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 [×4], C3, C4, C4 [×3], C22 [×5], C5, S3 [×3], C6, C6, C8 [×2], C2×C4 [×2], D4, D4 [×2], Q8 [×3], C23, D5 [×3], C10, C10, Dic3 [×2], C12, C12, D6 [×4], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20, C20, D10 [×4], C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3 [×2], D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, D15 [×2], D15, C30, C30, C2×SD16, C5⋊2C8, C40, Dic10, Dic10, C4×D5 [×2], 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 [×3], 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 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], SD16 [×2], C2×D4, D10 [×3], 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 55 11 45)(2 54 12 44)(3 53 13 43)(4 52 14 42)(5 51 15 41)(6 50 16 60)(7 49 17 59)(8 48 18 58)(9 47 19 57)(10 46 20 56)(21 72 31 62)(22 71 32 61)(23 70 33 80)(24 69 34 79)(25 68 35 78)(26 67 36 77)(27 66 37 76)(28 65 38 75)(29 64 39 74)(30 63 40 73)(81 120 91 110)(82 119 92 109)(83 118 93 108)(84 117 94 107)(85 116 95 106)(86 115 96 105)(87 114 97 104)(88 113 98 103)(89 112 99 102)(90 111 100 101)
(1 89 36)(2 100 37 12 90 27)(3 91 38)(4 82 39 14 92 29)(5 93 40)(6 84 21 16 94 31)(7 95 22)(8 86 23 18 96 33)(9 97 24)(10 88 25 20 98 35)(11 99 26)(13 81 28)(15 83 30)(17 85 32)(19 87 34)(41 113 63 56 118 78)(42 104 64 47 119 69)(43 115 65 58 120 80)(44 106 66 49 101 71)(45 117 67 60 102 62)(46 108 68 51 103 73)(48 110 70 53 105 75)(50 112 72 55 107 77)(52 114 74 57 109 79)(54 116 76 59 111 61)
(1 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 80)(58 79)(59 78)(60 77)(81 87)(82 86)(83 85)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(101 108)(102 107)(103 106)(104 105)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)
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), (1,55,11,45)(2,54,12,44)(3,53,13,43)(4,52,14,42)(5,51,15,41)(6,50,16,60)(7,49,17,59)(8,48,18,58)(9,47,19,57)(10,46,20,56)(21,72,31,62)(22,71,32,61)(23,70,33,80)(24,69,34,79)(25,68,35,78)(26,67,36,77)(27,66,37,76)(28,65,38,75)(29,64,39,74)(30,63,40,73)(81,120,91,110)(82,119,92,109)(83,118,93,108)(84,117,94,107)(85,116,95,106)(86,115,96,105)(87,114,97,104)(88,113,98,103)(89,112,99,102)(90,111,100,101), (1,89,36)(2,100,37,12,90,27)(3,91,38)(4,82,39,14,92,29)(5,93,40)(6,84,21,16,94,31)(7,95,22)(8,86,23,18,96,33)(9,97,24)(10,88,25,20,98,35)(11,99,26)(13,81,28)(15,83,30)(17,85,32)(19,87,34)(41,113,63,56,118,78)(42,104,64,47,119,69)(43,115,65,58,120,80)(44,106,66,49,101,71)(45,117,67,60,102,62)(46,108,68,51,103,73)(48,110,70,53,105,75)(50,112,72,55,107,77)(52,114,74,57,109,79)(54,116,76,59,111,61), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,80)(58,79)(59,78)(60,77)(81,87)(82,86)(83,85)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(101,108)(102,107)(103,106)(104,105)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)>;
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), (1,55,11,45)(2,54,12,44)(3,53,13,43)(4,52,14,42)(5,51,15,41)(6,50,16,60)(7,49,17,59)(8,48,18,58)(9,47,19,57)(10,46,20,56)(21,72,31,62)(22,71,32,61)(23,70,33,80)(24,69,34,79)(25,68,35,78)(26,67,36,77)(27,66,37,76)(28,65,38,75)(29,64,39,74)(30,63,40,73)(81,120,91,110)(82,119,92,109)(83,118,93,108)(84,117,94,107)(85,116,95,106)(86,115,96,105)(87,114,97,104)(88,113,98,103)(89,112,99,102)(90,111,100,101), (1,89,36)(2,100,37,12,90,27)(3,91,38)(4,82,39,14,92,29)(5,93,40)(6,84,21,16,94,31)(7,95,22)(8,86,23,18,96,33)(9,97,24)(10,88,25,20,98,35)(11,99,26)(13,81,28)(15,83,30)(17,85,32)(19,87,34)(41,113,63,56,118,78)(42,104,64,47,119,69)(43,115,65,58,120,80)(44,106,66,49,101,71)(45,117,67,60,102,62)(46,108,68,51,103,73)(48,110,70,53,105,75)(50,112,72,55,107,77)(52,114,74,57,109,79)(54,116,76,59,111,61), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,80)(58,79)(59,78)(60,77)(81,87)(82,86)(83,85)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(101,108)(102,107)(103,106)(104,105)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115) );
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)], [(1,55,11,45),(2,54,12,44),(3,53,13,43),(4,52,14,42),(5,51,15,41),(6,50,16,60),(7,49,17,59),(8,48,18,58),(9,47,19,57),(10,46,20,56),(21,72,31,62),(22,71,32,61),(23,70,33,80),(24,69,34,79),(25,68,35,78),(26,67,36,77),(27,66,37,76),(28,65,38,75),(29,64,39,74),(30,63,40,73),(81,120,91,110),(82,119,92,109),(83,118,93,108),(84,117,94,107),(85,116,95,106),(86,115,96,105),(87,114,97,104),(88,113,98,103),(89,112,99,102),(90,111,100,101)], [(1,89,36),(2,100,37,12,90,27),(3,91,38),(4,82,39,14,92,29),(5,93,40),(6,84,21,16,94,31),(7,95,22),(8,86,23,18,96,33),(9,97,24),(10,88,25,20,98,35),(11,99,26),(13,81,28),(15,83,30),(17,85,32),(19,87,34),(41,113,63,56,118,78),(42,104,64,47,119,69),(43,115,65,58,120,80),(44,106,66,49,101,71),(45,117,67,60,102,62),(46,108,68,51,103,73),(48,110,70,53,105,75),(50,112,72,55,107,77),(52,114,74,57,109,79),(54,116,76,59,111,61)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,21),(7,40),(8,39),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(41,76),(42,75),(43,74),(44,73),(45,72),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,80),(58,79),(59,78),(60,77),(81,87),(82,86),(83,85),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95),(101,108),(102,107),(103,106),(104,105),(109,120),(110,119),(111,118),(112,117),(113,116),(114,115)])
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