non-abelian, supersoluble, monomial
Aliases: 3- 1+2.Dic3, C3.He3⋊C4, (C3×C18).9S3, (C3×C9).2Dic3, C6.5(He3⋊C2), C3.5(He3⋊3C4), C32.4(C3⋊Dic3), (C2×3- 1+2).3S3, C2.(3- 1+2.S3), (C3×C6).4(C3⋊S3), (C2×C3.He3).C2, SmallGroup(324,25)
Series: Derived ►Chief ►Lower central ►Upper central
| C3.He3 — 3- 1+2.Dic3 |
Generators and relations for 3- 1+2.Dic3
G = < a,b,c,d | a9=b3=1, c6=a6, d2=a6c3, bab-1=a4, cac-1=a4b-1, dad-1=a-1, bc=cb, bd=db, dcd-1=a3c5 >
Character table of 3- 1+2.Dic3
| class | 1 | 2 | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 1 | 1 | 2 | 3 | 3 | 27 | 27 | 2 | 3 | 3 | 6 | 6 | 6 | 18 | 18 | 18 | 27 | 27 | 27 | 27 | 6 | 6 | 6 | 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 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ9 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -2 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ10 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -1 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ11 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ12 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | -2 | symplectic lifted from Dic3, Schur index 2 |
| ρ13 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 1 | 1 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ14 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1 | -1 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ15 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 1 | 1 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ16 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1 | -1 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ17 | 3 | -3 | 3 | -3+3√-3/2 | -3-3√-3/2 | i | -i | -3 | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊3C4 |
| ρ18 | 3 | -3 | 3 | -3+3√-3/2 | -3-3√-3/2 | -i | i | -3 | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊3C4 |
| ρ19 | 3 | -3 | 3 | -3-3√-3/2 | -3+3√-3/2 | i | -i | -3 | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊3C4 |
| ρ20 | 3 | -3 | 3 | -3-3√-3/2 | -3+3√-3/2 | -i | i | -3 | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊3C4 |
| ρ21 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ22 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ23 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ24 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ25 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ26 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 108 points
Generators in S108
(1 20 43 13 32 37 7 26 49)(2 33 44 14 27 38 8 21 50)(3 28 45 15 22 39 9 34 51)(4 23 46 16 35 40 10 29 52)(5 36 47 17 30 41 11 24 53)(6 31 48 18 25 42 12 19 54)(55 73 105 67 85 99 61 79 93)(56 74 100 68 86 94 62 80 106)(57 75 95 69 87 107 63 81 101)(58 76 108 70 88 102 64 82 96)(59 77 103 71 89 97 65 83 91)(60 78 98 72 90 92 66 84 104)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(37 43 49)(38 44 50)(39 45 51)(40 46 52)(41 47 53)(42 48 54)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)(73 85 79)(74 86 80)(75 87 81)(76 88 82)(77 89 83)(78 90 84)
(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)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)
(1 62 10 71)(2 61 11 70)(3 60 12 69)(4 59 13 68)(5 58 14 67)(6 57 15 66)(7 56 16 65)(8 55 17 64)(9 72 18 63)(19 95 28 104)(20 94 29 103)(21 93 30 102)(22 92 31 101)(23 91 32 100)(24 108 33 99)(25 107 34 98)(26 106 35 97)(27 105 36 96)(37 74 46 83)(38 73 47 82)(39 90 48 81)(40 89 49 80)(41 88 50 79)(42 87 51 78)(43 86 52 77)(44 85 53 76)(45 84 54 75)
G:=sub<Sym(108)| (1,20,43,13,32,37,7,26,49)(2,33,44,14,27,38,8,21,50)(3,28,45,15,22,39,9,34,51)(4,23,46,16,35,40,10,29,52)(5,36,47,17,30,41,11,24,53)(6,31,48,18,25,42,12,19,54)(55,73,105,67,85,99,61,79,93)(56,74,100,68,86,94,62,80,106)(57,75,95,69,87,107,63,81,101)(58,76,108,70,88,102,64,82,96)(59,77,103,71,89,97,65,83,91)(60,78,98,72,90,92,66,84,104), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(37,43,49)(38,44,50)(39,45,51)(40,46,52)(41,47,53)(42,48,54)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72)(73,85,79)(74,86,80)(75,87,81)(76,88,82)(77,89,83)(78,90,84), (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108), (1,62,10,71)(2,61,11,70)(3,60,12,69)(4,59,13,68)(5,58,14,67)(6,57,15,66)(7,56,16,65)(8,55,17,64)(9,72,18,63)(19,95,28,104)(20,94,29,103)(21,93,30,102)(22,92,31,101)(23,91,32,100)(24,108,33,99)(25,107,34,98)(26,106,35,97)(27,105,36,96)(37,74,46,83)(38,73,47,82)(39,90,48,81)(40,89,49,80)(41,88,50,79)(42,87,51,78)(43,86,52,77)(44,85,53,76)(45,84,54,75)>;
G:=Group( (1,20,43,13,32,37,7,26,49)(2,33,44,14,27,38,8,21,50)(3,28,45,15,22,39,9,34,51)(4,23,46,16,35,40,10,29,52)(5,36,47,17,30,41,11,24,53)(6,31,48,18,25,42,12,19,54)(55,73,105,67,85,99,61,79,93)(56,74,100,68,86,94,62,80,106)(57,75,95,69,87,107,63,81,101)(58,76,108,70,88,102,64,82,96)(59,77,103,71,89,97,65,83,91)(60,78,98,72,90,92,66,84,104), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(37,43,49)(38,44,50)(39,45,51)(40,46,52)(41,47,53)(42,48,54)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72)(73,85,79)(74,86,80)(75,87,81)(76,88,82)(77,89,83)(78,90,84), (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108), (1,62,10,71)(2,61,11,70)(3,60,12,69)(4,59,13,68)(5,58,14,67)(6,57,15,66)(7,56,16,65)(8,55,17,64)(9,72,18,63)(19,95,28,104)(20,94,29,103)(21,93,30,102)(22,92,31,101)(23,91,32,100)(24,108,33,99)(25,107,34,98)(26,106,35,97)(27,105,36,96)(37,74,46,83)(38,73,47,82)(39,90,48,81)(40,89,49,80)(41,88,50,79)(42,87,51,78)(43,86,52,77)(44,85,53,76)(45,84,54,75) );
G=PermutationGroup([[(1,20,43,13,32,37,7,26,49),(2,33,44,14,27,38,8,21,50),(3,28,45,15,22,39,9,34,51),(4,23,46,16,35,40,10,29,52),(5,36,47,17,30,41,11,24,53),(6,31,48,18,25,42,12,19,54),(55,73,105,67,85,99,61,79,93),(56,74,100,68,86,94,62,80,106),(57,75,95,69,87,107,63,81,101),(58,76,108,70,88,102,64,82,96),(59,77,103,71,89,97,65,83,91),(60,78,98,72,90,92,66,84,104)], [(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12),(37,43,49),(38,44,50),(39,45,51),(40,46,52),(41,47,53),(42,48,54),(55,61,67),(56,62,68),(57,63,69),(58,64,70),(59,65,71),(60,66,72),(73,85,79),(74,86,80),(75,87,81),(76,88,82),(77,89,83),(78,90,84)], [(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),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)], [(1,62,10,71),(2,61,11,70),(3,60,12,69),(4,59,13,68),(5,58,14,67),(6,57,15,66),(7,56,16,65),(8,55,17,64),(9,72,18,63),(19,95,28,104),(20,94,29,103),(21,93,30,102),(22,92,31,101),(23,91,32,100),(24,108,33,99),(25,107,34,98),(26,106,35,97),(27,105,36,96),(37,74,46,83),(38,73,47,82),(39,90,48,81),(40,89,49,80),(41,88,50,79),(42,87,51,78),(43,86,52,77),(44,85,53,76),(45,84,54,75)]])
Matrix representation of 3- 1+2.Dic3 ►in GL6(𝔽37)
| 0 | 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 36 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 36 | 36 | 0 | 0 | 0 | 0 |
| 36 | 36 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 31 | 0 | 0 | 0 | 0 |
| 6 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 31 | 0 | 0 |
| 0 | 0 | 6 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 17 |
| 0 | 0 | 0 | 0 | 20 | 26 |
| 0 | 0 | 7 | 14 | 0 | 0 |
| 0 | 0 | 7 | 30 | 0 | 0 |
| 7 | 14 | 0 | 0 | 0 | 0 |
| 7 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 30 |
| 0 | 0 | 0 | 0 | 7 | 14 |
G:=sub<GL(6,GF(37))| [0,0,0,0,0,36,0,0,0,0,1,36,36,1,0,0,0,0,36,0,0,0,0,0,0,0,36,1,0,0,0,0,36,0,0,0],[36,1,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,1,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,6,0,0,0,0,31,17,0,0,0,0,0,0,11,6,0,0,0,0,31,17,0,0,0,0,0,0,6,20,0,0,0,0,17,26],[0,0,7,7,0,0,0,0,14,30,0,0,7,7,0,0,0,0,14,30,0,0,0,0,0,0,0,0,23,7,0,0,0,0,30,14] >;
3- 1+2.Dic3 in GAP, Magma, Sage, TeX
3_-^{1+2}.{\rm Dic}_3 % in TeX
G:=Group("ES-(3,1).Dic3"); // GroupNames label
G:=SmallGroup(324,25);
// by ID
G=gap.SmallGroup(324,25);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,12,2090,986,3171,303,453,7564,1096,7781]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^3=1,c^6=a^6,d^2=a^6*c^3,b*a*b^-1=a^4,c*a*c^-1=a^4*b^-1,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^3*c^5>;
// generators/relations
Export
Subgroup lattice of 3- 1+2.Dic3 in TeX
Character table of 3- 1+2.Dic3 in TeX