metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40:1D6, C24:7D10, D6.4D20, D120:13C2, C120:6C22, Dic10:8D6, D20.19D6, D60:17C22, C60.93C23, Dic3.6D20, C3:C8:1D10, C8:3(S3xD5), (S3xD20):9C2, C40:C2:1S3, C8:S3:1D5, C5:1(Q8:3D6), C10.3(S3xD4), C30.7(C2xD4), C6.3(C2xD20), C2.8(S3xD20), C3:D40:10C2, C3:1(C8:D10), C15:3(C8:C22), (S3xC10).1D4, (C4xS3).1D10, D60:C2:8C2, (C5xDic3).1D4, C15:SD16:10C2, C12.66(C22xD5), (S3xC20).24C22, C20.143(C22xS3), (C3xD20).21C22, (C3xDic10):13C22, C4.92(C2xS3xD5), (C5xC8:S3):1C2, (C3xC40:C2):1C2, (C5xC3:C8):15C22, SmallGroup(480,329)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40:1D6
G = < a,b,c | a40=b6=c2=1, bab-1=a19, cac=a-1, cbc=b-1 >
Subgroups: 1068 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2xC4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, D6, C2xC6, C15, M4(2), D8, SD16, C2xD4, C4oD4, Dic5, C20, C20, D10, C2xC10, C3:C8, C24, C4xS3, C4xS3, D12, C3:D4, C3xD4, C3xQ8, C22xS3, C5xS3, C3xD5, D15, C30, C8:C22, C40, C40, Dic10, C4xD5, D20, D20, C5:D4, C2xC20, C22xD5, C8:S3, D24, D4:S3, Q8:2S3, C3xSD16, S3xD4, Q8:3S3, C5xDic3, C3xDic5, C60, S3xD5, C6xD5, S3xC10, D30, C40:C2, C40:C2, D40, C5xM4(2), C2xD20, C4oD20, Q8:3D6, C5xC3:C8, C120, D30.C2, C3:D20, C5:D12, C3xDic10, C3xD20, S3xC20, D60, C2xS3xD5, C8:D10, C3:D40, C15:SD16, C3xC40:C2, C5xC8:S3, D120, D60:C2, S3xD20, C40:1D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2xD4, D10, C22xS3, C8:C22, D20, C22xD5, S3xD4, S3xD5, C2xD20, Q8:3D6, C2xS3xD5, C8:D10, S3xD20, C40:1D6
►On 120 points
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 82 67)(2 101 68 20 83 46)(3 120 69 39 84 65)(4 99 70 18 85 44)(5 118 71 37 86 63)(6 97 72 16 87 42)(7 116 73 35 88 61)(8 95 74 14 89 80)(9 114 75 33 90 59)(10 93 76 12 91 78)(11 112 77 31 92 57)(13 110 79 29 94 55)(15 108 41 27 96 53)(17 106 43 25 98 51)(19 104 45 23 100 49)(21 102 47)(22 81 48 40 103 66)(24 119 50 38 105 64)(26 117 52 36 107 62)(28 115 54 34 109 60)(30 113 56 32 111 58)
(1 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 60)(9 59)(10 58)(11 57)(12 56)(13 55)(14 54)(15 53)(16 52)(17 51)(18 50)(19 49)(20 48)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 80)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 72)(37 71)(38 70)(39 69)(40 68)(81 83)(84 120)(85 119)(86 118)(87 117)(88 116)(89 115)(90 114)(91 113)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)
G:=sub<Sym(120)| (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,82,67)(2,101,68,20,83,46)(3,120,69,39,84,65)(4,99,70,18,85,44)(5,118,71,37,86,63)(6,97,72,16,87,42)(7,116,73,35,88,61)(8,95,74,14,89,80)(9,114,75,33,90,59)(10,93,76,12,91,78)(11,112,77,31,92,57)(13,110,79,29,94,55)(15,108,41,27,96,53)(17,106,43,25,98,51)(19,104,45,23,100,49)(21,102,47)(22,81,48,40,103,66)(24,119,50,38,105,64)(26,117,52,36,107,62)(28,115,54,34,109,60)(30,113,56,32,111,58), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,60)(9,59)(10,58)(11,57)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(81,83)(84,120)(85,119)(86,118)(87,117)(88,116)(89,115)(90,114)(91,113)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103)>;
G:=Group( (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,82,67)(2,101,68,20,83,46)(3,120,69,39,84,65)(4,99,70,18,85,44)(5,118,71,37,86,63)(6,97,72,16,87,42)(7,116,73,35,88,61)(8,95,74,14,89,80)(9,114,75,33,90,59)(10,93,76,12,91,78)(11,112,77,31,92,57)(13,110,79,29,94,55)(15,108,41,27,96,53)(17,106,43,25,98,51)(19,104,45,23,100,49)(21,102,47)(22,81,48,40,103,66)(24,119,50,38,105,64)(26,117,52,36,107,62)(28,115,54,34,109,60)(30,113,56,32,111,58), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,60)(9,59)(10,58)(11,57)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(81,83)(84,120)(85,119)(86,118)(87,117)(88,116)(89,115)(90,114)(91,113)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103) );
G=PermutationGroup([[(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),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,82,67),(2,101,68,20,83,46),(3,120,69,39,84,65),(4,99,70,18,85,44),(5,118,71,37,86,63),(6,97,72,16,87,42),(7,116,73,35,88,61),(8,95,74,14,89,80),(9,114,75,33,90,59),(10,93,76,12,91,78),(11,112,77,31,92,57),(13,110,79,29,94,55),(15,108,41,27,96,53),(17,106,43,25,98,51),(19,104,45,23,100,49),(21,102,47),(22,81,48,40,103,66),(24,119,50,38,105,64),(26,117,52,36,107,62),(28,115,54,34,109,60),(30,113,56,32,111,58)], [(1,67),(2,66),(3,65),(4,64),(5,63),(6,62),(7,61),(8,60),(9,59),(10,58),(11,57),(12,56),(13,55),(14,54),(15,53),(16,52),(17,51),(18,50),(19,49),(20,48),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(28,80),(29,79),(30,78),(31,77),(32,76),(33,75),(34,74),(35,73),(36,72),(37,71),(38,70),(39,69),(40,68),(81,83),(84,120),(85,119),(86,118),(87,117),(88,116),(89,115),(90,114),(91,113),(92,112),(93,111),(94,110),(95,109),(96,108),(97,107),(98,106),(99,105),(100,104),(101,103)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 40E | 40F | 40G | 40H | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 6 | 20 | 60 | 60 | 2 | 2 | 6 | 20 | 2 | 2 | 2 | 40 | 4 | 12 | 2 | 2 | 12 | 12 | 4 | 40 | 4 | 4 | 2 | 2 | 2 | 2 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | D20 | D20 | C8:C22 | S3xD4 | S3xD5 | Q8:3D6 | C2xS3xD5 | C8:D10 | S3xD20 | C40:1D6 |
| kernel | C40:1D6 | C3:D40 | C15:SD16 | C3xC40:C2 | C5xC8:S3 | D120 | D60:C2 | S3xD20 | C40:C2 | C5xDic3 | S3xC10 | C8:S3 | C40 | Dic10 | D20 | C3:C8 | C24 | C4xS3 | Dic3 | D6 | C15 | C10 | C8 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C40:1D6 ►in GL8(F241)
| 1 | 52 | 0 | 0 | 0 | 0 | 0 | 0 |
| 189 | 189 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 189 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 | 3 | 0 |
| 0 | 0 | 0 | 0 | 108 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 72 | 160 | 0 | 214 |
| 0 | 0 | 0 | 0 | 161 | 153 | 133 | 0 |
| 240 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 52 | 1 | 52 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 189 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 178 | 0 | 240 | 239 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 52 | 1 | 52 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 81 | 18 | 1 | 2 |
| 0 | 0 | 0 | 0 | 169 | 0 | 0 | 240 |
G:=sub<GL(8,GF(241))| [1,189,0,0,0,0,0,0,52,189,0,0,0,0,0,0,0,0,1,189,0,0,0,0,0,0,52,189,0,0,0,0,0,0,0,0,0,108,72,161,0,0,0,0,27,0,160,153,0,0,0,0,3,0,0,133,0,0,0,0,0,3,214,0],[240,52,1,189,0,0,0,0,0,1,0,240,0,0,0,0,240,52,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,240,178,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,239,1],[240,52,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,52,1,189,0,0,0,0,0,1,0,240,0,0,0,0,0,0,0,0,1,240,81,169,0,0,0,0,0,240,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,240] >;
C40:1D6 in GAP, Magma, Sage, TeX
C_{40}\rtimes_1D_6 % in TeX
G:=Group("C40:1D6"); // GroupNames label
G:=SmallGroup(480,329);
// by ID
G=gap.SmallGroup(480,329);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,58,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^40=b^6=c^2=1,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations