [(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)]])
Matrix representation of C34⋊C4 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 6 | 0 | 1 | 0 |
| 0 | 0 | 7 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 2 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 | 9 | 0 |
| 0 | 0 | 11 | 0 | 0 | 9 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 | 9 | 0 |
| 0 | 0 | 11 | 0 | 0 | 9 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 8 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,3,6,7,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,3,0,2,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,2,11,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,3,0,2,11,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[5,8,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,2,1,8,5,0,0,0,1,0,0] >;
C34⋊C4 in GAP, Magma, Sage, TeX
C_3^4\rtimes C_4
% in TeX
G:=Group("C3^4:C4"); // GroupNames label
G:=SmallGroup(324,163);
// by ID
G=gap.SmallGroup(324,163);
# by ID
G:=PCGroup([6,-2,-2,-3,3,-3,-3,12,506,80,771,297,2164,7781]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^4=1,e*b*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1*b,b*c=c*b,b*d=d*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations
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