metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊4D10, Dic10⋊4D6, D15⋊2SD16, D30.38D4, C60.11C23, D60.3C22, Dic15.12D4, C3⋊C8⋊16D10, C5⋊3(S3×SD16), C3⋊3(D5×SD16), C5⋊2C8⋊16D6, D15⋊Q8⋊1C2, D4.D5⋊2S3, D4.S3⋊2D5, (C5×D4).3D6, C6.69(D4×D5), (C3×D4).3D10, (D4×D15).1C2, D4.13(S3×D5), C10.70(S3×D4), C15⋊12(C2×SD16), D15⋊2C8⋊2C2, C30.173(C2×D4), Dic6⋊D5⋊2C2, C15⋊SD16⋊2C2, C20.11(C22×S3), (C5×Dic6)⋊3C22, (C4×D15).3C22, (D4×C15).5C22, C12.11(C22×D5), (C3×Dic10)⋊3C22, C2.22(D10⋊D6), C4.11(C2×S3×D5), (C5×C3⋊C8)⋊6C22, (C3×D4.D5)⋊3C2, (C5×D4.S3)⋊3C2, (C3×C5⋊2C8)⋊6C22, SmallGroup(480,563)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊D6
G = < a,b,c,d | a20=c6=d2=1, b2=a10, bab-1=dad=a-1, cac-1=a11, cbc-1=dbd=a5b, dcd=c-1 >
Subgroups: 956 in 136 conjugacy classes, 40 normal (38 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, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, D15, D15, C30, C30, C2×SD16, C5⋊2C8, C40, Dic10, Dic10, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, S3×C8, C24⋊C2, D4.S3, Q8⋊2S3, C3×SD16, S3×D4, S3×Q8, C5×Dic3, C3×Dic5, Dic15, C60, D30, D30, C2×C30, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, S3×SD16, C5×C3⋊C8, C3×C5⋊2C8, D30.C2, C15⋊Q8, C3×Dic10, C5×Dic6, C4×D15, D60, C15⋊7D4, D4×C15, C22×D15, D5×SD16, D15⋊2C8, C15⋊SD16, Dic6⋊D5, C3×D4.D5, C5×D4.S3, D15⋊Q8, D4×D15, Dic10⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, C22×S3, C2×SD16, C22×D5, S3×D4, S3×D5, D4×D5, S3×SD16, C2×S3×D5, D5×SD16, D10⋊D6, Dic10⋊D6
►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 57 11 47)(2 56 12 46)(3 55 13 45)(4 54 14 44)(5 53 15 43)(6 52 16 42)(7 51 17 41)(8 50 18 60)(9 49 19 59)(10 48 20 58)(21 73 31 63)(22 72 32 62)(23 71 33 61)(24 70 34 80)(25 69 35 79)(26 68 36 78)(27 67 37 77)(28 66 38 76)(29 65 39 75)(30 64 40 74)(81 117 91 107)(82 116 92 106)(83 115 93 105)(84 114 94 104)(85 113 95 103)(86 112 96 102)(87 111 97 101)(88 110 98 120)(89 109 99 119)(90 108 100 118)
(1 62 114)(2 73 115 12 63 105)(3 64 116)(4 75 117 14 65 107)(5 66 118)(6 77 119 16 67 109)(7 68 120)(8 79 101 18 69 111)(9 70 102)(10 61 103 20 71 113)(11 72 104)(13 74 106)(15 76 108)(17 78 110)(19 80 112)(21 98 56 26 93 41)(22 89 57 37 94 52)(23 100 58 28 95 43)(24 91 59 39 96 54)(25 82 60 30 97 45)(27 84 42 32 99 47)(29 86 44 34 81 49)(31 88 46 36 83 51)(33 90 48 38 85 53)(35 92 50 40 87 55)
(1 104)(2 103)(3 102)(4 101)(5 120)(6 119)(7 118)(8 117)(9 116)(10 115)(11 114)(12 113)(13 112)(14 111)(15 110)(16 109)(17 108)(18 107)(19 106)(20 105)(21 28)(22 27)(23 26)(24 25)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 95)(42 94)(43 93)(44 92)(45 91)(46 90)(47 89)(48 88)(49 87)(50 86)(51 85)(52 84)(53 83)(54 82)(55 81)(56 100)(57 99)(58 98)(59 97)(60 96)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)
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), (1,57,11,47)(2,56,12,46)(3,55,13,45)(4,54,14,44)(5,53,15,43)(6,52,16,42)(7,51,17,41)(8,50,18,60)(9,49,19,59)(10,48,20,58)(21,73,31,63)(22,72,32,62)(23,71,33,61)(24,70,34,80)(25,69,35,79)(26,68,36,78)(27,67,37,77)(28,66,38,76)(29,65,39,75)(30,64,40,74)(81,117,91,107)(82,116,92,106)(83,115,93,105)(84,114,94,104)(85,113,95,103)(86,112,96,102)(87,111,97,101)(88,110,98,120)(89,109,99,119)(90,108,100,118), (1,62,114)(2,73,115,12,63,105)(3,64,116)(4,75,117,14,65,107)(5,66,118)(6,77,119,16,67,109)(7,68,120)(8,79,101,18,69,111)(9,70,102)(10,61,103,20,71,113)(11,72,104)(13,74,106)(15,76,108)(17,78,110)(19,80,112)(21,98,56,26,93,41)(22,89,57,37,94,52)(23,100,58,28,95,43)(24,91,59,39,96,54)(25,82,60,30,97,45)(27,84,42,32,99,47)(29,86,44,34,81,49)(31,88,46,36,83,51)(33,90,48,38,85,53)(35,92,50,40,87,55), (1,104)(2,103)(3,102)(4,101)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,95)(42,94)(43,93)(44,92)(45,91)(46,90)(47,89)(48,88)(49,87)(50,86)(51,85)(52,84)(53,83)(54,82)(55,81)(56,100)(57,99)(58,98)(59,97)(60,96)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)>;
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), (1,57,11,47)(2,56,12,46)(3,55,13,45)(4,54,14,44)(5,53,15,43)(6,52,16,42)(7,51,17,41)(8,50,18,60)(9,49,19,59)(10,48,20,58)(21,73,31,63)(22,72,32,62)(23,71,33,61)(24,70,34,80)(25,69,35,79)(26,68,36,78)(27,67,37,77)(28,66,38,76)(29,65,39,75)(30,64,40,74)(81,117,91,107)(82,116,92,106)(83,115,93,105)(84,114,94,104)(85,113,95,103)(86,112,96,102)(87,111,97,101)(88,110,98,120)(89,109,99,119)(90,108,100,118), (1,62,114)(2,73,115,12,63,105)(3,64,116)(4,75,117,14,65,107)(5,66,118)(6,77,119,16,67,109)(7,68,120)(8,79,101,18,69,111)(9,70,102)(10,61,103,20,71,113)(11,72,104)(13,74,106)(15,76,108)(17,78,110)(19,80,112)(21,98,56,26,93,41)(22,89,57,37,94,52)(23,100,58,28,95,43)(24,91,59,39,96,54)(25,82,60,30,97,45)(27,84,42,32,99,47)(29,86,44,34,81,49)(31,88,46,36,83,51)(33,90,48,38,85,53)(35,92,50,40,87,55), (1,104)(2,103)(3,102)(4,101)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,95)(42,94)(43,93)(44,92)(45,91)(46,90)(47,89)(48,88)(49,87)(50,86)(51,85)(52,84)(53,83)(54,82)(55,81)(56,100)(57,99)(58,98)(59,97)(60,96)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78) );
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)], [(1,57,11,47),(2,56,12,46),(3,55,13,45),(4,54,14,44),(5,53,15,43),(6,52,16,42),(7,51,17,41),(8,50,18,60),(9,49,19,59),(10,48,20,58),(21,73,31,63),(22,72,32,62),(23,71,33,61),(24,70,34,80),(25,69,35,79),(26,68,36,78),(27,67,37,77),(28,66,38,76),(29,65,39,75),(30,64,40,74),(81,117,91,107),(82,116,92,106),(83,115,93,105),(84,114,94,104),(85,113,95,103),(86,112,96,102),(87,111,97,101),(88,110,98,120),(89,109,99,119),(90,108,100,118)], [(1,62,114),(2,73,115,12,63,105),(3,64,116),(4,75,117,14,65,107),(5,66,118),(6,77,119,16,67,109),(7,68,120),(8,79,101,18,69,111),(9,70,102),(10,61,103,20,71,113),(11,72,104),(13,74,106),(15,76,108),(17,78,110),(19,80,112),(21,98,56,26,93,41),(22,89,57,37,94,52),(23,100,58,28,95,43),(24,91,59,39,96,54),(25,82,60,30,97,45),(27,84,42,32,99,47),(29,86,44,34,81,49),(31,88,46,36,83,51),(33,90,48,38,85,53),(35,92,50,40,87,55)], [(1,104),(2,103),(3,102),(4,101),(5,120),(6,119),(7,118),(8,117),(9,116),(10,115),(11,114),(12,113),(13,112),(14,111),(15,110),(16,109),(17,108),(18,107),(19,106),(20,105),(21,28),(22,27),(23,26),(24,25),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(41,95),(42,94),(43,93),(44,92),(45,91),(46,90),(47,89),(48,88),(49,87),(50,86),(51,85),(52,84),(53,83),(54,82),(55,81),(56,100),(57,99),(58,98),(59,97),(60,96),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(74,80),(75,79),(76,78)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 15 | 15 | 60 | 2 | 2 | 12 | 20 | 30 | 2 | 2 | 2 | 8 | 6 | 6 | 10 | 10 | 2 | 2 | 8 | 8 | 4 | 40 | 4 | 4 | 4 | 4 | 24 | 24 | 20 | 20 | 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 | 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 | SD16 | D10 | D10 | D10 | S3×D4 | S3×D5 | D4×D5 | S3×SD16 | C2×S3×D5 | D5×SD16 | D10⋊D6 | Dic10⋊D6 |
| kernel | Dic10⋊D6 | D15⋊2C8 | C15⋊SD16 | Dic6⋊D5 | C3×D4.D5 | C5×D4.S3 | D15⋊Q8 | D4×D15 | D4.D5 | Dic15 | D30 | D4.S3 | C5⋊2C8 | Dic10 | C5×D4 | D15 | C3⋊C8 | Dic6 | C3×D4 | C10 | D4 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of Dic10⋊D6 ►in GL6(𝔽241)
| 189 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 0 | 0 | 0 | 0 | 2 | 1 |
| 52 | 189 | 0 | 0 | 0 | 0 |
| 1 | 189 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 19 |
| 0 | 0 | 0 | 0 | 38 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 239 | 54 | 0 | 0 |
| 0 | 0 | 174 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 189 | 52 | 0 | 0 | 0 | 0 |
| 240 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 174 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(6,GF(241))| [189,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,2,0,0,0,0,240,1],[52,1,0,0,0,0,189,189,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,38,0,0,0,0,19,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,239,174,0,0,0,0,54,1,0,0,0,0,0,0,1,0,0,0,0,0,1,240],[189,240,0,0,0,0,52,52,0,0,0,0,0,0,240,174,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,240] >;
Dic10⋊D6 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes D_6 % in TeX
G:=Group("Dic10:D6"); // GroupNames label
G:=SmallGroup(480,563);
// by ID
G=gap.SmallGroup(480,563);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,135,675,346,185,80,1356,18822]);
// 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^11,c*b*c^-1=d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations