metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊14D6, D60⋊9C22, Dic6⋊13D10, C60.54C23, C30.30C24, C15⋊52+ (1+4), D30.15C23, Dic15.17C23, (C4×D5)⋊4D6, (D4×D5)⋊5S3, C5⋊D4⋊6D6, (S3×D20)⋊5C2, (C5×D4)⋊14D6, (C4×S3)⋊4D10, (D4×D15)⋊6C2, D4⋊11(S3×D5), C3⋊D4⋊6D10, (C3×D4)⋊14D10, D4⋊2S3⋊5D5, C5⋊3(D4⋊6D6), (C22×D5)⋊7D6, C15⋊Q8⋊13C22, D20⋊S3⋊5C2, C12.28D10⋊5C2, D10⋊D6⋊4C2, C3⋊3(D4⋊8D10), (S3×C20)⋊5C22, (C2×Dic3)⋊8D10, (D5×C12)⋊5C22, (C4×D15)⋊5C22, C15⋊7D4⋊6C22, (C2×C30).6C23, C6.30(C23×D5), Dic5.D6⋊5C2, D6.D10⋊3C2, (D4×C15)⋊12C22, (C3×D20)⋊10C22, C15⋊D4⋊15C22, C3⋊D20⋊21C22, C5⋊D12⋊15C22, C10.30(S3×C23), C20.54(C22×S3), D30.C2⋊2C22, (C5×Dic6)⋊9C22, (D5×Dic3)⋊3C22, D6.14(C22×D5), (C6×D5).46C23, C12.54(C22×D5), (S3×C10).15C23, D10.15(C22×S3), (C10×Dic3)⋊14C22, (C22×D15)⋊12C22, (C5×Dic3).30C23, (C3×Dic5).15C23, Dic5.16(C22×S3), Dic3.27(C22×D5), (C3×D4×D5)⋊7C2, C4.54(C2×S3×D5), (D5×C3⋊D4)⋊5C2, (D5×C2×C6)⋊9C22, (C2×S3×D5)⋊5C22, C22.6(C2×S3×D5), (C5×D4⋊2S3)⋊7C2, (C2×C3⋊D20)⋊20C2, C2.33(C22×S3×D5), (C3×C5⋊D4)⋊6C22, (C5×C3⋊D4)⋊6C22, (C2×C6).6(C22×D5), (C2×C10).6(C22×S3), SmallGroup(480,1102)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1932 in 332 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×5], C22 [×2], C22 [×13], C5, S3 [×4], C6, C6 [×5], C2×C4 [×9], D4, D4 [×17], Q8 [×2], C23 [×6], D5 [×6], C10, C10 [×3], Dic3, Dic3 [×2], Dic3, C12, C12, D6, D6 [×7], C2×C6 [×2], C2×C6 [×5], C15, C2×D4 [×9], C4○D4 [×6], Dic5, Dic5, C20, C20 [×3], D10, D10 [×2], D10 [×9], C2×C10 [×2], C2×C10, Dic6, Dic6, C4×S3, C4×S3 [×3], D12 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×2], C3⋊D4 [×10], C2×C12, C3×D4, C3×D4 [×3], C22×S3 [×4], C22×C6 [×2], C5×S3, C3×D5 [×3], D15 [×3], C30, C30 [×2], 2+ (1+4), Dic10, C4×D5, C4×D5 [×5], D20, D20 [×8], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20 [×3], C5×D4, C5×D4 [×2], C5×Q8, C22×D5 [×2], C22×D5 [×4], C4○D12 [×2], S3×D4 [×4], D4⋊2S3, D4⋊2S3 [×3], C2×C3⋊D4 [×4], C6×D4, C5×Dic3, C5×Dic3 [×2], C3×Dic5, Dic15, C60, S3×D5 [×2], C6×D5, C6×D5 [×2], C6×D5 [×2], S3×C10, D30, D30 [×2], D30 [×2], C2×C30 [×2], C2×D20 [×3], C4○D20 [×3], D4×D5, D4×D5 [×5], Q8⋊2D5 [×2], C5×C4○D4, D4⋊6D6, D5×Dic3 [×2], D30.C2 [×2], C15⋊D4, C3⋊D20, C3⋊D20 [×6], C5⋊D12, C15⋊Q8, D5×C12, C3×D20, C3×C5⋊D4 [×2], C5×Dic6, S3×C20, C10×Dic3 [×2], C5×C3⋊D4 [×2], C4×D15, D60, C15⋊7D4 [×2], D4×C15, C2×S3×D5 [×2], D5×C2×C6 [×2], C22×D15 [×2], D4⋊8D10, D20⋊S3, D6.D10, C12.28D10, S3×D20, Dic5.D6 [×2], C2×C3⋊D20 [×2], D5×C3⋊D4 [×2], D10⋊D6 [×2], C3×D4×D5, C5×D4⋊2S3, D4×D15, D20⋊14D6
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⋊6D6, C2×S3×D5 [×3], D4⋊8D10, C22×S3×D5, D20⋊14D6
Generators and relations
G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, cac-1=a11, dad=a9, cbc-1=a10b, dbd=a18b, 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 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)(97 100)(98 99)(101 104)(102 103)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)
(1 34 99)(2 25 100 12 35 90)(3 36 81)(4 27 82 14 37 92)(5 38 83)(6 29 84 16 39 94)(7 40 85)(8 31 86 18 21 96)(9 22 87)(10 33 88 20 23 98)(11 24 89)(13 26 91)(15 28 93)(17 30 95)(19 32 97)(41 76 113 51 66 103)(42 67 114)(43 78 115 53 68 105)(44 69 116)(45 80 117 55 70 107)(46 71 118)(47 62 119 57 72 109)(48 73 120)(49 64 101 59 74 111)(50 75 102)(52 77 104)(54 79 106)(56 61 108)(58 63 110)(60 65 112)
(1 61)(2 70)(3 79)(4 68)(5 77)(6 66)(7 75)(8 64)(9 73)(10 62)(11 71)(12 80)(13 69)(14 78)(15 67)(16 76)(17 65)(18 74)(19 63)(20 72)(21 59)(22 48)(23 57)(24 46)(25 55)(26 44)(27 53)(28 42)(29 51)(30 60)(31 49)(32 58)(33 47)(34 56)(35 45)(36 54)(37 43)(38 52)(39 41)(40 50)(81 106)(82 115)(83 104)(84 113)(85 102)(86 111)(87 120)(88 109)(89 118)(90 107)(91 116)(92 105)(93 114)(94 103)(95 112)(96 101)(97 110)(98 119)(99 108)(100 117)
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)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(97,100)(98,99)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113), (1,34,99)(2,25,100,12,35,90)(3,36,81)(4,27,82,14,37,92)(5,38,83)(6,29,84,16,39,94)(7,40,85)(8,31,86,18,21,96)(9,22,87)(10,33,88,20,23,98)(11,24,89)(13,26,91)(15,28,93)(17,30,95)(19,32,97)(41,76,113,51,66,103)(42,67,114)(43,78,115,53,68,105)(44,69,116)(45,80,117,55,70,107)(46,71,118)(47,62,119,57,72,109)(48,73,120)(49,64,101,59,74,111)(50,75,102)(52,77,104)(54,79,106)(56,61,108)(58,63,110)(60,65,112), (1,61)(2,70)(3,79)(4,68)(5,77)(6,66)(7,75)(8,64)(9,73)(10,62)(11,71)(12,80)(13,69)(14,78)(15,67)(16,76)(17,65)(18,74)(19,63)(20,72)(21,59)(22,48)(23,57)(24,46)(25,55)(26,44)(27,53)(28,42)(29,51)(30,60)(31,49)(32,58)(33,47)(34,56)(35,45)(36,54)(37,43)(38,52)(39,41)(40,50)(81,106)(82,115)(83,104)(84,113)(85,102)(86,111)(87,120)(88,109)(89,118)(90,107)(91,116)(92,105)(93,114)(94,103)(95,112)(96,101)(97,110)(98,119)(99,108)(100,117)>;
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)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(97,100)(98,99)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113), (1,34,99)(2,25,100,12,35,90)(3,36,81)(4,27,82,14,37,92)(5,38,83)(6,29,84,16,39,94)(7,40,85)(8,31,86,18,21,96)(9,22,87)(10,33,88,20,23,98)(11,24,89)(13,26,91)(15,28,93)(17,30,95)(19,32,97)(41,76,113,51,66,103)(42,67,114)(43,78,115,53,68,105)(44,69,116)(45,80,117,55,70,107)(46,71,118)(47,62,119,57,72,109)(48,73,120)(49,64,101,59,74,111)(50,75,102)(52,77,104)(54,79,106)(56,61,108)(58,63,110)(60,65,112), (1,61)(2,70)(3,79)(4,68)(5,77)(6,66)(7,75)(8,64)(9,73)(10,62)(11,71)(12,80)(13,69)(14,78)(15,67)(16,76)(17,65)(18,74)(19,63)(20,72)(21,59)(22,48)(23,57)(24,46)(25,55)(26,44)(27,53)(28,42)(29,51)(30,60)(31,49)(32,58)(33,47)(34,56)(35,45)(36,54)(37,43)(38,52)(39,41)(40,50)(81,106)(82,115)(83,104)(84,113)(85,102)(86,111)(87,120)(88,109)(89,118)(90,107)(91,116)(92,105)(93,114)(94,103)(95,112)(96,101)(97,110)(98,119)(99,108)(100,117) );
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),(21,26),(22,25),(23,24),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,70),(62,69),(63,68),(64,67),(65,66),(71,80),(72,79),(73,78),(74,77),(75,76),(81,96),(82,95),(83,94),(84,93),(85,92),(86,91),(87,90),(88,89),(97,100),(98,99),(101,104),(102,103),(105,120),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113)], [(1,34,99),(2,25,100,12,35,90),(3,36,81),(4,27,82,14,37,92),(5,38,83),(6,29,84,16,39,94),(7,40,85),(8,31,86,18,21,96),(9,22,87),(10,33,88,20,23,98),(11,24,89),(13,26,91),(15,28,93),(17,30,95),(19,32,97),(41,76,113,51,66,103),(42,67,114),(43,78,115,53,68,105),(44,69,116),(45,80,117,55,70,107),(46,71,118),(47,62,119,57,72,109),(48,73,120),(49,64,101,59,74,111),(50,75,102),(52,77,104),(54,79,106),(56,61,108),(58,63,110),(60,65,112)], [(1,61),(2,70),(3,79),(4,68),(5,77),(6,66),(7,75),(8,64),(9,73),(10,62),(11,71),(12,80),(13,69),(14,78),(15,67),(16,76),(17,65),(18,74),(19,63),(20,72),(21,59),(22,48),(23,57),(24,46),(25,55),(26,44),(27,53),(28,42),(29,51),(30,60),(31,49),(32,58),(33,47),(34,56),(35,45),(36,54),(37,43),(38,52),(39,41),(40,50),(81,106),(82,115),(83,104),(84,113),(85,102),(86,111),(87,120),(88,109),(89,118),(90,107),(91,116),(92,105),(93,114),(94,103),(95,112),(96,101),(97,110),(98,119),(99,108),(100,117)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 17 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 44 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 27 | 24 | 0 | 0 | 0 | 0 |
| 51 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 54 | 0 | 0 |
| 0 | 0 | 59 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 54 |
| 0 | 0 | 0 | 0 | 59 | 32 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,60,44,0,0,0,60,0,0,0,0,1,17,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[0,60,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[27,51,0,0,0,0,24,34,0,0,0,0,0,0,29,59,0,0,0,0,54,32,0,0,0,0,0,0,29,59,0,0,0,0,54,32] >;
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 | 6E | 6F | 6G | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 30A | 30B | 30C | 30D | 30E | 30F | 60A | 60B |
| 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 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 2 | 2 | 6 | 10 | 10 | 10 | 30 | 30 | 30 | 2 | 2 | 6 | 6 | 6 | 10 | 30 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 4 | 20 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | D6 | D10 | D10 | D10 | D10 | D10 | 2+ (1+4) | S3×D5 | D4⋊6D6 | C2×S3×D5 | C2×S3×D5 | D4⋊8D10 | D20⋊14D6 |
| kernel | D20⋊14D6 | D20⋊S3 | D6.D10 | C12.28D10 | S3×D20 | Dic5.D6 | C2×C3⋊D20 | D5×C3⋊D4 | D10⋊D6 | C3×D4×D5 | C5×D4⋊2S3 | D4×D15 | D4×D5 | D4⋊2S3 | C4×D5 | D20 | C5⋊D4 | C5×D4 | C22×D5 | Dic6 | C4×S3 | C2×Dic3 | C3⋊D4 | C3×D4 | C15 | D4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_{14}D_6 % in TeX
G:=Group("D20:14D6"); // GroupNames label
G:=SmallGroup(480,1102);
// by ID
G=gap.SmallGroup(480,1102);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,675,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=a^11,d*a*d=a^9,c*b*c^-1=a^10*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations