metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5.4Dic6, C3⋊C8⋊2F5, C15⋊(C2.D8), C5⋊(C6.Q16), C15⋊3C8⋊2C4, C4⋊F5.4S3, C3⋊2(D5.D8), (C3×D5).3D8, C6.9(C4⋊F5), C4.16(S3×F5), C12.6(C2×F5), C30.2(C4⋊C4), C20.16(C4×S3), C60.16(C2×C4), (C6×D5).26D4, (C4×D5).61D6, (C3×D5).3Q16, C60⋊C4.4C2, D5.1(D4⋊S3), (C3×Dic5).4Q8, C2.5(Dic3⋊F5), D5.1(C3⋊Q16), D10.14(C3⋊D4), C10.2(Dic3⋊C4), (D5×C12).47C22, (C5×C3⋊C8)⋊2C4, (D5×C3⋊C8).3C2, (C3×C4⋊F5).4C2, SmallGroup(480,236)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.4Dic6
G = < a,b,c,d | a10=c12=1, b2=a5, d2=a5bc6, 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, C2.D8, 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, C6.Q16, C5×C3⋊C8, C15⋊3C8, D5×C12, C6×F5, C2×C3⋊F5, D5.D8, D5×C3⋊C8, C3×C4⋊F5, C60⋊C4, Dic5.4Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, D8, Q16, F5, Dic6, C4×S3, C3⋊D4, C2.D8, C2×F5, Dic3⋊C4, D4⋊S3, C3⋊Q16, C4⋊F5, C6.Q16, S3×F5, D5.D8, Dic3⋊F5, Dic5.4Dic6
►On 120 points
Generators in S120
(1 53 94 31 114 12 120 25 88 59)(2 32 109 60 95 7 89 54 115 26)(3 49 90 27 110 8 116 33 96 55)(4 28 117 56 91 9 85 50 111 34)(5 57 86 35 118 10 112 29 92 51)(6 36 113 52 87 11 93 58 119 30)(13 108 65 42 84 23 78 48 71 102)(14 43 79 103 66 24 72 97 73 37)(15 104 61 38 80 19 74 44 67 98)(16 39 75 99 62 20 68 105 81 45)(17 100 69 46 76 21 82 40 63 106)(18 47 83 107 70 22 64 101 77 41)
(1 20 12 16)(2 17 7 21)(3 22 8 18)(4 13 9 23)(5 24 10 14)(6 15 11 19)(25 81 94 99)(26 100 95 82)(27 83 96 101)(28 102 85 84)(29 73 86 103)(30 104 87 74)(31 75 88 105)(32 106 89 76)(33 77 90 107)(34 108 91 78)(35 79 92 97)(36 98 93 80)(37 57 66 112)(38 113 67 58)(39 59 68 114)(40 115 69 60)(41 49 70 116)(42 117 71 50)(43 51 72 118)(44 119 61 52)(45 53 62 120)(46 109 63 54)(47 55 64 110)(48 111 65 56)
(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 2 16 21 12 7 20 17)(3 6 18 19 8 11 22 15)(4 14 23 10 9 24 13 5)(25 54 105 69 94 109 75 40)(26 45 76 114 95 62 106 59)(27 52 107 67 96 119 77 38)(28 43 78 112 85 72 108 57)(29 50 97 65 86 117 79 48)(30 41 80 110 87 70 98 55)(31 60 99 63 88 115 81 46)(32 39 82 120 89 68 100 53)(33 58 101 61 90 113 83 44)(34 37 84 118 91 66 102 51)(35 56 103 71 92 111 73 42)(36 47 74 116 93 64 104 49)
G:=sub<Sym(120)| (1,53,94,31,114,12,120,25,88,59)(2,32,109,60,95,7,89,54,115,26)(3,49,90,27,110,8,116,33,96,55)(4,28,117,56,91,9,85,50,111,34)(5,57,86,35,118,10,112,29,92,51)(6,36,113,52,87,11,93,58,119,30)(13,108,65,42,84,23,78,48,71,102)(14,43,79,103,66,24,72,97,73,37)(15,104,61,38,80,19,74,44,67,98)(16,39,75,99,62,20,68,105,81,45)(17,100,69,46,76,21,82,40,63,106)(18,47,83,107,70,22,64,101,77,41), (1,20,12,16)(2,17,7,21)(3,22,8,18)(4,13,9,23)(5,24,10,14)(6,15,11,19)(25,81,94,99)(26,100,95,82)(27,83,96,101)(28,102,85,84)(29,73,86,103)(30,104,87,74)(31,75,88,105)(32,106,89,76)(33,77,90,107)(34,108,91,78)(35,79,92,97)(36,98,93,80)(37,57,66,112)(38,113,67,58)(39,59,68,114)(40,115,69,60)(41,49,70,116)(42,117,71,50)(43,51,72,118)(44,119,61,52)(45,53,62,120)(46,109,63,54)(47,55,64,110)(48,111,65,56), (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), (1,2,16,21,12,7,20,17)(3,6,18,19,8,11,22,15)(4,14,23,10,9,24,13,5)(25,54,105,69,94,109,75,40)(26,45,76,114,95,62,106,59)(27,52,107,67,96,119,77,38)(28,43,78,112,85,72,108,57)(29,50,97,65,86,117,79,48)(30,41,80,110,87,70,98,55)(31,60,99,63,88,115,81,46)(32,39,82,120,89,68,100,53)(33,58,101,61,90,113,83,44)(34,37,84,118,91,66,102,51)(35,56,103,71,92,111,73,42)(36,47,74,116,93,64,104,49)>;
G:=Group( (1,53,94,31,114,12,120,25,88,59)(2,32,109,60,95,7,89,54,115,26)(3,49,90,27,110,8,116,33,96,55)(4,28,117,56,91,9,85,50,111,34)(5,57,86,35,118,10,112,29,92,51)(6,36,113,52,87,11,93,58,119,30)(13,108,65,42,84,23,78,48,71,102)(14,43,79,103,66,24,72,97,73,37)(15,104,61,38,80,19,74,44,67,98)(16,39,75,99,62,20,68,105,81,45)(17,100,69,46,76,21,82,40,63,106)(18,47,83,107,70,22,64,101,77,41), (1,20,12,16)(2,17,7,21)(3,22,8,18)(4,13,9,23)(5,24,10,14)(6,15,11,19)(25,81,94,99)(26,100,95,82)(27,83,96,101)(28,102,85,84)(29,73,86,103)(30,104,87,74)(31,75,88,105)(32,106,89,76)(33,77,90,107)(34,108,91,78)(35,79,92,97)(36,98,93,80)(37,57,66,112)(38,113,67,58)(39,59,68,114)(40,115,69,60)(41,49,70,116)(42,117,71,50)(43,51,72,118)(44,119,61,52)(45,53,62,120)(46,109,63,54)(47,55,64,110)(48,111,65,56), (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), (1,2,16,21,12,7,20,17)(3,6,18,19,8,11,22,15)(4,14,23,10,9,24,13,5)(25,54,105,69,94,109,75,40)(26,45,76,114,95,62,106,59)(27,52,107,67,96,119,77,38)(28,43,78,112,85,72,108,57)(29,50,97,65,86,117,79,48)(30,41,80,110,87,70,98,55)(31,60,99,63,88,115,81,46)(32,39,82,120,89,68,100,53)(33,58,101,61,90,113,83,44)(34,37,84,118,91,66,102,51)(35,56,103,71,92,111,73,42)(36,47,74,116,93,64,104,49) );
G=PermutationGroup([[(1,53,94,31,114,12,120,25,88,59),(2,32,109,60,95,7,89,54,115,26),(3,49,90,27,110,8,116,33,96,55),(4,28,117,56,91,9,85,50,111,34),(5,57,86,35,118,10,112,29,92,51),(6,36,113,52,87,11,93,58,119,30),(13,108,65,42,84,23,78,48,71,102),(14,43,79,103,66,24,72,97,73,37),(15,104,61,38,80,19,74,44,67,98),(16,39,75,99,62,20,68,105,81,45),(17,100,69,46,76,21,82,40,63,106),(18,47,83,107,70,22,64,101,77,41)], [(1,20,12,16),(2,17,7,21),(3,22,8,18),(4,13,9,23),(5,24,10,14),(6,15,11,19),(25,81,94,99),(26,100,95,82),(27,83,96,101),(28,102,85,84),(29,73,86,103),(30,104,87,74),(31,75,88,105),(32,106,89,76),(33,77,90,107),(34,108,91,78),(35,79,92,97),(36,98,93,80),(37,57,66,112),(38,113,67,58),(39,59,68,114),(40,115,69,60),(41,49,70,116),(42,117,71,50),(43,51,72,118),(44,119,61,52),(45,53,62,120),(46,109,63,54),(47,55,64,110),(48,111,65,56)], [(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)], [(1,2,16,21,12,7,20,17),(3,6,18,19,8,11,22,15),(4,14,23,10,9,24,13,5),(25,54,105,69,94,109,75,40),(26,45,76,114,95,62,106,59),(27,52,107,67,96,119,77,38),(28,43,78,112,85,72,108,57),(29,50,97,65,86,117,79,48),(30,41,80,110,87,70,98,55),(31,60,99,63,88,115,81,46),(32,39,82,120,89,68,100,53),(33,58,101,61,90,113,83,44),(34,37,84,118,91,66,102,51),(35,56,103,71,92,111,73,42),(36,47,74,116,93,64,104,49)]])
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 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | - | + | + | + | - | - | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | Q8 | D4 | D6 | D8 | Q16 | Dic6 | C4×S3 | C3⋊D4 | F5 | C2×F5 | D4⋊S3 | C3⋊Q16 | C4⋊F5 | D5.D8 | S3×F5 | Dic3⋊F5 | Dic5.4Dic6 |
| kernel | Dic5.4Dic6 | D5×C3⋊C8 | C3×C4⋊F5 | C60⋊C4 | C5×C3⋊C8 | C15⋊3C8 | C4⋊F5 | C3×Dic5 | C6×D5 | C4×D5 | C3×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 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of Dic5.4Dic6 ►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 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 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 |
| 10 | 12 | 231 | 62 | 0 | 0 | 0 | 0 |
| 201 | 210 | 195 | 220 | 0 | 0 | 0 | 0 |
| 231 | 62 | 231 | 229 | 0 | 0 | 0 | 0 |
| 195 | 220 | 40 | 31 | 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 |
| 220 | 179 | 210 | 229 | 0 | 0 | 0 | 0 |
| 86 | 21 | 86 | 31 | 0 | 0 | 0 | 0 |
| 31 | 12 | 220 | 179 | 0 | 0 | 0 | 0 |
| 155 | 210 | 86 | 21 | 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],[0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0],[10,201,231,195,0,0,0,0,12,210,62,220,0,0,0,0,231,195,231,40,0,0,0,0,62,220,229,31,0,0,0,0,0,0,0,0,0,224,207,34,0,0,0,0,34,0,207,17,0,0,0,0,17,207,0,34,0,0,0,0,34,207,224,0],[220,86,31,155,0,0,0,0,179,21,12,210,0,0,0,0,210,86,220,86,0,0,0,0,229,31,179,21,0,0,0,0,0,0,0,0,224,34,0,207,0,0,0,0,0,17,34,207,0,0,0,0,207,34,17,0,0,0,0,0,207,0,34,224] >;
Dic5.4Dic6 in GAP, Magma, Sage, TeX
{\rm Dic}_5._4{\rm Dic}_6 % in TeX
G:=Group("Dic5.4Dic6"); // GroupNames label
G:=SmallGroup(480,236);
// by ID
G=gap.SmallGroup(480,236);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,176,675,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^12=1,b^2=a^5,d^2=a^5*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