metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5.3Dic6, C3⋊C8⋊1F5, C15⋊(C4.Q8), C15⋊3C8⋊1C4, C4⋊F5.3S3, C3⋊2(C40⋊C4), C6.8(C4⋊F5), C12.5(C2×F5), C4.15(S3×F5), C30.1(C4⋊C4), C5⋊(C12.Q8), C20.15(C4×S3), C60.15(C2×C4), (C6×D5).25D4, (C4×D5).60D6, C60⋊C4.3C2, (C3×Dic5).3Q8, (C3×D5).5SD16, D5.1(D4.S3), C2.4(Dic3⋊F5), D10.13(C3⋊D4), C10.1(Dic3⋊C4), D5.1(Q8⋊2S3), (D5×C12).46C22, (C5×C3⋊C8)⋊1C4, (D5×C3⋊C8).2C2, (C3×C4⋊F5).3C2, SmallGroup(480,235)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.Dic6
G = < a,b,c,d | a10=c12=1, b2=a5, d2=bc6, bab-1=a-1, cac-1=a3, ad=da, cbc-1=a5b, bd=db, dcd-1=a5bc-1 >
Subgroups: 404 in 72 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D5, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, F5, D10, C3⋊C8, C3⋊C8, C2×Dic3, C2×C12, C3×D5, C30, C4.Q8, C5⋊2C8, C40, C4×D5, C2×F5, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C3×Dic5, C60, C3×F5, C3⋊F5, C6×D5, C8×D5, C4⋊F5, C4⋊F5, C12.Q8, C5×C3⋊C8, C15⋊3C8, D5×C12, C6×F5, C2×C3⋊F5, C40⋊C4, D5×C3⋊C8, C3×C4⋊F5, C60⋊C4, Dic5.Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, SD16, F5, Dic6, C4×S3, C3⋊D4, C4.Q8, C2×F5, Dic3⋊C4, D4.S3, Q8⋊2S3, C4⋊F5, C12.Q8, S3×F5, C40⋊C4, Dic3⋊F5, Dic5.Dic6
►On 120 points
Generators in S120
(1 83 89 95 77 7 113 41 47 119)(2 96 114 120 90 8 48 84 78 42)(3 109 37 43 115 9 79 85 91 73)(4 44 80 74 38 10 92 110 116 86)(5 75 93 87 81 11 117 45 39 111)(6 88 118 112 94 12 40 76 82 46)(13 19 106 72 25 53 59 31 66 100)(14 61 60 101 107 54 67 20 26 32)(15 102 68 33 49 55 27 62 108 21)(16 34 28 22 69 56 97 103 50 63)(17 23 98 64 29 57 51 35 70 104)(18 65 52 105 99 58 71 24 30 36)
(1 72 7 66)(2 67 8 61)(3 62 9 68)(4 69 10 63)(5 64 11 70)(6 71 12 65)(13 47 53 95)(14 96 54 48)(15 37 55 85)(16 86 56 38)(17 39 57 87)(18 88 58 40)(19 41 59 89)(20 90 60 42)(21 43 49 91)(22 92 50 44)(23 45 51 93)(24 94 52 46)(25 77 100 119)(26 120 101 78)(27 79 102 109)(28 110 103 80)(29 81 104 111)(30 112 105 82)(31 83 106 113)(32 114 107 84)(33 73 108 115)(34 116 97 74)(35 75 98 117)(36 118 99 76)
(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 63 66 10 7 69 72 4)(2 3 61 68 8 9 67 62)(5 71 70 6 11 65 64 12)(13 110 41 97 53 80 89 34)(14 102 90 115 54 27 42 73)(15 120 43 107 55 78 91 32)(16 100 92 113 56 25 44 83)(17 118 45 105 57 76 93 30)(18 98 94 111 58 35 46 81)(19 116 47 103 59 74 95 28)(20 108 96 109 60 33 48 79)(21 114 37 101 49 84 85 26)(22 106 86 119 50 31 38 77)(23 112 39 99 51 82 87 36)(24 104 88 117 52 29 40 75)
G:=sub<Sym(120)| (1,83,89,95,77,7,113,41,47,119)(2,96,114,120,90,8,48,84,78,42)(3,109,37,43,115,9,79,85,91,73)(4,44,80,74,38,10,92,110,116,86)(5,75,93,87,81,11,117,45,39,111)(6,88,118,112,94,12,40,76,82,46)(13,19,106,72,25,53,59,31,66,100)(14,61,60,101,107,54,67,20,26,32)(15,102,68,33,49,55,27,62,108,21)(16,34,28,22,69,56,97,103,50,63)(17,23,98,64,29,57,51,35,70,104)(18,65,52,105,99,58,71,24,30,36), (1,72,7,66)(2,67,8,61)(3,62,9,68)(4,69,10,63)(5,64,11,70)(6,71,12,65)(13,47,53,95)(14,96,54,48)(15,37,55,85)(16,86,56,38)(17,39,57,87)(18,88,58,40)(19,41,59,89)(20,90,60,42)(21,43,49,91)(22,92,50,44)(23,45,51,93)(24,94,52,46)(25,77,100,119)(26,120,101,78)(27,79,102,109)(28,110,103,80)(29,81,104,111)(30,112,105,82)(31,83,106,113)(32,114,107,84)(33,73,108,115)(34,116,97,74)(35,75,98,117)(36,118,99,76), (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), (1,63,66,10,7,69,72,4)(2,3,61,68,8,9,67,62)(5,71,70,6,11,65,64,12)(13,110,41,97,53,80,89,34)(14,102,90,115,54,27,42,73)(15,120,43,107,55,78,91,32)(16,100,92,113,56,25,44,83)(17,118,45,105,57,76,93,30)(18,98,94,111,58,35,46,81)(19,116,47,103,59,74,95,28)(20,108,96,109,60,33,48,79)(21,114,37,101,49,84,85,26)(22,106,86,119,50,31,38,77)(23,112,39,99,51,82,87,36)(24,104,88,117,52,29,40,75)>;
G:=Group( (1,83,89,95,77,7,113,41,47,119)(2,96,114,120,90,8,48,84,78,42)(3,109,37,43,115,9,79,85,91,73)(4,44,80,74,38,10,92,110,116,86)(5,75,93,87,81,11,117,45,39,111)(6,88,118,112,94,12,40,76,82,46)(13,19,106,72,25,53,59,31,66,100)(14,61,60,101,107,54,67,20,26,32)(15,102,68,33,49,55,27,62,108,21)(16,34,28,22,69,56,97,103,50,63)(17,23,98,64,29,57,51,35,70,104)(18,65,52,105,99,58,71,24,30,36), (1,72,7,66)(2,67,8,61)(3,62,9,68)(4,69,10,63)(5,64,11,70)(6,71,12,65)(13,47,53,95)(14,96,54,48)(15,37,55,85)(16,86,56,38)(17,39,57,87)(18,88,58,40)(19,41,59,89)(20,90,60,42)(21,43,49,91)(22,92,50,44)(23,45,51,93)(24,94,52,46)(25,77,100,119)(26,120,101,78)(27,79,102,109)(28,110,103,80)(29,81,104,111)(30,112,105,82)(31,83,106,113)(32,114,107,84)(33,73,108,115)(34,116,97,74)(35,75,98,117)(36,118,99,76), (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), (1,63,66,10,7,69,72,4)(2,3,61,68,8,9,67,62)(5,71,70,6,11,65,64,12)(13,110,41,97,53,80,89,34)(14,102,90,115,54,27,42,73)(15,120,43,107,55,78,91,32)(16,100,92,113,56,25,44,83)(17,118,45,105,57,76,93,30)(18,98,94,111,58,35,46,81)(19,116,47,103,59,74,95,28)(20,108,96,109,60,33,48,79)(21,114,37,101,49,84,85,26)(22,106,86,119,50,31,38,77)(23,112,39,99,51,82,87,36)(24,104,88,117,52,29,40,75) );
G=PermutationGroup([[(1,83,89,95,77,7,113,41,47,119),(2,96,114,120,90,8,48,84,78,42),(3,109,37,43,115,9,79,85,91,73),(4,44,80,74,38,10,92,110,116,86),(5,75,93,87,81,11,117,45,39,111),(6,88,118,112,94,12,40,76,82,46),(13,19,106,72,25,53,59,31,66,100),(14,61,60,101,107,54,67,20,26,32),(15,102,68,33,49,55,27,62,108,21),(16,34,28,22,69,56,97,103,50,63),(17,23,98,64,29,57,51,35,70,104),(18,65,52,105,99,58,71,24,30,36)], [(1,72,7,66),(2,67,8,61),(3,62,9,68),(4,69,10,63),(5,64,11,70),(6,71,12,65),(13,47,53,95),(14,96,54,48),(15,37,55,85),(16,86,56,38),(17,39,57,87),(18,88,58,40),(19,41,59,89),(20,90,60,42),(21,43,49,91),(22,92,50,44),(23,45,51,93),(24,94,52,46),(25,77,100,119),(26,120,101,78),(27,79,102,109),(28,110,103,80),(29,81,104,111),(30,112,105,82),(31,83,106,113),(32,114,107,84),(33,73,108,115),(34,116,97,74),(35,75,98,117),(36,118,99,76)], [(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)], [(1,63,66,10,7,69,72,4),(2,3,61,68,8,9,67,62),(5,71,70,6,11,65,64,12),(13,110,41,97,53,80,89,34),(14,102,90,115,54,27,42,73),(15,120,43,107,55,78,91,32),(16,100,92,113,56,25,44,83),(17,118,45,105,57,76,93,30),(18,98,94,111,58,35,46,81),(19,116,47,103,59,74,95,28),(20,108,96,109,60,33,48,79),(21,114,37,101,49,84,85,26),(22,106,86,119,50,31,38,77),(23,112,39,99,51,82,87,36),(24,104,88,117,52,29,40,75)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10 | 12A | 12B | ··· | 12F | 15 | 20A | 20B | 30 | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | ··· | 12 | 15 | 20 | 20 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 10 | 20 | 20 | 60 | 60 | 4 | 2 | 10 | 10 | 6 | 6 | 30 | 30 | 4 | 4 | 20 | ··· | 20 | 8 | 4 | 4 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | - | + | + | - | + | + | - | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | Q8 | D4 | D6 | SD16 | Dic6 | C4×S3 | C3⋊D4 | F5 | C2×F5 | D4.S3 | Q8⋊2S3 | C4⋊F5 | C40⋊C4 | S3×F5 | Dic3⋊F5 | Dic5.Dic6 |
| kernel | Dic5.Dic6 | D5×C3⋊C8 | C3×C4⋊F5 | C60⋊C4 | C5×C3⋊C8 | C15⋊3C8 | C4⋊F5 | C3×Dic5 | C6×D5 | C4×D5 | C3×D5 | Dic5 | C20 | D10 | C3⋊C8 | C12 | D5 | D5 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of Dic5.Dic6 ►in GL8(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 240 | 240 | 240 |
| 156 | 0 | 5 | 236 | 0 | 0 | 0 | 0 |
| 0 | 154 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 208 | 87 | 0 | 0 | 0 | 0 | 0 |
| 192 | 208 | 2 | 85 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 78 | 29 | 143 | 194 | 0 | 0 | 0 | 0 |
| 172 | 173 | 163 | 212 | 0 | 0 | 0 | 0 |
| 69 | 181 | 122 | 187 | 0 | 0 | 0 | 0 |
| 143 | 44 | 63 | 109 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 34 | 17 | 34 |
| 0 | 0 | 0 | 0 | 224 | 0 | 207 | 207 |
| 0 | 0 | 0 | 0 | 207 | 207 | 0 | 224 |
| 0 | 0 | 0 | 0 | 34 | 17 | 34 | 0 |
| 19 | 157 | 194 | 51 | 0 | 0 | 0 | 0 |
| 233 | 57 | 49 | 78 | 0 | 0 | 0 | 0 |
| 6 | 49 | 126 | 67 | 0 | 0 | 0 | 0 |
| 72 | 90 | 115 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 224 | 0 | 207 | 207 |
| 0 | 0 | 0 | 0 | 34 | 17 | 34 | 0 |
| 0 | 0 | 0 | 0 | 0 | 34 | 17 | 34 |
| 0 | 0 | 0 | 0 | 207 | 207 | 0 | 224 |
G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240],[156,0,0,192,0,0,0,0,0,154,208,208,0,0,0,0,5,3,87,2,0,0,0,0,236,0,0,85,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0],[78,172,69,143,0,0,0,0,29,173,181,44,0,0,0,0,143,163,122,63,0,0,0,0,194,212,187,109,0,0,0,0,0,0,0,0,0,224,207,34,0,0,0,0,34,0,207,17,0,0,0,0,17,207,0,34,0,0,0,0,34,207,224,0],[19,233,6,72,0,0,0,0,157,57,49,90,0,0,0,0,194,49,126,115,0,0,0,0,51,78,67,39,0,0,0,0,0,0,0,0,224,34,0,207,0,0,0,0,0,17,34,207,0,0,0,0,207,34,17,0,0,0,0,0,207,0,34,224] >;
Dic5.Dic6 in GAP, Magma, Sage, TeX
{\rm Dic}_5.{\rm Dic}_6 % in TeX
G:=Group("Dic5.Dic6"); // GroupNames label
G:=SmallGroup(480,235);
// by ID
G=gap.SmallGroup(480,235);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,675,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^12=1,b^2=a^5,d^2=b*c^6,b*a*b^-1=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=a^5*b*c^-1>;
// generators/relations