metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.9D4, C20.6D6, C15⋊6SD16, Dic6⋊1D5, D60.4C2, C10.8D12, C12.24D10, C60.17C22, C5⋊2C8⋊3S3, C3⋊1(Q8⋊D5), C5⋊2(C24⋊C2), C4.10(S3×D5), (C5×Dic6)⋊1C2, C6.3(C5⋊D4), C2.6(C5⋊D12), (C3×C5⋊2C8)⋊3C2, SmallGroup(240,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊D5
G = < a,b,c,d | a12=c5=d2=1, b2=a6, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a3b, dcd=c-1 >
Smallest permutation representation of Dic6⋊D5
►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 91 7 85)(2 90 8 96)(3 89 9 95)(4 88 10 94)(5 87 11 93)(6 86 12 92)(13 107 19 101)(14 106 20 100)(15 105 21 99)(16 104 22 98)(17 103 23 97)(18 102 24 108)(25 57 31 51)(26 56 32 50)(27 55 33 49)(28 54 34 60)(29 53 35 59)(30 52 36 58)(37 67 43 61)(38 66 44 72)(39 65 45 71)(40 64 46 70)(41 63 47 69)(42 62 48 68)(73 110 79 116)(74 109 80 115)(75 120 81 114)(76 119 82 113)(77 118 83 112)(78 117 84 111)
(1 66 119 29 99)(2 67 120 30 100)(3 68 109 31 101)(4 69 110 32 102)(5 70 111 33 103)(6 71 112 34 104)(7 72 113 35 105)(8 61 114 36 106)(9 62 115 25 107)(10 63 116 26 108)(11 64 117 27 97)(12 65 118 28 98)(13 89 42 80 51)(14 90 43 81 52)(15 91 44 82 53)(16 92 45 83 54)(17 93 46 84 55)(18 94 47 73 56)(19 95 48 74 57)(20 96 37 75 58)(21 85 38 76 59)(22 86 39 77 60)(23 87 40 78 49)(24 88 41 79 50)
(1 99)(2 98)(3 97)(4 108)(5 107)(6 106)(7 105)(8 104)(9 103)(10 102)(11 101)(12 100)(13 90)(14 89)(15 88)(16 87)(17 86)(18 85)(19 96)(20 95)(21 94)(22 93)(23 92)(24 91)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 72)(36 71)(37 57)(38 56)(39 55)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(46 60)(47 59)(48 58)(73 76)(74 75)(77 84)(78 83)(79 82)(80 81)(109 117)(110 116)(111 115)(112 114)(118 120)
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), (1,91,7,85)(2,90,8,96)(3,89,9,95)(4,88,10,94)(5,87,11,93)(6,86,12,92)(13,107,19,101)(14,106,20,100)(15,105,21,99)(16,104,22,98)(17,103,23,97)(18,102,24,108)(25,57,31,51)(26,56,32,50)(27,55,33,49)(28,54,34,60)(29,53,35,59)(30,52,36,58)(37,67,43,61)(38,66,44,72)(39,65,45,71)(40,64,46,70)(41,63,47,69)(42,62,48,68)(73,110,79,116)(74,109,80,115)(75,120,81,114)(76,119,82,113)(77,118,83,112)(78,117,84,111), (1,66,119,29,99)(2,67,120,30,100)(3,68,109,31,101)(4,69,110,32,102)(5,70,111,33,103)(6,71,112,34,104)(7,72,113,35,105)(8,61,114,36,106)(9,62,115,25,107)(10,63,116,26,108)(11,64,117,27,97)(12,65,118,28,98)(13,89,42,80,51)(14,90,43,81,52)(15,91,44,82,53)(16,92,45,83,54)(17,93,46,84,55)(18,94,47,73,56)(19,95,48,74,57)(20,96,37,75,58)(21,85,38,76,59)(22,86,39,77,60)(23,87,40,78,49)(24,88,41,79,50), (1,99)(2,98)(3,97)(4,108)(5,107)(6,106)(7,105)(8,104)(9,103)(10,102)(11,101)(12,100)(13,90)(14,89)(15,88)(16,87)(17,86)(18,85)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,72)(36,71)(37,57)(38,56)(39,55)(40,54)(41,53)(42,52)(43,51)(44,50)(45,49)(46,60)(47,59)(48,58)(73,76)(74,75)(77,84)(78,83)(79,82)(80,81)(109,117)(110,116)(111,115)(112,114)(118,120)>;
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), (1,91,7,85)(2,90,8,96)(3,89,9,95)(4,88,10,94)(5,87,11,93)(6,86,12,92)(13,107,19,101)(14,106,20,100)(15,105,21,99)(16,104,22,98)(17,103,23,97)(18,102,24,108)(25,57,31,51)(26,56,32,50)(27,55,33,49)(28,54,34,60)(29,53,35,59)(30,52,36,58)(37,67,43,61)(38,66,44,72)(39,65,45,71)(40,64,46,70)(41,63,47,69)(42,62,48,68)(73,110,79,116)(74,109,80,115)(75,120,81,114)(76,119,82,113)(77,118,83,112)(78,117,84,111), (1,66,119,29,99)(2,67,120,30,100)(3,68,109,31,101)(4,69,110,32,102)(5,70,111,33,103)(6,71,112,34,104)(7,72,113,35,105)(8,61,114,36,106)(9,62,115,25,107)(10,63,116,26,108)(11,64,117,27,97)(12,65,118,28,98)(13,89,42,80,51)(14,90,43,81,52)(15,91,44,82,53)(16,92,45,83,54)(17,93,46,84,55)(18,94,47,73,56)(19,95,48,74,57)(20,96,37,75,58)(21,85,38,76,59)(22,86,39,77,60)(23,87,40,78,49)(24,88,41,79,50), (1,99)(2,98)(3,97)(4,108)(5,107)(6,106)(7,105)(8,104)(9,103)(10,102)(11,101)(12,100)(13,90)(14,89)(15,88)(16,87)(17,86)(18,85)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,72)(36,71)(37,57)(38,56)(39,55)(40,54)(41,53)(42,52)(43,51)(44,50)(45,49)(46,60)(47,59)(48,58)(73,76)(74,75)(77,84)(78,83)(79,82)(80,81)(109,117)(110,116)(111,115)(112,114)(118,120) );
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)], [(1,91,7,85),(2,90,8,96),(3,89,9,95),(4,88,10,94),(5,87,11,93),(6,86,12,92),(13,107,19,101),(14,106,20,100),(15,105,21,99),(16,104,22,98),(17,103,23,97),(18,102,24,108),(25,57,31,51),(26,56,32,50),(27,55,33,49),(28,54,34,60),(29,53,35,59),(30,52,36,58),(37,67,43,61),(38,66,44,72),(39,65,45,71),(40,64,46,70),(41,63,47,69),(42,62,48,68),(73,110,79,116),(74,109,80,115),(75,120,81,114),(76,119,82,113),(77,118,83,112),(78,117,84,111)], [(1,66,119,29,99),(2,67,120,30,100),(3,68,109,31,101),(4,69,110,32,102),(5,70,111,33,103),(6,71,112,34,104),(7,72,113,35,105),(8,61,114,36,106),(9,62,115,25,107),(10,63,116,26,108),(11,64,117,27,97),(12,65,118,28,98),(13,89,42,80,51),(14,90,43,81,52),(15,91,44,82,53),(16,92,45,83,54),(17,93,46,84,55),(18,94,47,73,56),(19,95,48,74,57),(20,96,37,75,58),(21,85,38,76,59),(22,86,39,77,60),(23,87,40,78,49),(24,88,41,79,50)], [(1,99),(2,98),(3,97),(4,108),(5,107),(6,106),(7,105),(8,104),(9,103),(10,102),(11,101),(12,100),(13,90),(14,89),(15,88),(16,87),(17,86),(18,85),(19,96),(20,95),(21,94),(22,93),(23,92),(24,91),(25,70),(26,69),(27,68),(28,67),(29,66),(30,65),(31,64),(32,63),(33,62),(34,61),(35,72),(36,71),(37,57),(38,56),(39,55),(40,54),(41,53),(42,52),(43,51),(44,50),(45,49),(46,60),(47,59),(48,58),(73,76),(74,75),(77,84),(78,83),(79,82),(80,81),(109,117),(110,116),(111,115),(112,114),(118,120)]])
Dic6⋊D5 is a maximal subgroup of
D5×C24⋊C2 C24⋊D10 C24.2D10 D120⋊C2 C20.60D12 C60.38D4 C20.D12 D60.C22 Dic10⋊D6 C60.19C23 Dic6⋊D10 S3×Q8⋊D5 C60.C23 C60.44C23 D20.16D6
Dic6⋊D5 is a maximal quotient of
D60⋊15C4 C10.Dic12 C60.7Q8
33 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 60 | 2 | 2 | 12 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | SD16 | D10 | D12 | C5⋊D4 | C24⋊C2 | S3×D5 | Q8⋊D5 | C5⋊D12 | Dic6⋊D5 |
| kernel | Dic6⋊D5 | C3×C5⋊2C8 | C5×Dic6 | D60 | C5⋊2C8 | C30 | Dic6 | C20 | C15 | C12 | C10 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of Dic6⋊D5 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 43 | 142 |
| 0 | 0 | 99 | 142 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 213 | 147 |
| 0 | 0 | 175 | 28 |
| 0 | 1 | 0 | 0 |
| 240 | 189 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 240 | 240 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,43,99,0,0,142,142],[1,0,0,0,0,1,0,0,0,0,213,175,0,0,147,28],[0,240,0,0,1,189,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,240,0,0,0,240] >;
Dic6⋊D5 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes D_5 % in TeX
G:=Group("Dic6:D5"); // GroupNames label
G:=SmallGroup(240,21);
// by ID
G=gap.SmallGroup(240,21);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,169,55,116,50,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^5=d^2=1,b^2=a^6,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations
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