metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊14D10, Dic10⋊13D6, D60⋊10C22, C30.31C24, C60.55C23, C15⋊62+ (1+4), D30.16C23, Dic15.18C23, (C4×D5)⋊5D6, (S3×D4)⋊5D5, C5⋊D4⋊7D6, (C4×S3)⋊5D10, (D4×D15)⋊7C2, (D5×D12)⋊5C2, D4⋊12(S3×D5), (C5×D4)⋊15D6, C5⋊3(D4○D12), C3⋊D4⋊7D10, (C3×D4)⋊15D10, D4⋊2D5⋊5S3, (C2×Dic5)⋊8D6, C15⋊Q8⋊14C22, D10⋊D6⋊5C2, D12⋊D5⋊5C2, D60⋊C2⋊5C2, C3⋊3(D4⋊6D10), (S3×C20)⋊6C22, (C22×S3)⋊6D10, (C4×D15)⋊6C22, (D5×C12)⋊6C22, C15⋊7D4⋊7C22, (C2×C30).7C23, C6.31(C23×D5), D6.D10⋊4C2, Dic3.D10⋊5C2, (C5×D12)⋊10C22, (D4×C15)⋊13C22, C5⋊D12⋊16C22, C3⋊D20⋊15C22, C15⋊D4⋊16C22, C20.55(C22×S3), C10.31(S3×C23), D30.C2⋊3C22, (S3×Dic5)⋊3C22, D6.27(C22×D5), (C6×D5).13C23, C12.55(C22×D5), (S3×C10).16C23, (C3×Dic10)⋊9C22, (C6×Dic5)⋊14C22, D10.16(C22×S3), (C22×D15)⋊13C22, (C5×Dic3).17C23, Dic3.16(C22×D5), (C3×Dic5).46C23, Dic5.17(C22×S3), (C5×S3×D4)⋊7C2, C4.55(C2×S3×D5), (S3×C5⋊D4)⋊5C2, (C2×S3×D5)⋊6C22, C22.7(C2×S3×D5), (S3×C2×C10)⋊9C22, (C3×D4⋊2D5)⋊7C2, (C2×C5⋊D12)⋊20C2, C2.34(C22×S3×D5), (C5×C3⋊D4)⋊7C22, (C3×C5⋊D4)⋊7C22, (C2×C6).7(C22×D5), (C2×C10).7(C22×S3), SmallGroup(480,1103)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1884 in 332 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×5], C22 [×2], C22 [×13], C5, S3 [×6], C6, C6 [×3], C2×C4 [×9], D4, D4 [×17], Q8 [×2], C23 [×6], D5 [×4], C10, C10 [×5], Dic3, Dic3, C12, C12 [×3], D6, D6 [×2], D6 [×9], C2×C6 [×2], C2×C6, C15, C2×D4 [×9], C4○D4 [×6], Dic5, Dic5 [×2], Dic5, C20, C20, D10, D10 [×7], C2×C10 [×2], C2×C10 [×5], Dic6, C4×S3, C4×S3 [×5], D12, D12 [×8], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12 [×3], C3×D4, C3×D4 [×2], C3×Q8, C22×S3 [×2], C22×S3 [×4], C5×S3 [×3], C3×D5, D15 [×3], C30, C30 [×2], 2+ (1+4), Dic10, Dic10, C4×D5, C4×D5 [×3], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×2], C5⋊D4 [×10], C2×C20, C5×D4, C5×D4 [×3], C22×D5 [×4], C22×C10 [×2], C2×D12 [×3], C4○D12 [×3], S3×D4, S3×D4 [×5], Q8⋊3S3 [×2], C3×C4○D4, C5×Dic3, C3×Dic5, C3×Dic5 [×2], Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, S3×C10 [×2], S3×C10 [×2], D30, D30 [×2], D30 [×2], C2×C30 [×2], C4○D20 [×2], D4×D5 [×4], D4⋊2D5, D4⋊2D5 [×3], C2×C5⋊D4 [×4], D4×C10, D4○D12, S3×Dic5 [×2], D30.C2 [×2], C15⋊D4, C3⋊D20, C5⋊D12, C5⋊D12 [×6], C15⋊Q8, C3×Dic10, D5×C12, C6×Dic5 [×2], C3×C5⋊D4 [×2], S3×C20, C5×D12, C5×C3⋊D4 [×2], C4×D15, D60, C15⋊7D4 [×2], D4×C15, C2×S3×D5 [×2], S3×C2×C10 [×2], C22×D15 [×2], D4⋊6D10, D12⋊D5, D60⋊C2, D6.D10, D5×D12, Dic3.D10 [×2], C2×C5⋊D12 [×2], S3×C5⋊D4 [×2], D10⋊D6 [×2], C3×D4⋊2D5, C5×S3×D4, D4×D15, D12⋊14D10
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⋊6D10, C22×S3×D5, D12⋊14D10
Generators and relations
G = < a,b,c,d | a12=b2=c10=d2=1, bab=a-1, cac-1=a7, dad=a5, cbc-1=a6b, dbd=a10b, 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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 34)(26 33)(27 32)(28 31)(29 30)(35 36)(37 38)(39 48)(40 47)(41 46)(42 45)(43 44)(49 58)(50 57)(51 56)(52 55)(53 54)(59 60)(61 68)(62 67)(63 66)(64 65)(69 72)(70 71)(73 74)(75 84)(76 83)(77 82)(78 81)(79 80)(85 88)(86 87)(89 96)(90 95)(91 94)(92 93)(97 98)(99 108)(100 107)(101 106)(102 105)(103 104)(109 110)(111 120)(112 119)(113 118)(114 117)(115 116)
(1 17 54 36 98)(2 24 55 31 99 8 18 49 25 105)(3 19 56 26 100)(4 14 57 33 101 10 20 51 27 107)(5 21 58 28 102)(6 16 59 35 103 12 22 53 29 97)(7 23 60 30 104)(9 13 50 32 106)(11 15 52 34 108)(37 64 92 73 109)(38 71 93 80 110 44 65 87 74 116)(39 66 94 75 111)(40 61 95 82 112 46 67 89 76 118)(41 68 96 77 113)(42 63 85 84 114 48 69 91 78 120)(43 70 86 79 115)(45 72 88 81 117)(47 62 90 83 119)
(1 68)(2 61)(3 66)(4 71)(5 64)(6 69)(7 62)(8 67)(9 72)(10 65)(11 70)(12 63)(13 45)(14 38)(15 43)(16 48)(17 41)(18 46)(19 39)(20 44)(21 37)(22 42)(23 47)(24 40)(25 82)(26 75)(27 80)(28 73)(29 78)(30 83)(31 76)(32 81)(33 74)(34 79)(35 84)(36 77)(49 112)(50 117)(51 110)(52 115)(53 120)(54 113)(55 118)(56 111)(57 116)(58 109)(59 114)(60 119)(85 103)(86 108)(87 101)(88 106)(89 99)(90 104)(91 97)(92 102)(93 107)(94 100)(95 105)(96 98)
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,34)(26,33)(27,32)(28,31)(29,30)(35,36)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,68)(62,67)(63,66)(64,65)(69,72)(70,71)(73,74)(75,84)(76,83)(77,82)(78,81)(79,80)(85,88)(86,87)(89,96)(90,95)(91,94)(92,93)(97,98)(99,108)(100,107)(101,106)(102,105)(103,104)(109,110)(111,120)(112,119)(113,118)(114,117)(115,116), (1,17,54,36,98)(2,24,55,31,99,8,18,49,25,105)(3,19,56,26,100)(4,14,57,33,101,10,20,51,27,107)(5,21,58,28,102)(6,16,59,35,103,12,22,53,29,97)(7,23,60,30,104)(9,13,50,32,106)(11,15,52,34,108)(37,64,92,73,109)(38,71,93,80,110,44,65,87,74,116)(39,66,94,75,111)(40,61,95,82,112,46,67,89,76,118)(41,68,96,77,113)(42,63,85,84,114,48,69,91,78,120)(43,70,86,79,115)(45,72,88,81,117)(47,62,90,83,119), (1,68)(2,61)(3,66)(4,71)(5,64)(6,69)(7,62)(8,67)(9,72)(10,65)(11,70)(12,63)(13,45)(14,38)(15,43)(16,48)(17,41)(18,46)(19,39)(20,44)(21,37)(22,42)(23,47)(24,40)(25,82)(26,75)(27,80)(28,73)(29,78)(30,83)(31,76)(32,81)(33,74)(34,79)(35,84)(36,77)(49,112)(50,117)(51,110)(52,115)(53,120)(54,113)(55,118)(56,111)(57,116)(58,109)(59,114)(60,119)(85,103)(86,108)(87,101)(88,106)(89,99)(90,104)(91,97)(92,102)(93,107)(94,100)(95,105)(96,98)>;
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,34)(26,33)(27,32)(28,31)(29,30)(35,36)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,68)(62,67)(63,66)(64,65)(69,72)(70,71)(73,74)(75,84)(76,83)(77,82)(78,81)(79,80)(85,88)(86,87)(89,96)(90,95)(91,94)(92,93)(97,98)(99,108)(100,107)(101,106)(102,105)(103,104)(109,110)(111,120)(112,119)(113,118)(114,117)(115,116), (1,17,54,36,98)(2,24,55,31,99,8,18,49,25,105)(3,19,56,26,100)(4,14,57,33,101,10,20,51,27,107)(5,21,58,28,102)(6,16,59,35,103,12,22,53,29,97)(7,23,60,30,104)(9,13,50,32,106)(11,15,52,34,108)(37,64,92,73,109)(38,71,93,80,110,44,65,87,74,116)(39,66,94,75,111)(40,61,95,82,112,46,67,89,76,118)(41,68,96,77,113)(42,63,85,84,114,48,69,91,78,120)(43,70,86,79,115)(45,72,88,81,117)(47,62,90,83,119), (1,68)(2,61)(3,66)(4,71)(5,64)(6,69)(7,62)(8,67)(9,72)(10,65)(11,70)(12,63)(13,45)(14,38)(15,43)(16,48)(17,41)(18,46)(19,39)(20,44)(21,37)(22,42)(23,47)(24,40)(25,82)(26,75)(27,80)(28,73)(29,78)(30,83)(31,76)(32,81)(33,74)(34,79)(35,84)(36,77)(49,112)(50,117)(51,110)(52,115)(53,120)(54,113)(55,118)(56,111)(57,116)(58,109)(59,114)(60,119)(85,103)(86,108)(87,101)(88,106)(89,99)(90,104)(91,97)(92,102)(93,107)(94,100)(95,105)(96,98) );
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,20),(14,19),(15,18),(16,17),(21,24),(22,23),(25,34),(26,33),(27,32),(28,31),(29,30),(35,36),(37,38),(39,48),(40,47),(41,46),(42,45),(43,44),(49,58),(50,57),(51,56),(52,55),(53,54),(59,60),(61,68),(62,67),(63,66),(64,65),(69,72),(70,71),(73,74),(75,84),(76,83),(77,82),(78,81),(79,80),(85,88),(86,87),(89,96),(90,95),(91,94),(92,93),(97,98),(99,108),(100,107),(101,106),(102,105),(103,104),(109,110),(111,120),(112,119),(113,118),(114,117),(115,116)], [(1,17,54,36,98),(2,24,55,31,99,8,18,49,25,105),(3,19,56,26,100),(4,14,57,33,101,10,20,51,27,107),(5,21,58,28,102),(6,16,59,35,103,12,22,53,29,97),(7,23,60,30,104),(9,13,50,32,106),(11,15,52,34,108),(37,64,92,73,109),(38,71,93,80,110,44,65,87,74,116),(39,66,94,75,111),(40,61,95,82,112,46,67,89,76,118),(41,68,96,77,113),(42,63,85,84,114,48,69,91,78,120),(43,70,86,79,115),(45,72,88,81,117),(47,62,90,83,119)], [(1,68),(2,61),(3,66),(4,71),(5,64),(6,69),(7,62),(8,67),(9,72),(10,65),(11,70),(12,63),(13,45),(14,38),(15,43),(16,48),(17,41),(18,46),(19,39),(20,44),(21,37),(22,42),(23,47),(24,40),(25,82),(26,75),(27,80),(28,73),(29,78),(30,83),(31,76),(32,81),(33,74),(34,79),(35,84),(36,77),(49,112),(50,117),(51,110),(52,115),(53,120),(54,113),(55,118),(56,111),(57,116),(58,109),(59,114),(60,119),(85,103),(86,108),(87,101),(88,106),(89,99),(90,104),(91,97),(92,102),(93,107),(94,100),(95,105),(96,98)])
Matrix representation ►G ⊆ GL8(𝔽61)
| 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 60 | 44 | 0 | 0 | 0 | 0 | 0 | 0 |
| 17 | 44 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 34 |
| 30 | 47 | 0 | 0 | 0 | 0 | 0 | 0 |
| 25 | 31 | 0 | 0 | 0 | 0 | 0 | 0 |
| 31 | 14 | 31 | 14 | 0 | 0 | 0 | 0 |
| 36 | 30 | 36 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 0 | 0 | 0 |
G:=sub<GL(8,GF(61))| [1,0,60,0,0,0,0,0,0,1,0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,60,0,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[60,17,0,0,0,0,0,0,44,44,0,0,0,0,0,0,0,0,60,17,0,0,0,0,0,0,44,44,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,34],[30,25,31,36,0,0,0,0,47,31,14,30,0,0,0,0,0,0,31,36,0,0,0,0,0,0,14,30,0,0,0,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,9,0,0,0,0,0,0,34,0,0,0,0,0,0,27,0,0,0] >;
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 | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 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 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 10 | 30 | 30 | 30 | 2 | 2 | 6 | 10 | 10 | 10 | 30 | 2 | 2 | 2 | 4 | 4 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 4 | 10 | 10 | 20 | 20 | 4 | 4 | 4 | 4 | 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○D12 | C2×S3×D5 | C2×S3×D5 | D4⋊6D10 | D12⋊14D10 |
| kernel | D12⋊14D10 | D12⋊D5 | D60⋊C2 | D6.D10 | D5×D12 | Dic3.D10 | C2×C5⋊D12 | S3×C5⋊D4 | D10⋊D6 | C3×D4⋊2D5 | C5×S3×D4 | D4×D15 | D4⋊2D5 | S3×D4 | Dic10 | C4×D5 | C2×Dic5 | C5⋊D4 | C5×D4 | C4×S3 | D12 | C3⋊D4 | C3×D4 | C22×S3 | 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 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 1 | 2 | 2 | 2 | 4 | 4 | 2 |
In GAP, Magma, Sage, TeX
D_{12}\rtimes_{14}D_{10} % in TeX
G:=Group("D12:14D10"); // GroupNames label
G:=SmallGroup(480,1103);
// by ID
G=gap.SmallGroup(480,1103);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,100,675,185,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^10=d^2=1,b*a*b=a^-1,c*a*c^-1=a^7,d*a*d=a^5,c*b*c^-1=a^6*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations