metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊5D6, Dic6⋊5D10, D30.39D4, C60.22C23, D60.6C22, Dic15.13D4, C3⋊C8⋊9D10, D4⋊D5⋊6S3, C5⋊2C8⋊9D6, (D4×D15)⋊2C2, C3⋊D40⋊4C2, C5⋊4(D8⋊S3), D4.S3⋊6D5, C6.72(D4×D5), C3⋊3(D40⋊C2), D4.16(S3×D5), (C5×D4).10D6, C10.73(S3×D4), D20⋊S3⋊2C2, C15⋊20(C8⋊C22), (C3×D4).10D10, C30.184(C2×D4), Dic6⋊D5⋊4C2, (C3×D20)⋊5C22, D30.5C4⋊3C2, C20.22(C22×S3), (C5×Dic6)⋊5C22, (C4×D15).6C22, C12.22(C22×D5), (D4×C15).16C22, C2.25(D10⋊D6), C4.22(C2×S3×D5), (C3×D4⋊D5)⋊8C2, (C5×C3⋊C8)⋊9C22, (C5×D4.S3)⋊8C2, (C3×C5⋊2C8)⋊9C22, SmallGroup(480,574)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊D10
G = < a,b,c,d | a12=c10=d2=1, b2=a6, bab-1=dad=a-1, cac-1=a7, cbc-1=a3b, dbd=a9b, dcd=c-1 >
Subgroups: 1004 in 136 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×3], C10, C10, Dic3 [×2], C12, D6 [×4], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10 [×5], C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4, C3×D4, C22×S3, C3×D5, D15 [×2], C30, C30, C8⋊C22, C5⋊2C8, C40, C4×D5 [×2], D20, D20 [×2], C5⋊D4, C5×D4, C5×Q8, C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, C5×Dic3, Dic15, C60, C6×D5, D30, D30 [×3], C2×C30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q8⋊2D5, D8⋊S3, C5×C3⋊C8, C3×C5⋊2C8, D5×Dic3, C3⋊D20, C3×D20, C5×Dic6, C4×D15, D60, C15⋊7D4, D4×C15, C22×D15, D40⋊C2, D30.5C4, C3⋊D40, Dic6⋊D5, C3×D4⋊D5, C5×D4.S3, D20⋊S3, D4×D15, Dic6⋊D10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, D8⋊S3, C2×S3×D5, D40⋊C2, D10⋊D6, Dic6⋊D10
►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 37 7 43)(2 48 8 42)(3 47 9 41)(4 46 10 40)(5 45 11 39)(6 44 12 38)(13 89 19 95)(14 88 20 94)(15 87 21 93)(16 86 22 92)(17 85 23 91)(18 96 24 90)(25 103 31 97)(26 102 32 108)(27 101 33 107)(28 100 34 106)(29 99 35 105)(30 98 36 104)(49 62 55 68)(50 61 56 67)(51 72 57 66)(52 71 58 65)(53 70 59 64)(54 69 60 63)(73 115 79 109)(74 114 80 120)(75 113 81 119)(76 112 82 118)(77 111 83 117)(78 110 84 116)
(1 102 53 89 81 4 99 56 86 84)(2 97 54 96 82 11 100 51 87 79)(3 104 55 91 83 6 101 58 88 74)(5 106 57 93 73 8 103 60 90 76)(7 108 59 95 75 10 105 50 92 78)(9 98 49 85 77 12 107 52 94 80)(13 110 40 26 61)(14 117 41 33 62 20 111 47 27 68)(15 112 42 28 63)(16 119 43 35 64 22 113 37 29 70)(17 114 44 30 65)(18 109 45 25 66 24 115 39 31 72)(19 116 46 32 67)(21 118 48 34 69)(23 120 38 36 71)
(1 84)(2 83)(3 82)(4 81)(5 80)(6 79)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 36)(22 35)(23 34)(24 33)(37 119)(38 118)(39 117)(40 116)(41 115)(42 114)(43 113)(44 112)(45 111)(46 110)(47 109)(48 120)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 67)(62 66)(63 65)(68 72)(69 71)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)(92 108)(93 107)(94 106)(95 105)(96 104)
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), (1,37,7,43)(2,48,8,42)(3,47,9,41)(4,46,10,40)(5,45,11,39)(6,44,12,38)(13,89,19,95)(14,88,20,94)(15,87,21,93)(16,86,22,92)(17,85,23,91)(18,96,24,90)(25,103,31,97)(26,102,32,108)(27,101,33,107)(28,100,34,106)(29,99,35,105)(30,98,36,104)(49,62,55,68)(50,61,56,67)(51,72,57,66)(52,71,58,65)(53,70,59,64)(54,69,60,63)(73,115,79,109)(74,114,80,120)(75,113,81,119)(76,112,82,118)(77,111,83,117)(78,110,84,116), (1,102,53,89,81,4,99,56,86,84)(2,97,54,96,82,11,100,51,87,79)(3,104,55,91,83,6,101,58,88,74)(5,106,57,93,73,8,103,60,90,76)(7,108,59,95,75,10,105,50,92,78)(9,98,49,85,77,12,107,52,94,80)(13,110,40,26,61)(14,117,41,33,62,20,111,47,27,68)(15,112,42,28,63)(16,119,43,35,64,22,113,37,29,70)(17,114,44,30,65)(18,109,45,25,66,24,115,39,31,72)(19,116,46,32,67)(21,118,48,34,69)(23,120,38,36,71), (1,84)(2,83)(3,82)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,36)(22,35)(23,34)(24,33)(37,119)(38,118)(39,117)(40,116)(41,115)(42,114)(43,113)(44,112)(45,111)(46,110)(47,109)(48,120)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,67)(62,66)(63,65)(68,72)(69,71)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,108)(93,107)(94,106)(95,105)(96,104)>;
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), (1,37,7,43)(2,48,8,42)(3,47,9,41)(4,46,10,40)(5,45,11,39)(6,44,12,38)(13,89,19,95)(14,88,20,94)(15,87,21,93)(16,86,22,92)(17,85,23,91)(18,96,24,90)(25,103,31,97)(26,102,32,108)(27,101,33,107)(28,100,34,106)(29,99,35,105)(30,98,36,104)(49,62,55,68)(50,61,56,67)(51,72,57,66)(52,71,58,65)(53,70,59,64)(54,69,60,63)(73,115,79,109)(74,114,80,120)(75,113,81,119)(76,112,82,118)(77,111,83,117)(78,110,84,116), (1,102,53,89,81,4,99,56,86,84)(2,97,54,96,82,11,100,51,87,79)(3,104,55,91,83,6,101,58,88,74)(5,106,57,93,73,8,103,60,90,76)(7,108,59,95,75,10,105,50,92,78)(9,98,49,85,77,12,107,52,94,80)(13,110,40,26,61)(14,117,41,33,62,20,111,47,27,68)(15,112,42,28,63)(16,119,43,35,64,22,113,37,29,70)(17,114,44,30,65)(18,109,45,25,66,24,115,39,31,72)(19,116,46,32,67)(21,118,48,34,69)(23,120,38,36,71), (1,84)(2,83)(3,82)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,36)(22,35)(23,34)(24,33)(37,119)(38,118)(39,117)(40,116)(41,115)(42,114)(43,113)(44,112)(45,111)(46,110)(47,109)(48,120)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,67)(62,66)(63,65)(68,72)(69,71)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,108)(93,107)(94,106)(95,105)(96,104) );
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)], [(1,37,7,43),(2,48,8,42),(3,47,9,41),(4,46,10,40),(5,45,11,39),(6,44,12,38),(13,89,19,95),(14,88,20,94),(15,87,21,93),(16,86,22,92),(17,85,23,91),(18,96,24,90),(25,103,31,97),(26,102,32,108),(27,101,33,107),(28,100,34,106),(29,99,35,105),(30,98,36,104),(49,62,55,68),(50,61,56,67),(51,72,57,66),(52,71,58,65),(53,70,59,64),(54,69,60,63),(73,115,79,109),(74,114,80,120),(75,113,81,119),(76,112,82,118),(77,111,83,117),(78,110,84,116)], [(1,102,53,89,81,4,99,56,86,84),(2,97,54,96,82,11,100,51,87,79),(3,104,55,91,83,6,101,58,88,74),(5,106,57,93,73,8,103,60,90,76),(7,108,59,95,75,10,105,50,92,78),(9,98,49,85,77,12,107,52,94,80),(13,110,40,26,61),(14,117,41,33,62,20,111,47,27,68),(15,112,42,28,63),(16,119,43,35,64,22,113,37,29,70),(17,114,44,30,65),(18,109,45,25,66,24,115,39,31,72),(19,116,46,32,67),(21,118,48,34,69),(23,120,38,36,71)], [(1,84),(2,83),(3,82),(4,81),(5,80),(6,79),(7,78),(8,77),(9,76),(10,75),(11,74),(12,73),(13,32),(14,31),(15,30),(16,29),(17,28),(18,27),(19,26),(20,25),(21,36),(22,35),(23,34),(24,33),(37,119),(38,118),(39,117),(40,116),(41,115),(42,114),(43,113),(44,112),(45,111),(46,110),(47,109),(48,120),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,67),(62,66),(63,65),(68,72),(69,71),(85,103),(86,102),(87,101),(88,100),(89,99),(90,98),(91,97),(92,108),(93,107),(94,106),(95,105),(96,104)])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 12 | 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 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 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 | 20 | 30 | 60 | 2 | 2 | 12 | 30 | 2 | 2 | 2 | 8 | 40 | 12 | 20 | 2 | 2 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 24 | 24 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | C8⋊C22 | S3×D4 | S3×D5 | D4×D5 | D8⋊S3 | C2×S3×D5 | D40⋊C2 | D10⋊D6 | Dic6⋊D10 |
| kernel | Dic6⋊D10 | D30.5C4 | C3⋊D40 | Dic6⋊D5 | C3×D4⋊D5 | C5×D4.S3 | D20⋊S3 | D4×D15 | D4⋊D5 | Dic15 | D30 | D4.S3 | C5⋊2C8 | D20 | C5×D4 | C3⋊C8 | Dic6 | C3×D4 | C15 | C10 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of Dic6⋊D10 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 94 | 194 | 147 |
| 0 | 0 | 47 | 194 | 194 | 47 |
| 0 | 0 | 194 | 147 | 194 | 147 |
| 0 | 0 | 194 | 47 | 194 | 47 |
| 190 | 190 | 0 | 0 | 0 | 0 |
| 51 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 190 | 190 | 0 | 0 | 0 | 0 |
| 240 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,240,1,0,0,0,240,0,0,0,0,1,240,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,47,47,194,194,0,0,94,194,147,47,0,0,194,194,194,194,0,0,147,47,147,47],[190,51,0,0,0,0,190,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[190,240,0,0,0,0,190,51,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,240,0,0,1,240,0,0,0,0,0,240,0,0] >;
Dic6⋊D10 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes D_{10} % in TeX
G:=Group("Dic6:D10"); // GroupNames label
G:=SmallGroup(480,574);
// by ID
G=gap.SmallGroup(480,574);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,303,675,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^10=d^2=1,b^2=a^6,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^3*b,d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations