Aliases: C33⋊1(C4⋊C4), C33⋊5C4⋊2C4, Dic3⋊(C32⋊C4), C3⋊S3.3Dic6, (C32×Dic3)⋊2C4, C32⋊7(Dic3⋊C4), (C3×C3⋊S3).8D4, C2.5(S3×C32⋊C4), C6.3(C2×C32⋊C4), (C3×C3⋊S3).2Q8, (C3×C6).28(C4×S3), (C2×C3⋊S3).26D6, (C2×C32⋊C4).1S3, (C6×C32⋊C4).1C2, C3⋊2(C4⋊(C32⋊C4)), C3⋊S3.6(C3⋊D4), (C6×C3⋊S3).3C22, (Dic3×C3⋊S3).4C2, (C32×C6).3(C2×C4), (C2×C33⋊C4).2C2, SmallGroup(432,569)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33⋊(C4⋊C4)
G = < a,b,c,d,e | a3=b3=c3=d4=e4=1, ab=ba, ac=ca, dad-1=a-1, eae-1=ab-1, bc=cb, dbd-1=b-1, ebe-1=a-1b-1, dcd-1=c-1, ce=ec, ede-1=d-1 >
Subgroups: 704 in 96 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C32⋊C4, S3×C6, C2×C3⋊S3, Dic3⋊C4, C3×C3⋊S3, C32×C6, S3×Dic3, C4×C3⋊S3, C2×C32⋊C4, C2×C32⋊C4, C32×Dic3, C33⋊5C4, C3×C32⋊C4, C33⋊C4, C6×C3⋊S3, C4⋊(C32⋊C4), Dic3×C3⋊S3, C6×C32⋊C4, C2×C33⋊C4, C33⋊(C4⋊C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, Dic6, C4×S3, C3⋊D4, C32⋊C4, Dic3⋊C4, C2×C32⋊C4, C4⋊(C32⋊C4), S3×C32⋊C4, C33⋊(C4⋊C4)
Character table of C33⋊(C4⋊C4)
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 9 | 9 | 2 | 4 | 4 | 8 | 8 | 6 | 18 | 18 | 54 | 54 | 54 | 2 | 4 | 4 | 8 | 8 | 18 | 18 | 12 | 12 | 12 | 12 | 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 | -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 | 1 | 1 | -1 | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | i | -i | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | i | -i | i | -i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | -i | i | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | -i | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | 0 | -2 | -2 | 0 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | -2 | -2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -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 | 2 | -1 | -1 | 0 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | -2 | 2 | -2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 2 | -2 | 2 | -2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | √3 | √3 | -√3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ14 | 2 | -2 | 2 | -2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -√3 | -√3 | √3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -2 | -1 | 2 | 2 | -1 | -1 | 0 | -2i | 2i | 0 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | i | -i | i | -i | complex lifted from C4×S3 |
| ρ16 | 2 | 2 | -2 | -2 | -1 | 2 | 2 | -1 | -1 | 0 | 2i | -2i | 0 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -i | i | -i | i | complex lifted from C4×S3 |
| ρ17 | 2 | -2 | -2 | 2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -√-3 | √-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | -2 | -2 | 2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | √-3 | -√-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ19 | 4 | 4 | 0 | 0 | 4 | -2 | 1 | 1 | -2 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 1 | -2 | 1 | -2 | 0 | 0 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C32⋊C4 |
| ρ20 | 4 | 4 | 0 | 0 | 4 | 1 | -2 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | -2 | 1 | -2 | 1 | 0 | 0 | -1 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C32⋊C4 |
| ρ21 | 4 | 4 | 0 | 0 | 4 | 1 | -2 | -2 | 1 | 4 | 0 | 0 | 0 | 0 | 0 | 4 | -2 | 1 | -2 | 1 | 0 | 0 | 1 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ22 | 4 | 4 | 0 | 0 | 4 | -2 | 1 | 1 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 4 | 1 | -2 | 1 | -2 | 0 | 0 | -2 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ23 | 4 | -4 | 0 | 0 | 4 | 1 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | -1 | 2 | -1 | 0 | 0 | 3i | 0 | 0 | -3i | 0 | 0 | 0 | 0 | complex lifted from C4⋊(C32⋊C4) |
| ρ24 | 4 | -4 | 0 | 0 | 4 | -2 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | -1 | 2 | -1 | 2 | 0 | 0 | 0 | -3i | 3i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4⋊(C32⋊C4) |
| ρ25 | 4 | -4 | 0 | 0 | 4 | 1 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | -1 | 2 | -1 | 0 | 0 | -3i | 0 | 0 | 3i | 0 | 0 | 0 | 0 | complex lifted from C4⋊(C32⋊C4) |
| ρ26 | 4 | -4 | 0 | 0 | 4 | -2 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | -1 | 2 | -1 | 2 | 0 | 0 | 0 | 3i | -3i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4⋊(C32⋊C4) |
| ρ27 | 8 | 8 | 0 | 0 | -4 | 2 | -4 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | -4 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×C32⋊C4 |
| ρ28 | 8 | 8 | 0 | 0 | -4 | -4 | 2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | -4 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×C32⋊C4 |
| ρ29 | 8 | -8 | 0 | 0 | -4 | 2 | -4 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ30 | 8 | -8 | 0 | 0 | -4 | -4 | 2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -2 | 4 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(5 34 29)(6 30 35)(7 36 31)(8 32 33)(21 37 46)(22 47 38)(23 39 48)(24 45 40)
(1 16 17)(2 18 13)(3 14 19)(4 20 15)(5 34 29)(6 30 35)(7 36 31)(8 32 33)(9 25 42)(10 43 26)(11 27 44)(12 41 28)(21 37 46)(22 47 38)(23 39 48)(24 45 40)
(1 17 16)(2 13 18)(3 19 14)(4 15 20)(5 29 34)(6 35 30)(7 31 36)(8 33 32)(9 25 42)(10 43 26)(11 27 44)(12 41 28)(21 37 46)(22 47 38)(23 39 48)(24 45 40)
(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 47 43 8)(2 46 44 7)(3 45 41 6)(4 48 42 5)(9 29 15 23)(10 32 16 22)(11 31 13 21)(12 30 14 24)(17 38 26 33)(18 37 27 36)(19 40 28 35)(20 39 25 34)
G:=sub<Sym(48)| (5,34,29)(6,30,35)(7,36,31)(8,32,33)(21,37,46)(22,47,38)(23,39,48)(24,45,40), (1,16,17)(2,18,13)(3,14,19)(4,20,15)(5,34,29)(6,30,35)(7,36,31)(8,32,33)(9,25,42)(10,43,26)(11,27,44)(12,41,28)(21,37,46)(22,47,38)(23,39,48)(24,45,40), (1,17,16)(2,13,18)(3,19,14)(4,15,20)(5,29,34)(6,35,30)(7,31,36)(8,33,32)(9,25,42)(10,43,26)(11,27,44)(12,41,28)(21,37,46)(22,47,38)(23,39,48)(24,45,40), (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), (1,47,43,8)(2,46,44,7)(3,45,41,6)(4,48,42,5)(9,29,15,23)(10,32,16,22)(11,31,13,21)(12,30,14,24)(17,38,26,33)(18,37,27,36)(19,40,28,35)(20,39,25,34)>;
G:=Group( (5,34,29)(6,30,35)(7,36,31)(8,32,33)(21,37,46)(22,47,38)(23,39,48)(24,45,40), (1,16,17)(2,18,13)(3,14,19)(4,20,15)(5,34,29)(6,30,35)(7,36,31)(8,32,33)(9,25,42)(10,43,26)(11,27,44)(12,41,28)(21,37,46)(22,47,38)(23,39,48)(24,45,40), (1,17,16)(2,13,18)(3,19,14)(4,15,20)(5,29,34)(6,35,30)(7,31,36)(8,33,32)(9,25,42)(10,43,26)(11,27,44)(12,41,28)(21,37,46)(22,47,38)(23,39,48)(24,45,40), (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), (1,47,43,8)(2,46,44,7)(3,45,41,6)(4,48,42,5)(9,29,15,23)(10,32,16,22)(11,31,13,21)(12,30,14,24)(17,38,26,33)(18,37,27,36)(19,40,28,35)(20,39,25,34) );
G=PermutationGroup([[(5,34,29),(6,30,35),(7,36,31),(8,32,33),(21,37,46),(22,47,38),(23,39,48),(24,45,40)], [(1,16,17),(2,18,13),(3,14,19),(4,20,15),(5,34,29),(6,30,35),(7,36,31),(8,32,33),(9,25,42),(10,43,26),(11,27,44),(12,41,28),(21,37,46),(22,47,38),(23,39,48),(24,45,40)], [(1,17,16),(2,13,18),(3,19,14),(4,15,20),(5,29,34),(6,35,30),(7,31,36),(8,33,32),(9,25,42),(10,43,26),(11,27,44),(12,41,28),(21,37,46),(22,47,38),(23,39,48),(24,45,40)], [(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)], [(1,47,43,8),(2,46,44,7),(3,45,41,6),(4,48,42,5),(9,29,15,23),(10,32,16,22),(11,31,13,21),(12,30,14,24),(17,38,26,33),(18,37,27,36),(19,40,28,35),(20,39,25,34)]])
Matrix representation of C33⋊(C4⋊C4) ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 | 0 | 12 |
| 0 | 0 | 1 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 12 | 1 |
| 0 | 0 | 1 | 2 | 11 | 2 |
| 0 | 0 | 1 | 1 | 12 | 2 |
| 0 | 0 | 1 | 11 | 1 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 11 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 12 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 | 12 |
| 11 | 9 | 0 | 0 | 0 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 1 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 2 | 12 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,1,0,0,12,0,12,0,0,0,0,12,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,0,0,2,2,1,11,0,0,12,11,12,1,0,0,1,2,2,0],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,11,0,0,0,0,2,9,0,0,0,0,0,0,0,0,12,0,0,0,2,1,1,12,0,0,12,0,0,0,0,0,1,0,12,12],[11,4,0,0,0,0,9,2,0,0,0,0,0,0,1,0,12,12,0,0,12,12,0,2,0,0,0,0,0,12,0,0,1,0,0,0] >;
C33⋊(C4⋊C4) in GAP, Magma, Sage, TeX
C_3^3\rtimes (C_4\rtimes C_4)
% in TeX
G:=Group("C3^3:(C4:C4)"); // GroupNames label
G:=SmallGroup(432,569);
// by ID
G=gap.SmallGroup(432,569);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,141,36,1411,298,1356,1027,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,e*a*e^-1=a*b^-1,b*c=c*b,d*b*d^-1=b^-1,e*b*e^-1=a^-1*b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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