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, (D4×D15)⋊7C2, (D5×D12)⋊5C2, (C4×S3)⋊5D10, 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, D12⋊D5⋊5C2, D10⋊D6⋊5C2, D60⋊C2⋊5C2, C3⋊3(D4⋊6D10), (S3×C20)⋊6C22, (C4×D15)⋊6C22, (D5×C12)⋊6C22, (C22×S3)⋊6D10, C15⋊7D4⋊7C22, (C2×C30).7C23, C6.31(C23×D5), D6.D10⋊4C2, Dic3.D10⋊5C2, (D4×C15)⋊13C22, (C5×D12)⋊10C22, C5⋊D12⋊16C22, C3⋊D20⋊15C22, C15⋊D4⋊16C22, C20.55(C22×S3), C10.31(S3×C23), D30.C2⋊3C22, (S3×Dic5)⋊3C22, (C6×D5).13C23, D6.27(C22×D5), C12.55(C22×D5), (S3×C10).16C23, (C6×Dic5)⋊14C22, (C3×Dic10)⋊9C22, D10.16(C22×S3), (C22×D15)⋊13C22, Dic5.17(C22×S3), Dic3.16(C22×D5), (C3×Dic5).46C23, (C5×Dic3).17C23, (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
Generators and relations for D12⋊14D10
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 >
Subgroups: 1884 in 332 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×C6, C15, C2×D4, C4○D4, Dic5, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, Dic6, C4×S3, C4×S3, D12, D12, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C5×S3, C3×D5, D15, C30, C30, 2+ 1+4, Dic10, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C2×D12, C4○D12, S3×D4, S3×D4, Q8⋊3S3, C3×C4○D4, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, S3×C10, S3×C10, D30, D30, D30, C2×C30, C4○D20, D4×D5, D4⋊2D5, D4⋊2D5, C2×C5⋊D4, D4×C10, D4○D12, S3×Dic5, D30.C2, C15⋊D4, C3⋊D20, C5⋊D12, C5⋊D12, C15⋊Q8, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, S3×C20, C5×D12, C5×C3⋊D4, C4×D15, D60, C15⋊7D4, D4×C15, C2×S3×D5, S3×C2×C10, C22×D15, D4⋊6D10, 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, D12⋊14D10
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, 2+ 1+4, C22×D5, S3×C23, S3×D5, C23×D5, D4○D12, C2×S3×D5, D4⋊6D10, C22×S3×D5, D12⋊14D10
►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 26)(27 36)(28 35)(29 34)(30 33)(31 32)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 54)(50 53)(51 52)(55 60)(56 59)(57 58)(61 66)(62 65)(63 64)(67 72)(68 71)(69 70)(73 78)(74 77)(75 76)(79 84)(80 83)(81 82)(85 88)(86 87)(89 96)(90 95)(91 94)(92 93)(97 98)(99 108)(100 107)(101 106)(102 105)(103 104)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)
(1 45 109 76 23)(2 40 110 83 24 8 46 116 77 18)(3 47 111 78 13)(4 42 112 73 14 10 48 118 79 20)(5 37 113 80 15)(6 44 114 75 16 12 38 120 81 22)(7 39 115 82 17)(9 41 117 84 19)(11 43 119 74 21)(25 86 51 103 69)(26 93 52 98 70 32 87 58 104 64)(27 88 53 105 71)(28 95 54 100 72 34 89 60 106 66)(29 90 55 107 61)(30 85 56 102 62 36 91 50 108 68)(31 92 57 97 63)(33 94 59 99 65)(35 96 49 101 67)
(1 96)(2 89)(3 94)(4 87)(5 92)(6 85)(7 90)(8 95)(9 88)(10 93)(11 86)(12 91)(13 59)(14 52)(15 57)(16 50)(17 55)(18 60)(19 53)(20 58)(21 51)(22 56)(23 49)(24 54)(25 43)(26 48)(27 41)(28 46)(29 39)(30 44)(31 37)(32 42)(33 47)(34 40)(35 45)(36 38)(61 115)(62 120)(63 113)(64 118)(65 111)(66 116)(67 109)(68 114)(69 119)(70 112)(71 117)(72 110)(73 98)(74 103)(75 108)(76 101)(77 106)(78 99)(79 104)(80 97)(81 102)(82 107)(83 100)(84 105)
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)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,54)(50,53)(51,52)(55,60)(56,59)(57,58)(61,66)(62,65)(63,64)(67,72)(68,71)(69,70)(73,78)(74,77)(75,76)(79,84)(80,83)(81,82)(85,88)(86,87)(89,96)(90,95)(91,94)(92,93)(97,98)(99,108)(100,107)(101,106)(102,105)(103,104)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115), (1,45,109,76,23)(2,40,110,83,24,8,46,116,77,18)(3,47,111,78,13)(4,42,112,73,14,10,48,118,79,20)(5,37,113,80,15)(6,44,114,75,16,12,38,120,81,22)(7,39,115,82,17)(9,41,117,84,19)(11,43,119,74,21)(25,86,51,103,69)(26,93,52,98,70,32,87,58,104,64)(27,88,53,105,71)(28,95,54,100,72,34,89,60,106,66)(29,90,55,107,61)(30,85,56,102,62,36,91,50,108,68)(31,92,57,97,63)(33,94,59,99,65)(35,96,49,101,67), (1,96)(2,89)(3,94)(4,87)(5,92)(6,85)(7,90)(8,95)(9,88)(10,93)(11,86)(12,91)(13,59)(14,52)(15,57)(16,50)(17,55)(18,60)(19,53)(20,58)(21,51)(22,56)(23,49)(24,54)(25,43)(26,48)(27,41)(28,46)(29,39)(30,44)(31,37)(32,42)(33,47)(34,40)(35,45)(36,38)(61,115)(62,120)(63,113)(64,118)(65,111)(66,116)(67,109)(68,114)(69,119)(70,112)(71,117)(72,110)(73,98)(74,103)(75,108)(76,101)(77,106)(78,99)(79,104)(80,97)(81,102)(82,107)(83,100)(84,105)>;
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)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,54)(50,53)(51,52)(55,60)(56,59)(57,58)(61,66)(62,65)(63,64)(67,72)(68,71)(69,70)(73,78)(74,77)(75,76)(79,84)(80,83)(81,82)(85,88)(86,87)(89,96)(90,95)(91,94)(92,93)(97,98)(99,108)(100,107)(101,106)(102,105)(103,104)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115), (1,45,109,76,23)(2,40,110,83,24,8,46,116,77,18)(3,47,111,78,13)(4,42,112,73,14,10,48,118,79,20)(5,37,113,80,15)(6,44,114,75,16,12,38,120,81,22)(7,39,115,82,17)(9,41,117,84,19)(11,43,119,74,21)(25,86,51,103,69)(26,93,52,98,70,32,87,58,104,64)(27,88,53,105,71)(28,95,54,100,72,34,89,60,106,66)(29,90,55,107,61)(30,85,56,102,62,36,91,50,108,68)(31,92,57,97,63)(33,94,59,99,65)(35,96,49,101,67), (1,96)(2,89)(3,94)(4,87)(5,92)(6,85)(7,90)(8,95)(9,88)(10,93)(11,86)(12,91)(13,59)(14,52)(15,57)(16,50)(17,55)(18,60)(19,53)(20,58)(21,51)(22,56)(23,49)(24,54)(25,43)(26,48)(27,41)(28,46)(29,39)(30,44)(31,37)(32,42)(33,47)(34,40)(35,45)(36,38)(61,115)(62,120)(63,113)(64,118)(65,111)(66,116)(67,109)(68,114)(69,119)(70,112)(71,117)(72,110)(73,98)(74,103)(75,108)(76,101)(77,106)(78,99)(79,104)(80,97)(81,102)(82,107)(83,100)(84,105) );
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),(25,26),(27,36),(28,35),(29,34),(30,33),(31,32),(37,40),(38,39),(41,48),(42,47),(43,46),(44,45),(49,54),(50,53),(51,52),(55,60),(56,59),(57,58),(61,66),(62,65),(63,64),(67,72),(68,71),(69,70),(73,78),(74,77),(75,76),(79,84),(80,83),(81,82),(85,88),(86,87),(89,96),(90,95),(91,94),(92,93),(97,98),(99,108),(100,107),(101,106),(102,105),(103,104),(109,120),(110,119),(111,118),(112,117),(113,116),(114,115)], [(1,45,109,76,23),(2,40,110,83,24,8,46,116,77,18),(3,47,111,78,13),(4,42,112,73,14,10,48,118,79,20),(5,37,113,80,15),(6,44,114,75,16,12,38,120,81,22),(7,39,115,82,17),(9,41,117,84,19),(11,43,119,74,21),(25,86,51,103,69),(26,93,52,98,70,32,87,58,104,64),(27,88,53,105,71),(28,95,54,100,72,34,89,60,106,66),(29,90,55,107,61),(30,85,56,102,62,36,91,50,108,68),(31,92,57,97,63),(33,94,59,99,65),(35,96,49,101,67)], [(1,96),(2,89),(3,94),(4,87),(5,92),(6,85),(7,90),(8,95),(9,88),(10,93),(11,86),(12,91),(13,59),(14,52),(15,57),(16,50),(17,55),(18,60),(19,53),(20,58),(21,51),(22,56),(23,49),(24,54),(25,43),(26,48),(27,41),(28,46),(29,39),(30,44),(31,37),(32,42),(33,47),(34,40),(35,45),(36,38),(61,115),(62,120),(63,113),(64,118),(65,111),(66,116),(67,109),(68,114),(69,119),(70,112),(71,117),(72,110),(73,98),(74,103),(75,108),(76,101),(77,106),(78,99),(79,104),(80,97),(81,102),(82,107),(83,100),(84,105)]])
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 |
Matrix representation of D12⋊14D10 ►in 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] >;
D12⋊14D10 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