metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊17D6, D12⋊17D10, D60⋊13C22, C30.39C24, C60.63C23, C15⋊72+ (1+4), D30.20C23, Dic15.42C23, (C4×D5)⋊6D6, (C4×S3)⋊6D10, (D5×D12)⋊7C2, (S3×D20)⋊7C2, C5⋊4(D4○D12), (C5×Q8)⋊16D6, Q8⋊11(S3×D5), C20⋊D6⋊7C2, Q8⋊2D5⋊8S3, Q8⋊3S3⋊5D5, (C3×Q8)⋊13D10, C15⋊Q8⋊19C22, Q8⋊3D15⋊7C2, C3⋊4(D4⋊8D10), (S3×C20)⋊9C22, (C4×D15)⋊9C22, (D5×C12)⋊9C22, C6.39(C23×D5), D6.D10⋊8C2, (C5×D12)⋊13C22, (C3×D20)⋊13C22, C5⋊D12⋊17C22, C3⋊D20⋊16C22, C15⋊D4⋊17C22, C20.63(C22×S3), C10.39(S3×C23), (Q8×C15)⋊11C22, (C6×D5).17C23, D6.18(C22×D5), C12.63(C22×D5), (S3×C10).20C23, D10.20(C22×S3), (C3×Dic5).50C23, (C5×Dic3).34C23, Dic3.29(C22×D5), Dic5.45(C22×S3), C4.63(C2×S3×D5), (C2×S3×D5)⋊7C22, C2.42(C22×S3×D5), (C5×Q8⋊3S3)⋊7C2, (C3×Q8⋊2D5)⋊7C2, SmallGroup(480,1111)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2044 in 332 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2 [×9], C3, C4 [×3], C4 [×3], C22 [×15], C5, S3 [×6], C6, C6 [×3], C2×C4 [×9], D4 [×18], Q8, Q8, C23 [×6], D5 [×6], C10, C10 [×3], Dic3, Dic3, C12 [×3], C12, D6 [×3], D6 [×9], C2×C6 [×3], C15, C2×D4 [×9], C4○D4 [×6], Dic5, Dic5, C20 [×3], C20, D10 [×3], D10 [×9], C2×C10 [×3], Dic6, C4×S3 [×3], C4×S3 [×3], D12 [×3], D12 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×6], C5×S3 [×3], C3×D5 [×3], D15 [×3], C30, 2+ (1+4), Dic10, C4×D5 [×3], C4×D5 [×3], D20 [×3], D20 [×6], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×6], C2×D12 [×3], C4○D12 [×3], S3×D4 [×6], Q8⋊3S3, Q8⋊3S3, C3×C4○D4, C5×Dic3, C3×Dic5, Dic15, C60 [×3], S3×D5 [×6], C6×D5 [×3], S3×C10 [×3], D30 [×3], C2×D20 [×3], C4○D20 [×3], D4×D5 [×6], Q8⋊2D5, Q8⋊2D5, C5×C4○D4, D4○D12, C15⋊D4 [×3], C3⋊D20 [×3], C5⋊D12 [×3], C15⋊Q8, D5×C12 [×3], C3×D20 [×3], S3×C20 [×3], C5×D12 [×3], C4×D15 [×3], D60 [×3], Q8×C15, C2×S3×D5 [×6], D4⋊8D10, D6.D10 [×3], D5×D12 [×3], S3×D20 [×3], C20⋊D6 [×3], C3×Q8⋊2D5, C5×Q8⋊3S3, Q8⋊3D15, D20⋊17D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2+ (1+4), C22×D5 [×7], S3×C23, S3×D5, C23×D5, D4○D12, C2×S3×D5 [×3], D4⋊8D10, C22×S3×D5, D20⋊17D6
Generators and relations
G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a18b, dbd=a8b, dcd=c-1 >
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)(95 100)(96 99)(97 98)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)
(1 106 98 34 71 56)(2 115 99 23 72 45)(3 104 100 32 73 54)(4 113 81 21 74 43)(5 102 82 30 75 52)(6 111 83 39 76 41)(7 120 84 28 77 50)(8 109 85 37 78 59)(9 118 86 26 79 48)(10 107 87 35 80 57)(11 116 88 24 61 46)(12 105 89 33 62 55)(13 114 90 22 63 44)(14 103 91 31 64 53)(15 112 92 40 65 42)(16 101 93 29 66 51)(17 110 94 38 67 60)(18 119 95 27 68 49)(19 108 96 36 69 58)(20 117 97 25 70 47)
(1 93)(2 92)(3 91)(4 90)(5 89)(6 88)(7 87)(8 86)(9 85)(10 84)(11 83)(12 82)(13 81)(14 100)(15 99)(16 98)(17 97)(18 96)(19 95)(20 94)(21 44)(22 43)(23 42)(24 41)(25 60)(26 59)(27 58)(28 57)(29 56)(30 55)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)(101 106)(102 105)(103 104)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(95,100)(96,99)(97,98)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111), (1,106,98,34,71,56)(2,115,99,23,72,45)(3,104,100,32,73,54)(4,113,81,21,74,43)(5,102,82,30,75,52)(6,111,83,39,76,41)(7,120,84,28,77,50)(8,109,85,37,78,59)(9,118,86,26,79,48)(10,107,87,35,80,57)(11,116,88,24,61,46)(12,105,89,33,62,55)(13,114,90,22,63,44)(14,103,91,31,64,53)(15,112,92,40,65,42)(16,101,93,29,66,51)(17,110,94,38,67,60)(18,119,95,27,68,49)(19,108,96,36,69,58)(20,117,97,25,70,47), (1,93)(2,92)(3,91)(4,90)(5,89)(6,88)(7,87)(8,86)(9,85)(10,84)(11,83)(12,82)(13,81)(14,100)(15,99)(16,98)(17,97)(18,96)(19,95)(20,94)(21,44)(22,43)(23,42)(24,41)(25,60)(26,59)(27,58)(28,57)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79)(101,106)(102,105)(103,104)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(95,100)(96,99)(97,98)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111), (1,106,98,34,71,56)(2,115,99,23,72,45)(3,104,100,32,73,54)(4,113,81,21,74,43)(5,102,82,30,75,52)(6,111,83,39,76,41)(7,120,84,28,77,50)(8,109,85,37,78,59)(9,118,86,26,79,48)(10,107,87,35,80,57)(11,116,88,24,61,46)(12,105,89,33,62,55)(13,114,90,22,63,44)(14,103,91,31,64,53)(15,112,92,40,65,42)(16,101,93,29,66,51)(17,110,94,38,67,60)(18,119,95,27,68,49)(19,108,96,36,69,58)(20,117,97,25,70,47), (1,93)(2,92)(3,91)(4,90)(5,89)(6,88)(7,87)(8,86)(9,85)(10,84)(11,83)(12,82)(13,81)(14,100)(15,99)(16,98)(17,97)(18,96)(19,95)(20,94)(21,44)(22,43)(23,42)(24,41)(25,60)(26,59)(27,58)(28,57)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79)(101,106)(102,105)(103,104)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(37,40),(38,39),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(81,94),(82,93),(83,92),(84,91),(85,90),(86,89),(87,88),(95,100),(96,99),(97,98),(101,120),(102,119),(103,118),(104,117),(105,116),(106,115),(107,114),(108,113),(109,112),(110,111)], [(1,106,98,34,71,56),(2,115,99,23,72,45),(3,104,100,32,73,54),(4,113,81,21,74,43),(5,102,82,30,75,52),(6,111,83,39,76,41),(7,120,84,28,77,50),(8,109,85,37,78,59),(9,118,86,26,79,48),(10,107,87,35,80,57),(11,116,88,24,61,46),(12,105,89,33,62,55),(13,114,90,22,63,44),(14,103,91,31,64,53),(15,112,92,40,65,42),(16,101,93,29,66,51),(17,110,94,38,67,60),(18,119,95,27,68,49),(19,108,96,36,69,58),(20,117,97,25,70,47)], [(1,93),(2,92),(3,91),(4,90),(5,89),(6,88),(7,87),(8,86),(9,85),(10,84),(11,83),(12,82),(13,81),(14,100),(15,99),(16,98),(17,97),(18,96),(19,95),(20,94),(21,44),(22,43),(23,42),(24,41),(25,60),(26,59),(27,58),(28,57),(29,56),(30,55),(31,54),(32,53),(33,52),(34,51),(35,50),(36,49),(37,48),(38,47),(39,46),(40,45),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(77,80),(78,79),(101,106),(102,105),(103,104),(107,120),(108,119),(109,118),(110,117),(111,116),(112,115),(113,114)])
Matrix representation ►G ⊆ GL8(𝔽61)
| 18 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 42 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 36 | 57 | 25 |
| 0 | 0 | 0 | 0 | 25 | 27 | 36 | 34 |
| 60 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 57 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 36 |
| 0 | 0 | 0 | 0 | 0 | 0 | 25 | 57 |
| 25 | 27 | 46 | 26 | 0 | 0 | 0 | 0 |
| 38 | 36 | 41 | 35 | 0 | 0 | 0 | 0 |
| 21 | 21 | 60 | 0 | 0 | 0 | 0 | 0 |
| 12 | 52 | 44 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 | 25 | 59 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 43 | 60 |
| 36 | 34 | 15 | 35 | 0 | 0 | 0 | 0 |
| 23 | 25 | 20 | 26 | 0 | 0 | 0 | 0 |
| 40 | 40 | 2 | 0 | 0 | 0 | 0 | 0 |
| 49 | 9 | 34 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 57 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 57 | 36 | 4 |
| 0 | 0 | 0 | 0 | 34 | 36 | 27 | 25 |
G:=sub<GL(8,GF(61))| [18,42,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,17,0,0,0,0,0,0,0,0,4,25,4,25,0,0,0,0,36,27,36,27,0,0,0,0,0,0,57,36,0,0,0,0,0,0,25,34],[60,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,25,0,0,0,0,0,0,36,57,0,0,0,0,0,0,0,0,4,25,0,0,0,0,0,0,36,57],[25,38,21,12,0,0,0,0,27,36,21,52,0,0,0,0,46,41,60,44,0,0,0,0,26,35,0,1,0,0,0,0,0,0,0,0,60,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,25,1,43,0,0,0,0,0,59,0,60],[36,23,40,49,0,0,0,0,34,25,40,9,0,0,0,0,15,20,2,34,0,0,0,0,35,26,0,59,0,0,0,0,0,0,0,0,25,34,25,34,0,0,0,0,57,36,57,36,0,0,0,0,0,0,36,27,0,0,0,0,0,0,4,25] >;
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | ··· | 20F | 20G | 20H | 20I | 20J | 30A | 30B | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 6 | 6 | 10 | 10 | 10 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 6 | 10 | 30 | 2 | 2 | 2 | 20 | 20 | 20 | 2 | 2 | 12 | ··· | 12 | 4 | 4 | 4 | 10 | 10 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | ··· | 8 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | 2+ (1+4) | S3×D5 | D4○D12 | C2×S3×D5 | D4⋊8D10 | D20⋊17D6 |
| kernel | D20⋊17D6 | D6.D10 | D5×D12 | S3×D20 | C20⋊D6 | C3×Q8⋊2D5 | C5×Q8⋊3S3 | Q8⋊3D15 | Q8⋊2D5 | Q8⋊3S3 | C4×D5 | D20 | C5×Q8 | C4×S3 | D12 | C3×Q8 | C15 | Q8 | C5 | C4 | C3 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 1 | 6 | 6 | 2 | 1 | 2 | 2 | 6 | 4 | 2 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_{17}D_6 % in TeX
G:=Group("D20:17D6"); // GroupNames label
G:=SmallGroup(480,1111);
// by ID
G=gap.SmallGroup(480,1111);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,100,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^18*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations