non-abelian, supersoluble, monomial
Aliases: He3.3Dic3, 3- 1+2⋊2Dic3, (C3×C18).7S3, (C3×C9)⋊5Dic3, He3.C3⋊1C4, (C2×He3).5S3, C6.3(He3⋊C2), C3.3(He3⋊3C4), C2.(He3.3S3), C32.2(C3⋊Dic3), (C2×3- 1+2).2S3, (C3×C6).2(C3⋊S3), (C2×He3.C3).1C2, SmallGroup(324,23)
Series: Derived ►Chief ►Lower central ►Upper central
| He3.C3 — He3.3Dic3 |
Generators and relations for He3.3Dic3
G = < a,b,c,d,e | a3=b3=c3=1, d6=ebe-1=b-1, e2=b-1d3, ab=ba, cac-1=eae-1=ab-1, ad=da, bc=cb, bd=db, dcd-1=ab-1c, ece-1=c-1, ede-1=bd5 >
Character table of He3.3Dic3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 9A | 9B | 9C | 9D | 9E | 12A | 12B | 12C | 12D | 18A | 18B | 18C | 18D | 18E | |
| size | 1 | 1 | 2 | 3 | 3 | 18 | 27 | 27 | 2 | 3 | 3 | 18 | 6 | 6 | 6 | 18 | 18 | 27 | 27 | 27 | 27 | 6 | 6 | 6 | 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 | 1 | i | -i | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | -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 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | -2 | 2 | 2 | 2 | -1 | 0 | 0 | -2 | -2 | -2 | 1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -2 | symplectic lifted from Dic3, Schur index 2 |
| ρ10 | 2 | -2 | 2 | 2 | 2 | -1 | 0 | 0 | -2 | -2 | -2 | 1 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ11 | 2 | -2 | 2 | 2 | 2 | -1 | 0 | 0 | -2 | -2 | -2 | 1 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ12 | 2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ13 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | -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 | complex lifted from He3⋊C2 |
| ρ14 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | -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 | complex lifted from He3⋊C2 |
| ρ15 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 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 | complex lifted from He3⋊C2 |
| ρ16 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 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 | complex lifted from He3⋊C2 |
| ρ17 | 3 | -3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | -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 | complex lifted from He3⋊3C4 |
| ρ18 | 3 | -3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | -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 | complex lifted from He3⋊3C4 |
| ρ19 | 3 | -3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 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 | complex lifted from He3⋊3C4 |
| ρ20 | 3 | -3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 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 | complex lifted from He3⋊3C4 |
| ρ21 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | 0 | 0 | orthogonal lifted from He3.3S3 |
| ρ22 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | 0 | 0 | orthogonal lifted from He3.3S3 |
| ρ23 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 0 | 0 | orthogonal lifted from He3.3S3 |
| ρ24 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ25 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | ζ98+ζ94-ζ92+2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ26 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | ζ98+ζ94-ζ92+2ζ9 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ97-ζ94+2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 108 points
Generators in S108
(1 26 106)(2 27 107)(3 28 108)(4 29 91)(5 30 92)(6 31 93)(7 32 94)(8 33 95)(9 34 96)(10 35 97)(11 36 98)(12 19 99)(13 20 100)(14 21 101)(15 22 102)(16 23 103)(17 24 104)(18 25 105)(37 90 64)(38 73 65)(39 74 66)(40 75 67)(41 76 68)(42 77 69)(43 78 70)(44 79 71)(45 80 72)(46 81 55)(47 82 56)(48 83 57)(49 84 58)(50 85 59)(51 86 60)(52 87 61)(53 88 62)(54 89 63)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 67 61)(56 68 62)(57 69 63)(58 70 64)(59 71 65)(60 72 66)(73 85 79)(74 86 80)(75 87 81)(76 88 82)(77 89 83)(78 90 84)(91 103 97)(92 104 98)(93 105 99)(94 106 100)(95 107 101)(96 108 102)
(2 27 101)(3 108 34)(5 30 104)(6 93 19)(8 33 107)(9 96 22)(11 36 92)(12 99 25)(14 21 95)(15 102 28)(17 24 98)(18 105 31)(20 32 26)(23 35 29)(37 78 70)(38 71 73)(39 45 51)(40 81 55)(41 56 76)(42 48 54)(43 84 58)(44 59 79)(46 87 61)(47 62 82)(49 90 64)(50 65 85)(52 75 67)(53 68 88)(57 69 63)(60 72 66)(91 97 103)(94 100 106)
(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 80 10 89)(2 79 11 88)(3 78 12 87)(4 77 13 86)(5 76 14 85)(6 75 15 84)(7 74 16 83)(8 73 17 82)(9 90 18 81)(19 67 28 58)(20 66 29 57)(21 65 30 56)(22 64 31 55)(23 63 32 72)(24 62 33 71)(25 61 34 70)(26 60 35 69)(27 59 36 68)(37 99 46 108)(38 98 47 107)(39 97 48 106)(40 96 49 105)(41 95 50 104)(42 94 51 103)(43 93 52 102)(44 92 53 101)(45 91 54 100)
G:=sub<Sym(108)| (1,26,106)(2,27,107)(3,28,108)(4,29,91)(5,30,92)(6,31,93)(7,32,94)(8,33,95)(9,34,96)(10,35,97)(11,36,98)(12,19,99)(13,20,100)(14,21,101)(15,22,102)(16,23,103)(17,24,104)(18,25,105)(37,90,64)(38,73,65)(39,74,66)(40,75,67)(41,76,68)(42,77,69)(43,78,70)(44,79,71)(45,80,72)(46,81,55)(47,82,56)(48,83,57)(49,84,58)(50,85,59)(51,86,60)(52,87,61)(53,88,62)(54,89,63), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,67,61)(56,68,62)(57,69,63)(58,70,64)(59,71,65)(60,72,66)(73,85,79)(74,86,80)(75,87,81)(76,88,82)(77,89,83)(78,90,84)(91,103,97)(92,104,98)(93,105,99)(94,106,100)(95,107,101)(96,108,102), (2,27,101)(3,108,34)(5,30,104)(6,93,19)(8,33,107)(9,96,22)(11,36,92)(12,99,25)(14,21,95)(15,102,28)(17,24,98)(18,105,31)(20,32,26)(23,35,29)(37,78,70)(38,71,73)(39,45,51)(40,81,55)(41,56,76)(42,48,54)(43,84,58)(44,59,79)(46,87,61)(47,62,82)(49,90,64)(50,65,85)(52,75,67)(53,68,88)(57,69,63)(60,72,66)(91,97,103)(94,100,106), (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), (1,80,10,89)(2,79,11,88)(3,78,12,87)(4,77,13,86)(5,76,14,85)(6,75,15,84)(7,74,16,83)(8,73,17,82)(9,90,18,81)(19,67,28,58)(20,66,29,57)(21,65,30,56)(22,64,31,55)(23,63,32,72)(24,62,33,71)(25,61,34,70)(26,60,35,69)(27,59,36,68)(37,99,46,108)(38,98,47,107)(39,97,48,106)(40,96,49,105)(41,95,50,104)(42,94,51,103)(43,93,52,102)(44,92,53,101)(45,91,54,100)>;
G:=Group( (1,26,106)(2,27,107)(3,28,108)(4,29,91)(5,30,92)(6,31,93)(7,32,94)(8,33,95)(9,34,96)(10,35,97)(11,36,98)(12,19,99)(13,20,100)(14,21,101)(15,22,102)(16,23,103)(17,24,104)(18,25,105)(37,90,64)(38,73,65)(39,74,66)(40,75,67)(41,76,68)(42,77,69)(43,78,70)(44,79,71)(45,80,72)(46,81,55)(47,82,56)(48,83,57)(49,84,58)(50,85,59)(51,86,60)(52,87,61)(53,88,62)(54,89,63), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,67,61)(56,68,62)(57,69,63)(58,70,64)(59,71,65)(60,72,66)(73,85,79)(74,86,80)(75,87,81)(76,88,82)(77,89,83)(78,90,84)(91,103,97)(92,104,98)(93,105,99)(94,106,100)(95,107,101)(96,108,102), (2,27,101)(3,108,34)(5,30,104)(6,93,19)(8,33,107)(9,96,22)(11,36,92)(12,99,25)(14,21,95)(15,102,28)(17,24,98)(18,105,31)(20,32,26)(23,35,29)(37,78,70)(38,71,73)(39,45,51)(40,81,55)(41,56,76)(42,48,54)(43,84,58)(44,59,79)(46,87,61)(47,62,82)(49,90,64)(50,65,85)(52,75,67)(53,68,88)(57,69,63)(60,72,66)(91,97,103)(94,100,106), (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), (1,80,10,89)(2,79,11,88)(3,78,12,87)(4,77,13,86)(5,76,14,85)(6,75,15,84)(7,74,16,83)(8,73,17,82)(9,90,18,81)(19,67,28,58)(20,66,29,57)(21,65,30,56)(22,64,31,55)(23,63,32,72)(24,62,33,71)(25,61,34,70)(26,60,35,69)(27,59,36,68)(37,99,46,108)(38,98,47,107)(39,97,48,106)(40,96,49,105)(41,95,50,104)(42,94,51,103)(43,93,52,102)(44,92,53,101)(45,91,54,100) );
G=PermutationGroup([[(1,26,106),(2,27,107),(3,28,108),(4,29,91),(5,30,92),(6,31,93),(7,32,94),(8,33,95),(9,34,96),(10,35,97),(11,36,98),(12,19,99),(13,20,100),(14,21,101),(15,22,102),(16,23,103),(17,24,104),(18,25,105),(37,90,64),(38,73,65),(39,74,66),(40,75,67),(41,76,68),(42,77,69),(43,78,70),(44,79,71),(45,80,72),(46,81,55),(47,82,56),(48,83,57),(49,84,58),(50,85,59),(51,86,60),(52,87,61),(53,88,62),(54,89,63)], [(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12),(19,31,25),(20,32,26),(21,33,27),(22,34,28),(23,35,29),(24,36,30),(37,49,43),(38,50,44),(39,51,45),(40,52,46),(41,53,47),(42,54,48),(55,67,61),(56,68,62),(57,69,63),(58,70,64),(59,71,65),(60,72,66),(73,85,79),(74,86,80),(75,87,81),(76,88,82),(77,89,83),(78,90,84),(91,103,97),(92,104,98),(93,105,99),(94,106,100),(95,107,101),(96,108,102)], [(2,27,101),(3,108,34),(5,30,104),(6,93,19),(8,33,107),(9,96,22),(11,36,92),(12,99,25),(14,21,95),(15,102,28),(17,24,98),(18,105,31),(20,32,26),(23,35,29),(37,78,70),(38,71,73),(39,45,51),(40,81,55),(41,56,76),(42,48,54),(43,84,58),(44,59,79),(46,87,61),(47,62,82),(49,90,64),(50,65,85),(52,75,67),(53,68,88),(57,69,63),(60,72,66),(91,97,103),(94,100,106)], [(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)], [(1,80,10,89),(2,79,11,88),(3,78,12,87),(4,77,13,86),(5,76,14,85),(6,75,15,84),(7,74,16,83),(8,73,17,82),(9,90,18,81),(19,67,28,58),(20,66,29,57),(21,65,30,56),(22,64,31,55),(23,63,32,72),(24,62,33,71),(25,61,34,70),(26,60,35,69),(27,59,36,68),(37,99,46,108),(38,98,47,107),(39,97,48,106),(40,96,49,105),(41,95,50,104),(42,94,51,103),(43,93,52,102),(44,92,53,101),(45,91,54,100)]])
Matrix representation of He3.3Dic3 ►in GL9(𝔽37)
| 26 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 36 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 27 | 0 | 26 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 36 | 36 |
| 12 | 34 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 25 | 36 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 14 | 34 | 20 | 23 | 14 | 34 |
| 0 | 0 | 0 | 3 | 17 | 14 | 34 | 3 | 17 |
| 0 | 0 | 0 | 14 | 34 | 14 | 34 | 20 | 23 |
| 0 | 0 | 0 | 3 | 17 | 3 | 17 | 14 | 34 |
| 0 | 0 | 0 | 20 | 23 | 14 | 34 | 14 | 34 |
| 0 | 0 | 0 | 14 | 34 | 3 | 17 | 3 | 17 |
| 31 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 0 | 31 | 0 | 0 | 0 | 0 | 0 | 0 |
| 29 | 31 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 36 | 36 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(9,GF(37))| [26,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36],[1,1,27,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36],[12,4,2,0,0,0,0,0,0,34,25,21,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,14,3,14,3,20,14,0,0,0,34,17,34,17,23,34,0,0,0,20,14,14,3,14,3,0,0,0,23,34,34,17,34,17,0,0,0,14,3,20,14,14,3,0,0,0,34,17,23,34,34,17],[31,8,29,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,1] >;
He3.3Dic3 in GAP, Magma, Sage, TeX
{\rm He}_3._3{\rm Dic}_3 % in TeX
G:=Group("He3.3Dic3"); // GroupNames label
G:=SmallGroup(324,23);
// by ID
G=gap.SmallGroup(324,23);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,12,146,5763,303,237,7564,1096,7781]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=1,d^6=e*b*e^-1=b^-1,e^2=b^-1*d^3,a*b=b*a,c*a*c^-1=e*a*e^-1=a*b^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*b^-1*c,e*c*e^-1=c^-1,e*d*e^-1=b*d^5>;
// generators/relations
Export
Subgroup lattice of He3.3Dic3 in TeX
Character table of He3.3Dic3 in TeX