metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.8D6, D12⋊10D10, C60.13C23, Dic30⋊4C22, C3⋊C8⋊7D10, D4⋊S3⋊4D5, (D4×D5)⋊2S3, C15⋊D8⋊3C2, (C5×D4).5D6, (C4×D5).7D6, D4.9(S3×D5), C3⋊6(D8⋊D5), D4.D15⋊5C2, (C6×D5).62D4, C6.142(D4×D5), D12⋊5D5⋊1C2, C15⋊16(C8⋊C22), C15⋊3C8⋊7C22, (C3×D4).20D10, C30.175(C2×D4), C6.D20⋊3C2, C5⋊2(D12⋊6C22), (C5×D12)⋊4C22, C20.32D6⋊3C2, C20.13(C22×S3), (C3×Dic5).13D4, (D5×C12).5C22, (C3×D20).4C22, (D4×C15).7C22, C12.13(C22×D5), D10.28(C3⋊D4), Dic5.22(C3⋊D4), (C3×D4×D5)⋊2C2, C4.13(C2×S3×D5), (C5×D4⋊S3)⋊5C2, (C5×C3⋊C8)⋊7C22, C2.24(D5×C3⋊D4), C10.45(C2×C3⋊D4), SmallGroup(480,565)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊10D10
G = < a,b,c,d | a12=b2=c10=d2=1, bab=a-1, cac-1=dad=a7, cbc-1=a3b, dbd=a9b, dcd=c-1 >
Subgroups: 748 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×C6, C5×S3, C3×D5, C30, C30, C8⋊C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C5×D4, C22×D5, C4.Dic3, D4⋊S3, D4⋊S3, D4.S3, C4○D12, C6×D4, C3×Dic5, Dic15, C60, C6×D5, C6×D5, S3×C10, C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D4⋊2D5, D12⋊6C22, C5×C3⋊C8, C15⋊3C8, S3×Dic5, C15⋊D4, D5×C12, C3×D20, C3×C5⋊D4, C5×D12, Dic30, D4×C15, D5×C2×C6, D8⋊D5, C20.32D6, C15⋊D8, C6.D20, C5×D4⋊S3, D4.D15, D12⋊5D5, C3×D4×D5, D12⋊10D10
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C8⋊C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, D12⋊6C22, C2×S3×D5, D8⋊D5, D5×C3⋊D4, D12⋊10D10
►On 120 points
Generators in S120
(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 51)(2 50)(3 49)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 52)(13 88)(14 87)(15 86)(16 85)(17 96)(18 95)(19 94)(20 93)(21 92)(22 91)(23 90)(24 89)(25 77)(26 76)(27 75)(28 74)(29 73)(30 84)(31 83)(32 82)(33 81)(34 80)(35 79)(36 78)(37 66)(38 65)(39 64)(40 63)(41 62)(42 61)(43 72)(44 71)(45 70)(46 69)(47 68)(48 67)(97 111)(98 110)(99 109)(100 120)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)
(1 65 25 21 119 4 62 28 18 110)(2 72 26 16 120 11 63 35 19 117)(3 67 27 23 109 6 64 30 20 112)(5 69 29 13 111 8 66 32 22 114)(7 71 31 15 113 10 68 34 24 116)(9 61 33 17 115 12 70 36 14 118)(37 79 91 103 59 43 73 85 97 53)(38 74 92 98 60)(39 81 93 105 49 45 75 87 99 55)(40 76 94 100 50)(41 83 95 107 51 47 77 89 101 57)(42 78 96 102 52)(44 80 86 104 54)(46 82 88 106 56)(48 84 90 108 58)
(1 110)(2 117)(3 112)(4 119)(5 114)(6 109)(7 116)(8 111)(9 118)(10 113)(11 120)(12 115)(13 66)(14 61)(15 68)(16 63)(17 70)(18 65)(19 72)(20 67)(21 62)(22 69)(23 64)(24 71)(25 28)(26 35)(27 30)(29 32)(31 34)(33 36)(37 91)(38 86)(39 93)(40 88)(41 95)(42 90)(43 85)(44 92)(45 87)(46 94)(47 89)(48 96)(49 99)(50 106)(51 101)(52 108)(53 103)(54 98)(55 105)(56 100)(57 107)(58 102)(59 97)(60 104)(74 80)(76 82)(78 84)
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), (1,51)(2,50)(3,49)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,88)(14,87)(15,86)(16,85)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,77)(26,76)(27,75)(28,74)(29,73)(30,84)(31,83)(32,82)(33,81)(34,80)(35,79)(36,78)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(97,111)(98,110)(99,109)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112), (1,65,25,21,119,4,62,28,18,110)(2,72,26,16,120,11,63,35,19,117)(3,67,27,23,109,6,64,30,20,112)(5,69,29,13,111,8,66,32,22,114)(7,71,31,15,113,10,68,34,24,116)(9,61,33,17,115,12,70,36,14,118)(37,79,91,103,59,43,73,85,97,53)(38,74,92,98,60)(39,81,93,105,49,45,75,87,99,55)(40,76,94,100,50)(41,83,95,107,51,47,77,89,101,57)(42,78,96,102,52)(44,80,86,104,54)(46,82,88,106,56)(48,84,90,108,58), (1,110)(2,117)(3,112)(4,119)(5,114)(6,109)(7,116)(8,111)(9,118)(10,113)(11,120)(12,115)(13,66)(14,61)(15,68)(16,63)(17,70)(18,65)(19,72)(20,67)(21,62)(22,69)(23,64)(24,71)(25,28)(26,35)(27,30)(29,32)(31,34)(33,36)(37,91)(38,86)(39,93)(40,88)(41,95)(42,90)(43,85)(44,92)(45,87)(46,94)(47,89)(48,96)(49,99)(50,106)(51,101)(52,108)(53,103)(54,98)(55,105)(56,100)(57,107)(58,102)(59,97)(60,104)(74,80)(76,82)(78,84)>;
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), (1,51)(2,50)(3,49)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,88)(14,87)(15,86)(16,85)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,77)(26,76)(27,75)(28,74)(29,73)(30,84)(31,83)(32,82)(33,81)(34,80)(35,79)(36,78)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(97,111)(98,110)(99,109)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112), (1,65,25,21,119,4,62,28,18,110)(2,72,26,16,120,11,63,35,19,117)(3,67,27,23,109,6,64,30,20,112)(5,69,29,13,111,8,66,32,22,114)(7,71,31,15,113,10,68,34,24,116)(9,61,33,17,115,12,70,36,14,118)(37,79,91,103,59,43,73,85,97,53)(38,74,92,98,60)(39,81,93,105,49,45,75,87,99,55)(40,76,94,100,50)(41,83,95,107,51,47,77,89,101,57)(42,78,96,102,52)(44,80,86,104,54)(46,82,88,106,56)(48,84,90,108,58), (1,110)(2,117)(3,112)(4,119)(5,114)(6,109)(7,116)(8,111)(9,118)(10,113)(11,120)(12,115)(13,66)(14,61)(15,68)(16,63)(17,70)(18,65)(19,72)(20,67)(21,62)(22,69)(23,64)(24,71)(25,28)(26,35)(27,30)(29,32)(31,34)(33,36)(37,91)(38,86)(39,93)(40,88)(41,95)(42,90)(43,85)(44,92)(45,87)(46,94)(47,89)(48,96)(49,99)(50,106)(51,101)(52,108)(53,103)(54,98)(55,105)(56,100)(57,107)(58,102)(59,97)(60,104)(74,80)(76,82)(78,84) );
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)], [(1,51),(2,50),(3,49),(4,60),(5,59),(6,58),(7,57),(8,56),(9,55),(10,54),(11,53),(12,52),(13,88),(14,87),(15,86),(16,85),(17,96),(18,95),(19,94),(20,93),(21,92),(22,91),(23,90),(24,89),(25,77),(26,76),(27,75),(28,74),(29,73),(30,84),(31,83),(32,82),(33,81),(34,80),(35,79),(36,78),(37,66),(38,65),(39,64),(40,63),(41,62),(42,61),(43,72),(44,71),(45,70),(46,69),(47,68),(48,67),(97,111),(98,110),(99,109),(100,120),(101,119),(102,118),(103,117),(104,116),(105,115),(106,114),(107,113),(108,112)], [(1,65,25,21,119,4,62,28,18,110),(2,72,26,16,120,11,63,35,19,117),(3,67,27,23,109,6,64,30,20,112),(5,69,29,13,111,8,66,32,22,114),(7,71,31,15,113,10,68,34,24,116),(9,61,33,17,115,12,70,36,14,118),(37,79,91,103,59,43,73,85,97,53),(38,74,92,98,60),(39,81,93,105,49,45,75,87,99,55),(40,76,94,100,50),(41,83,95,107,51,47,77,89,101,57),(42,78,96,102,52),(44,80,86,104,54),(46,82,88,106,56),(48,84,90,108,58)], [(1,110),(2,117),(3,112),(4,119),(5,114),(6,109),(7,116),(8,111),(9,118),(10,113),(11,120),(12,115),(13,66),(14,61),(15,68),(16,63),(17,70),(18,65),(19,72),(20,67),(21,62),(22,69),(23,64),(24,71),(25,28),(26,35),(27,30),(29,32),(31,34),(33,36),(37,91),(38,86),(39,93),(40,88),(41,95),(42,90),(43,85),(44,92),(45,87),(46,94),(47,89),(48,96),(49,99),(50,106),(51,101),(52,108),(53,103),(54,98),(55,105),(56,100),(57,107),(58,102),(59,97),(60,104),(74,80),(76,82),(78,84)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 20A | 20B | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 10 | 12 | 20 | 2 | 2 | 10 | 60 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 12 | 60 | 2 | 2 | 8 | 8 | 24 | 24 | 4 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
45 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 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C3⋊D4 | C3⋊D4 | C8⋊C22 | S3×D5 | D4×D5 | D12⋊6C22 | C2×S3×D5 | D8⋊D5 | D5×C3⋊D4 | D12⋊10D10 |
| kernel | D12⋊10D10 | C20.32D6 | C15⋊D8 | C6.D20 | C5×D4⋊S3 | D4.D15 | D12⋊5D5 | C3×D4×D5 | D4×D5 | C3×Dic5 | C6×D5 | D4⋊S3 | C4×D5 | D20 | C5×D4 | C3⋊C8 | D12 | C3×D4 | Dic5 | D10 | C15 | D4 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D12⋊10D10 ►in GL6(𝔽241)
| 15 | 0 | 0 | 0 | 0 | 0 |
| 239 | 225 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 131 | 223 | 0 | 0 | 0 | 0 |
| 150 | 110 | 0 | 0 | 0 | 0 |
| 0 | 0 | 113 | 57 | 128 | 184 |
| 0 | 0 | 184 | 128 | 57 | 113 |
| 0 | 0 | 128 | 184 | 128 | 184 |
| 0 | 0 | 57 | 113 | 57 | 113 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 190 | 190 |
| 0 | 0 | 0 | 0 | 51 | 240 |
| 0 | 0 | 190 | 190 | 0 | 0 |
| 0 | 0 | 51 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 190 | 190 |
| 0 | 0 | 0 | 0 | 240 | 51 |
| 0 | 0 | 190 | 190 | 0 | 0 |
| 0 | 0 | 240 | 51 | 0 | 0 |
G:=sub<GL(6,GF(241))| [15,239,0,0,0,0,0,225,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[131,150,0,0,0,0,223,110,0,0,0,0,0,0,113,184,128,57,0,0,57,128,184,113,0,0,128,57,128,57,0,0,184,113,184,113],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,190,51,0,0,0,0,190,240,0,0,190,51,0,0,0,0,190,240,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,190,240,0,0,0,0,190,51,0,0,190,240,0,0,0,0,190,51,0,0] >;
D12⋊10D10 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{10}D_{10} % in TeX
G:=Group("D12:10D10"); // GroupNames label
G:=SmallGroup(480,565);
// by ID
G=gap.SmallGroup(480,565);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,135,346,185,80,1356,18822]);
// 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=d*a*d=a^7,c*b*c^-1=a^3*b,d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations