metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊3D6, C60.2C23, D12.23D10, D60.1C22, C3⋊C8⋊5D10, D4⋊S3⋊1D5, D4⋊4(S3×D5), (C5×D4)⋊2D6, D4⋊D15⋊2C2, (C3×D4)⋊8D10, (D5×D12)⋊2C2, (C6×D5).8D4, (C4×D5).5D6, D4⋊2D5⋊1S3, C3⋊7(D8⋊D5), C5⋊2(D4⋊D6), C6.139(D4×D5), C15⋊13(C8⋊C22), C15⋊3C8⋊2C22, C20.D6⋊1C2, C15⋊SD16⋊1C2, C30.164(C2×D4), (D4×C15)⋊2C22, C20.2(C22×S3), C12.2(C22×D5), C20.32D6⋊1C2, (C3×Dic5).66D4, (D5×C12).2C22, (C5×D12).1C22, D10.17(C3⋊D4), (C3×Dic10)⋊1C22, Dic5.31(C3⋊D4), C4.2(C2×S3×D5), (C5×D4⋊S3)⋊2C2, (C5×C3⋊C8)⋊2C22, C2.21(D5×C3⋊D4), (C3×D4⋊2D5)⋊1C2, C10.42(C2×C3⋊D4), SmallGroup(480,554)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊3D6
G = < a,b,c,d | a20=c6=d2=1, b2=a10, bab-1=dad=a-1, cac-1=a9, cbc-1=a10b, dbd=a5b, dcd=c-1 >
Subgroups: 892 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, 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, D12, D12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, 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, Q8⋊2S3, C2×D12, C3×C4○D4, C3×Dic5, C3×Dic5, C60, S3×D5, C6×D5, S3×C10, D30, C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D4⋊2D5, D4⋊D6, C5×C3⋊C8, C15⋊3C8, C5⋊D12, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, C5×D12, D60, D4×C15, C2×S3×D5, D8⋊D5, C20.32D6, C20.D6, C15⋊SD16, C5×D4⋊S3, D4⋊D15, D5×D12, C3×D4⋊2D5, Dic10⋊3D6
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, D4⋊D6, C2×S3×D5, D8⋊D5, D5×C3⋊D4, Dic10⋊3D6
►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 73 11 63)(2 72 12 62)(3 71 13 61)(4 70 14 80)(5 69 15 79)(6 68 16 78)(7 67 17 77)(8 66 18 76)(9 65 19 75)(10 64 20 74)(21 99 31 89)(22 98 32 88)(23 97 33 87)(24 96 34 86)(25 95 35 85)(26 94 36 84)(27 93 37 83)(28 92 38 82)(29 91 39 81)(30 90 40 100)(41 112 51 102)(42 111 52 101)(43 110 53 120)(44 109 54 119)(45 108 55 118)(46 107 56 117)(47 106 57 116)(48 105 58 115)(49 104 59 114)(50 103 60 113)
(1 21 55)(2 30 56 10 22 44)(3 39 57 19 23 53)(4 28 58 8 24 42)(5 37 59 17 25 51)(6 26 60)(7 35 41 15 27 49)(9 33 43 13 29 47)(11 31 45)(12 40 46 20 32 54)(14 38 48 18 34 52)(16 36 50)(61 81 106 75 87 120)(62 90 107 64 88 109)(63 99 108 73 89 118)(65 97 110 71 91 116)(66 86 111 80 92 105)(67 95 112 69 93 114)(68 84 113 78 94 103)(70 82 115 76 96 101)(72 100 117 74 98 119)(77 85 102 79 83 104)
(1 50)(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 112)(80 111)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(98 100)
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), (1,73,11,63)(2,72,12,62)(3,71,13,61)(4,70,14,80)(5,69,15,79)(6,68,16,78)(7,67,17,77)(8,66,18,76)(9,65,19,75)(10,64,20,74)(21,99,31,89)(22,98,32,88)(23,97,33,87)(24,96,34,86)(25,95,35,85)(26,94,36,84)(27,93,37,83)(28,92,38,82)(29,91,39,81)(30,90,40,100)(41,112,51,102)(42,111,52,101)(43,110,53,120)(44,109,54,119)(45,108,55,118)(46,107,56,117)(47,106,57,116)(48,105,58,115)(49,104,59,114)(50,103,60,113), (1,21,55)(2,30,56,10,22,44)(3,39,57,19,23,53)(4,28,58,8,24,42)(5,37,59,17,25,51)(6,26,60)(7,35,41,15,27,49)(9,33,43,13,29,47)(11,31,45)(12,40,46,20,32,54)(14,38,48,18,34,52)(16,36,50)(61,81,106,75,87,120)(62,90,107,64,88,109)(63,99,108,73,89,118)(65,97,110,71,91,116)(66,86,111,80,92,105)(67,95,112,69,93,114)(68,84,113,78,94,103)(70,82,115,76,96,101)(72,100,117,74,98,119)(77,85,102,79,83,104), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100)>;
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), (1,73,11,63)(2,72,12,62)(3,71,13,61)(4,70,14,80)(5,69,15,79)(6,68,16,78)(7,67,17,77)(8,66,18,76)(9,65,19,75)(10,64,20,74)(21,99,31,89)(22,98,32,88)(23,97,33,87)(24,96,34,86)(25,95,35,85)(26,94,36,84)(27,93,37,83)(28,92,38,82)(29,91,39,81)(30,90,40,100)(41,112,51,102)(42,111,52,101)(43,110,53,120)(44,109,54,119)(45,108,55,118)(46,107,56,117)(47,106,57,116)(48,105,58,115)(49,104,59,114)(50,103,60,113), (1,21,55)(2,30,56,10,22,44)(3,39,57,19,23,53)(4,28,58,8,24,42)(5,37,59,17,25,51)(6,26,60)(7,35,41,15,27,49)(9,33,43,13,29,47)(11,31,45)(12,40,46,20,32,54)(14,38,48,18,34,52)(16,36,50)(61,81,106,75,87,120)(62,90,107,64,88,109)(63,99,108,73,89,118)(65,97,110,71,91,116)(66,86,111,80,92,105)(67,95,112,69,93,114)(68,84,113,78,94,103)(70,82,115,76,96,101)(72,100,117,74,98,119)(77,85,102,79,83,104), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,100) );
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)], [(1,73,11,63),(2,72,12,62),(3,71,13,61),(4,70,14,80),(5,69,15,79),(6,68,16,78),(7,67,17,77),(8,66,18,76),(9,65,19,75),(10,64,20,74),(21,99,31,89),(22,98,32,88),(23,97,33,87),(24,96,34,86),(25,95,35,85),(26,94,36,84),(27,93,37,83),(28,92,38,82),(29,91,39,81),(30,90,40,100),(41,112,51,102),(42,111,52,101),(43,110,53,120),(44,109,54,119),(45,108,55,118),(46,107,56,117),(47,106,57,116),(48,105,58,115),(49,104,59,114),(50,103,60,113)], [(1,21,55),(2,30,56,10,22,44),(3,39,57,19,23,53),(4,28,58,8,24,42),(5,37,59,17,25,51),(6,26,60),(7,35,41,15,27,49),(9,33,43,13,29,47),(11,31,45),(12,40,46,20,32,54),(14,38,48,18,34,52),(16,36,50),(61,81,106,75,87,120),(62,90,107,64,88,109),(63,99,108,73,89,118),(65,97,110,71,91,116),(66,86,111,80,92,105),(67,95,112,69,93,114),(68,84,113,78,94,103),(70,82,115,76,96,101),(72,100,117,74,98,119),(77,85,102,79,83,104)], [(1,50),(2,49),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,42),(10,41),(11,60),(12,59),(13,58),(14,57),(15,56),(16,55),(17,54),(18,53),(19,52),(20,51),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(37,40),(38,39),(61,110),(62,109),(63,108),(64,107),(65,106),(66,105),(67,104),(68,103),(69,102),(70,101),(71,120),(72,119),(73,118),(74,117),(75,116),(76,115),(77,114),(78,113),(79,112),(80,111),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(98,100)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 12D | 12E | 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 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 10 | 12 | 60 | 2 | 2 | 10 | 20 | 2 | 2 | 2 | 4 | 4 | 20 | 12 | 60 | 2 | 2 | 8 | 8 | 24 | 24 | 4 | 10 | 10 | 20 | 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 | D4⋊D6 | C2×S3×D5 | D8⋊D5 | D5×C3⋊D4 | Dic10⋊3D6 |
| kernel | Dic10⋊3D6 | C20.32D6 | C20.D6 | C15⋊SD16 | C5×D4⋊S3 | D4⋊D15 | D5×D12 | C3×D4⋊2D5 | D4⋊2D5 | C3×Dic5 | C6×D5 | D4⋊S3 | Dic10 | C4×D5 | 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 Dic10⋊3D6 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 1 | 240 | 188 |
| 0 | 0 | 240 | 190 | 52 | 51 |
| 0 | 0 | 233 | 125 | 0 | 240 |
| 0 | 0 | 234 | 124 | 0 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 148 | 225 |
| 0 | 0 | 164 | 0 | 148 | 77 |
| 0 | 0 | 15 | 169 | 0 | 0 |
| 0 | 0 | 200 | 57 | 148 | 225 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 1 | 0 | 0 |
| 0 | 0 | 51 | 190 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 52 | 1 | 189 | 240 |
| 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 1 | 188 | 240 |
| 0 | 0 | 240 | 190 | 51 | 52 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 51 | 1 | 240 | 0 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,52,240,233,234,0,0,1,190,125,124,0,0,240,52,0,0,0,0,188,51,240,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,16,164,15,200,0,0,0,0,169,57,0,0,148,148,0,148,0,0,225,77,0,225],[0,1,0,0,0,0,240,240,0,0,0,0,0,0,51,51,0,52,0,0,1,190,0,1,0,0,0,0,1,189,0,0,0,0,0,240],[1,0,0,0,0,0,240,240,0,0,0,0,0,0,52,240,0,51,0,0,1,190,0,1,0,0,188,51,240,240,0,0,240,52,0,0] >;
Dic10⋊3D6 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_3D_6 % in TeX
G:=Group("Dic10:3D6"); // GroupNames label
G:=SmallGroup(480,554);
// by ID
G=gap.SmallGroup(480,554);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^6=d^2=1,b^2=a^10,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^10*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations