Aliases: C42⋊1C18, C42⋊C9⋊1C2, C42⋊2C2⋊C9, (C4×C12).1C6, (C22×C6).2A4, C3.(C42⋊C6), C23.1(C3.A4), (C2×C6).7(C2×A4), (C3×C42⋊2C2).C3, C22.3(C2×C3.A4), SmallGroup(288,74)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C42⋊C18 |
Generators and relations for C42⋊C18
G = < a,b,c | a4=b4=c18=1, ab=ba, cac-1=b-1, cbc-1=a-1b >
Character table of C42⋊C18
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 1 | 3 | 4 | 1 | 1 | 6 | 6 | 12 | 3 | 3 | 4 | 4 | 16 | 16 | 16 | 16 | 16 | 16 | 6 | 6 | 6 | 6 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | |
| 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 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | -1 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | -1 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ98 | ζ95 | ζ97 | ζ9 | ζ94 | ζ92 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ95 | ζ97 | ζ9 | ζ98 | ζ92 | ζ94 | linear of order 9 |
| ρ8 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ92 | ζ98 | ζ94 | ζ97 | ζ9 | ζ95 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ98 | ζ94 | ζ97 | ζ92 | ζ95 | ζ9 | linear of order 9 |
| ρ9 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ94 | ζ97 | ζ98 | ζ95 | ζ92 | ζ9 | ζ3 | ζ3 | ζ32 | ζ32 | ζ65 | ζ6 | -ζ97 | -ζ98 | -ζ95 | -ζ94 | -ζ9 | -ζ92 | linear of order 18 |
| ρ10 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ94 | ζ97 | ζ98 | ζ95 | ζ92 | ζ9 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ97 | ζ98 | ζ95 | ζ94 | ζ9 | ζ92 | linear of order 9 |
| ρ11 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ95 | ζ92 | ζ9 | ζ94 | ζ97 | ζ98 | ζ32 | ζ32 | ζ3 | ζ3 | ζ6 | ζ65 | -ζ92 | -ζ9 | -ζ94 | -ζ95 | -ζ98 | -ζ97 | linear of order 18 |
| ρ12 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ9 | ζ94 | ζ92 | ζ98 | ζ95 | ζ97 | ζ3 | ζ3 | ζ32 | ζ32 | ζ65 | ζ6 | -ζ94 | -ζ92 | -ζ98 | -ζ9 | -ζ97 | -ζ95 | linear of order 18 |
| ρ13 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ9 | ζ94 | ζ92 | ζ98 | ζ95 | ζ97 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ94 | ζ92 | ζ98 | ζ9 | ζ97 | ζ95 | linear of order 9 |
| ρ14 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ92 | ζ98 | ζ94 | ζ97 | ζ9 | ζ95 | ζ32 | ζ32 | ζ3 | ζ3 | ζ6 | ζ65 | -ζ98 | -ζ94 | -ζ97 | -ζ92 | -ζ95 | -ζ9 | linear of order 18 |
| ρ15 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ98 | ζ95 | ζ97 | ζ9 | ζ94 | ζ92 | ζ32 | ζ32 | ζ3 | ζ3 | ζ6 | ζ65 | -ζ95 | -ζ97 | -ζ9 | -ζ98 | -ζ92 | -ζ94 | linear of order 18 |
| ρ16 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ97 | ζ9 | ζ95 | ζ92 | ζ98 | ζ94 | ζ3 | ζ3 | ζ32 | ζ32 | ζ65 | ζ6 | -ζ9 | -ζ95 | -ζ92 | -ζ97 | -ζ94 | -ζ98 | linear of order 18 |
| ρ17 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ95 | ζ92 | ζ9 | ζ94 | ζ97 | ζ98 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ92 | ζ9 | ζ94 | ζ95 | ζ98 | ζ97 | linear of order 9 |
| ρ18 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ97 | ζ9 | ζ95 | ζ92 | ζ98 | ζ94 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ9 | ζ95 | ζ92 | ζ97 | ζ94 | ζ98 | linear of order 9 |
| ρ19 | 3 | 3 | -3 | 3 | 3 | -1 | -1 | 1 | 3 | 3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ20 | 3 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ21 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3.A4 |
| ρ22 | 3 | 3 | -3 | -3-3√-3/2 | -3+3√-3/2 | -1 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | ζ6 | ζ6 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×C3.A4 |
| ρ23 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3.A4 |
| ρ24 | 3 | 3 | -3 | -3+3√-3/2 | -3-3√-3/2 | -1 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | ζ65 | ζ65 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×C3.A4 |
| ρ25 | 6 | -2 | 0 | 6 | 6 | 2i | -2i | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C6 |
| ρ26 | 6 | -2 | 0 | 6 | 6 | -2i | 2i | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C6 |
| ρ27 | 6 | -2 | 0 | -3+3√-3 | -3-3√-3 | -2i | 2i | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ4ζ32 | 2ζ43ζ32 | 2ζ4ζ3 | 2ζ43ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ28 | 6 | -2 | 0 | -3-3√-3 | -3+3√-3 | 2i | -2i | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ43ζ3 | 2ζ4ζ3 | 2ζ43ζ32 | 2ζ4ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ29 | 6 | -2 | 0 | -3+3√-3 | -3-3√-3 | 2i | -2i | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ43ζ32 | 2ζ4ζ32 | 2ζ43ζ3 | 2ζ4ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ30 | 6 | -2 | 0 | -3-3√-3 | -3+3√-3 | -2i | 2i | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ4ζ3 | 2ζ43ζ3 | 2ζ4ζ32 | 2ζ43ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 72 points
Generators in S72
(1 33 24 10)(2 61 25 42)(3 52)(4 13 27 36)(5 64 28 45)(6 56)(7 21 30 16)(8 67 31 48)(9 40)(11 70 34 51)(12 62)(14 55 19 54)(15 46)(17 58 22 39)(18 68)(20 65)(23 49)(26 71)(29 37)(32 59)(35 43)(38 47 57 66)(41 69 60 50)(44 53 63 72)
(1 41 24 60)(2 51)(3 35 26 12)(4 44 27 63)(5 55)(6 15 29 20)(7 47 30 66)(8 39)(9 23 32 18)(10 50 33 69)(11 61)(13 53 36 72)(14 45)(16 38 21 57)(17 67)(19 64)(22 48)(25 70)(28 54)(31 58)(34 42)(37 65 56 46)(40 49 59 68)(43 71 62 52)
(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 68 69 70 71 72)
G:=sub<Sym(72)| (1,33,24,10)(2,61,25,42)(3,52)(4,13,27,36)(5,64,28,45)(6,56)(7,21,30,16)(8,67,31,48)(9,40)(11,70,34,51)(12,62)(14,55,19,54)(15,46)(17,58,22,39)(18,68)(20,65)(23,49)(26,71)(29,37)(32,59)(35,43)(38,47,57,66)(41,69,60,50)(44,53,63,72), (1,41,24,60)(2,51)(3,35,26,12)(4,44,27,63)(5,55)(6,15,29,20)(7,47,30,66)(8,39)(9,23,32,18)(10,50,33,69)(11,61)(13,53,36,72)(14,45)(16,38,21,57)(17,67)(19,64)(22,48)(25,70)(28,54)(31,58)(34,42)(37,65,56,46)(40,49,59,68)(43,71,62,52), (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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)>;
G:=Group( (1,33,24,10)(2,61,25,42)(3,52)(4,13,27,36)(5,64,28,45)(6,56)(7,21,30,16)(8,67,31,48)(9,40)(11,70,34,51)(12,62)(14,55,19,54)(15,46)(17,58,22,39)(18,68)(20,65)(23,49)(26,71)(29,37)(32,59)(35,43)(38,47,57,66)(41,69,60,50)(44,53,63,72), (1,41,24,60)(2,51)(3,35,26,12)(4,44,27,63)(5,55)(6,15,29,20)(7,47,30,66)(8,39)(9,23,32,18)(10,50,33,69)(11,61)(13,53,36,72)(14,45)(16,38,21,57)(17,67)(19,64)(22,48)(25,70)(28,54)(31,58)(34,42)(37,65,56,46)(40,49,59,68)(43,71,62,52), (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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72) );
G=PermutationGroup([[(1,33,24,10),(2,61,25,42),(3,52),(4,13,27,36),(5,64,28,45),(6,56),(7,21,30,16),(8,67,31,48),(9,40),(11,70,34,51),(12,62),(14,55,19,54),(15,46),(17,58,22,39),(18,68),(20,65),(23,49),(26,71),(29,37),(32,59),(35,43),(38,47,57,66),(41,69,60,50),(44,53,63,72)], [(1,41,24,60),(2,51),(3,35,26,12),(4,44,27,63),(5,55),(6,15,29,20),(7,47,30,66),(8,39),(9,23,32,18),(10,50,33,69),(11,61),(13,53,36,72),(14,45),(16,38,21,57),(17,67),(19,64),(22,48),(25,70),(28,54),(31,58),(34,42),(37,65,56,46),(40,49,59,68),(43,71,62,52)], [(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),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)]])
Matrix representation of C42⋊C18 ►in GL6(𝔽37)
| 1 | 35 | 0 | 0 | 0 | 0 |
| 1 | 36 | 0 | 0 | 0 | 0 |
| 9 | 11 | 6 | 0 | 0 | 0 |
| 9 | 11 | 0 | 6 | 0 | 0 |
| 12 | 27 | 0 | 0 | 0 | 6 |
| 35 | 27 | 0 | 0 | 31 | 0 |
| 31 | 0 | 0 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 31 | 0 | 0 |
| 29 | 0 | 6 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 36 | 0 |
| 1 | 0 | 0 | 0 | 0 | 22 |
| 0 | 0 | 0 | 0 | 26 | 11 |
| 8 | 10 | 0 | 0 | 0 | 27 |
| 8 | 0 | 0 | 0 | 0 | 27 |
| 23 | 0 | 0 | 10 | 0 | 36 |
| 23 | 0 | 10 | 0 | 0 | 36 |
G:=sub<GL(6,GF(37))| [1,1,9,9,12,35,35,36,11,11,27,27,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,31,0,0,0,0,6,0],[31,0,0,29,2,12,0,31,0,0,0,0,0,0,0,6,0,0,0,0,31,0,0,0,0,0,0,0,0,36,0,0,0,0,1,0],[1,0,8,8,23,23,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,10,0,0,26,0,0,0,0,22,11,27,27,36,36] >;
C42⋊C18 in GAP, Magma, Sage, TeX
C_4^2\rtimes C_{18} % in TeX
G:=Group("C4^2:C18"); // GroupNames label
G:=SmallGroup(288,74);
// by ID
G=gap.SmallGroup(288,74);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,50,2523,514,360,6304,3476,102,3036,5305]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^18=1,a*b=b*a,c*a*c^-1=b^-1,c*b*c^-1=a^-1*b>;
// generators/relations
Export
Subgroup lattice of C42⋊C18 in TeX
Character table of C42⋊C18 in TeX