metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10:1Dic3, C15:11C4wrC2, D4:2(C3:F5), (C3xD4):2F5, (D4xC15):2C4, C60.34(C2xC4), C3:3(D4:F5), (C5xD4):2Dic3, (C6xD5).33D4, (C4xD5).28D6, C12.10(C2xF5), C12.F5:7C2, (C3xDic10):1C4, D4:2D5.2S3, C20.2(C2xDic3), C5:1(Q8:3Dic3), D10.6(C3:D4), (C3xDic5).82D4, C6.21(C22:F5), C30.21(C22:C4), (D5xC12).69C22, Dic5.37(C3:D4), C10.6(C6.D4), C2.7(D10.D6), (C4xC3:F5):6C2, C4.2(C2xC3:F5), (C3xD4:2D5).2C2, SmallGroup(480,313)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10:Dic3
G = < a,b,c,d | a20=c6=1, b2=a10, d2=c3, bab-1=a-1, cac-1=a9, dad-1=a13, cbc-1=a10b, dbd-1=a15b, dcd-1=c-1 >
Subgroups: 460 in 88 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2xC4, D4, D4, Q8, D5, C10, C10, Dic3, C12, C12, C2xC6, C15, C42, M4(2), C4oD4, Dic5, Dic5, C20, F5, D10, C2xC10, C3:C8, C2xDic3, C2xC12, C3xD4, C3xD4, C3xQ8, C3xD5, C30, C30, C4wrC2, C5:C8, Dic10, C4xD5, C2xDic5, C5:D4, C5xD4, C2xF5, C4.Dic3, C4xDic3, C3xC4oD4, C3xDic5, C3xDic5, C60, C3:F5, C6xD5, C2xC30, C4.F5, C4xF5, D4:2D5, Q8:3Dic3, C15:C8, C3xDic10, D5xC12, C6xDic5, C3xC5:D4, D4xC15, C2xC3:F5, D4:F5, C12.F5, C4xC3:F5, C3xD4:2D5, Dic10:Dic3
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Dic3, D6, C22:C4, F5, C2xDic3, C3:D4, C4wrC2, C2xF5, C6.D4, C3:F5, C22:F5, Q8:3Dic3, C2xC3:F5, D4:F5, D10.D6, Dic10:Dic3
►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 70 11 80)(2 69 12 79)(3 68 13 78)(4 67 14 77)(5 66 15 76)(6 65 16 75)(7 64 17 74)(8 63 18 73)(9 62 19 72)(10 61 20 71)(21 59 31 49)(22 58 32 48)(23 57 33 47)(24 56 34 46)(25 55 35 45)(26 54 36 44)(27 53 37 43)(28 52 38 42)(29 51 39 41)(30 50 40 60)(81 109 91 119)(82 108 92 118)(83 107 93 117)(84 106 94 116)(85 105 95 115)(86 104 96 114)(87 103 97 113)(88 102 98 112)(89 101 99 111)(90 120 100 110)
(1 90 31)(2 99 32 10 91 40)(3 88 33 19 92 29)(4 97 34 8 93 38)(5 86 35 17 94 27)(6 95 36)(7 84 37 15 96 25)(9 82 39 13 98 23)(11 100 21)(12 89 22 20 81 30)(14 87 24 18 83 28)(16 85 26)(41 68 112 47 62 118)(42 77 113 56 63 107)(43 66 114 45 64 116)(44 75 115 54 65 105)(46 73 117 52 67 103)(48 71 119 50 69 101)(49 80 120 59 70 110)(51 78 102 57 72 108)(53 76 104 55 74 106)(58 61 109 60 79 111)
(1 11)(2 8 10 4)(3 5 19 17)(6 16)(7 13 15 9)(12 18 20 14)(21 90)(22 87 30 83)(23 84 39 96)(24 81 28 89)(25 98 37 82)(26 95)(27 92 35 88)(29 86 33 94)(31 100)(32 97 40 93)(34 91 38 99)(36 85)(41 109 47 111)(42 106 56 104)(43 103 45 117)(44 120 54 110)(46 114 52 116)(48 108 50 102)(49 105 59 115)(51 119 57 101)(53 113 55 107)(58 118 60 112)(61 62 79 68)(63 76 77 74)(64 73 66 67)(65 70 75 80)(69 78 71 72)
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), (1,70,11,80)(2,69,12,79)(3,68,13,78)(4,67,14,77)(5,66,15,76)(6,65,16,75)(7,64,17,74)(8,63,18,73)(9,62,19,72)(10,61,20,71)(21,59,31,49)(22,58,32,48)(23,57,33,47)(24,56,34,46)(25,55,35,45)(26,54,36,44)(27,53,37,43)(28,52,38,42)(29,51,39,41)(30,50,40,60)(81,109,91,119)(82,108,92,118)(83,107,93,117)(84,106,94,116)(85,105,95,115)(86,104,96,114)(87,103,97,113)(88,102,98,112)(89,101,99,111)(90,120,100,110), (1,90,31)(2,99,32,10,91,40)(3,88,33,19,92,29)(4,97,34,8,93,38)(5,86,35,17,94,27)(6,95,36)(7,84,37,15,96,25)(9,82,39,13,98,23)(11,100,21)(12,89,22,20,81,30)(14,87,24,18,83,28)(16,85,26)(41,68,112,47,62,118)(42,77,113,56,63,107)(43,66,114,45,64,116)(44,75,115,54,65,105)(46,73,117,52,67,103)(48,71,119,50,69,101)(49,80,120,59,70,110)(51,78,102,57,72,108)(53,76,104,55,74,106)(58,61,109,60,79,111), (1,11)(2,8,10,4)(3,5,19,17)(6,16)(7,13,15,9)(12,18,20,14)(21,90)(22,87,30,83)(23,84,39,96)(24,81,28,89)(25,98,37,82)(26,95)(27,92,35,88)(29,86,33,94)(31,100)(32,97,40,93)(34,91,38,99)(36,85)(41,109,47,111)(42,106,56,104)(43,103,45,117)(44,120,54,110)(46,114,52,116)(48,108,50,102)(49,105,59,115)(51,119,57,101)(53,113,55,107)(58,118,60,112)(61,62,79,68)(63,76,77,74)(64,73,66,67)(65,70,75,80)(69,78,71,72)>;
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), (1,70,11,80)(2,69,12,79)(3,68,13,78)(4,67,14,77)(5,66,15,76)(6,65,16,75)(7,64,17,74)(8,63,18,73)(9,62,19,72)(10,61,20,71)(21,59,31,49)(22,58,32,48)(23,57,33,47)(24,56,34,46)(25,55,35,45)(26,54,36,44)(27,53,37,43)(28,52,38,42)(29,51,39,41)(30,50,40,60)(81,109,91,119)(82,108,92,118)(83,107,93,117)(84,106,94,116)(85,105,95,115)(86,104,96,114)(87,103,97,113)(88,102,98,112)(89,101,99,111)(90,120,100,110), (1,90,31)(2,99,32,10,91,40)(3,88,33,19,92,29)(4,97,34,8,93,38)(5,86,35,17,94,27)(6,95,36)(7,84,37,15,96,25)(9,82,39,13,98,23)(11,100,21)(12,89,22,20,81,30)(14,87,24,18,83,28)(16,85,26)(41,68,112,47,62,118)(42,77,113,56,63,107)(43,66,114,45,64,116)(44,75,115,54,65,105)(46,73,117,52,67,103)(48,71,119,50,69,101)(49,80,120,59,70,110)(51,78,102,57,72,108)(53,76,104,55,74,106)(58,61,109,60,79,111), (1,11)(2,8,10,4)(3,5,19,17)(6,16)(7,13,15,9)(12,18,20,14)(21,90)(22,87,30,83)(23,84,39,96)(24,81,28,89)(25,98,37,82)(26,95)(27,92,35,88)(29,86,33,94)(31,100)(32,97,40,93)(34,91,38,99)(36,85)(41,109,47,111)(42,106,56,104)(43,103,45,117)(44,120,54,110)(46,114,52,116)(48,108,50,102)(49,105,59,115)(51,119,57,101)(53,113,55,107)(58,118,60,112)(61,62,79,68)(63,76,77,74)(64,73,66,67)(65,70,75,80)(69,78,71,72) );
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)], [(1,70,11,80),(2,69,12,79),(3,68,13,78),(4,67,14,77),(5,66,15,76),(6,65,16,75),(7,64,17,74),(8,63,18,73),(9,62,19,72),(10,61,20,71),(21,59,31,49),(22,58,32,48),(23,57,33,47),(24,56,34,46),(25,55,35,45),(26,54,36,44),(27,53,37,43),(28,52,38,42),(29,51,39,41),(30,50,40,60),(81,109,91,119),(82,108,92,118),(83,107,93,117),(84,106,94,116),(85,105,95,115),(86,104,96,114),(87,103,97,113),(88,102,98,112),(89,101,99,111),(90,120,100,110)], [(1,90,31),(2,99,32,10,91,40),(3,88,33,19,92,29),(4,97,34,8,93,38),(5,86,35,17,94,27),(6,95,36),(7,84,37,15,96,25),(9,82,39,13,98,23),(11,100,21),(12,89,22,20,81,30),(14,87,24,18,83,28),(16,85,26),(41,68,112,47,62,118),(42,77,113,56,63,107),(43,66,114,45,64,116),(44,75,115,54,65,105),(46,73,117,52,67,103),(48,71,119,50,69,101),(49,80,120,59,70,110),(51,78,102,57,72,108),(53,76,104,55,74,106),(58,61,109,60,79,111)], [(1,11),(2,8,10,4),(3,5,19,17),(6,16),(7,13,15,9),(12,18,20,14),(21,90),(22,87,30,83),(23,84,39,96),(24,81,28,89),(25,98,37,82),(26,95),(27,92,35,88),(29,86,33,94),(31,100),(32,97,40,93),(34,91,38,99),(36,85),(41,109,47,111),(42,106,56,104),(43,103,45,117),(44,120,54,110),(46,114,52,116),(48,108,50,102),(49,105,59,115),(51,119,57,101),(53,113,55,107),(58,118,60,112),(61,62,79,68),(63,76,77,74),(64,73,66,67),(65,70,75,80),(69,78,71,72)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20 | 30A | 30B | 30C | 30D | 30E | 30F | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 4 | 10 | 2 | 2 | 5 | 5 | 20 | 30 | 30 | 30 | 30 | 4 | 2 | 4 | 4 | 20 | 60 | 60 | 4 | 8 | 8 | 4 | 10 | 10 | 20 | 20 | 4 | 4 | 8 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | - | + | - | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | Dic3 | D6 | Dic3 | C3:D4 | C3:D4 | C4wrC2 | F5 | C2xF5 | C3:F5 | C22:F5 | Q8:3Dic3 | C2xC3:F5 | D10.D6 | D4:F5 | Dic10:Dic3 |
| kernel | Dic10:Dic3 | C12.F5 | C4xC3:F5 | C3xD4:2D5 | C3xDic10 | D4xC15 | D4:2D5 | C3xDic5 | C6xD5 | Dic10 | C4xD5 | C5xD4 | Dic5 | D10 | C15 | C3xD4 | C12 | D4 | C6 | C5 | C4 | C2 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 2 |
Matrix representation of Dic10:Dic3 ►in GL6(F241)
| 177 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 126 | 0 | 12 | 12 |
| 0 | 0 | 115 | 127 | 127 | 115 |
| 0 | 0 | 12 | 12 | 0 | 126 |
| 0 | 0 | 0 | 229 | 114 | 229 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 240 | 240 | 240 |
G:=sub<GL(6,GF(241))| [177,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[0,64,0,0,0,0,64,0,0,0,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,0,0,240,0,1,0,0,0,240,1,0],[1,0,0,0,0,0,0,240,0,0,0,0,0,0,126,115,12,0,0,0,0,127,12,229,0,0,12,127,0,114,0,0,12,115,126,229],[240,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,240,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,1,0,240] >;
Dic10:Dic3 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes {\rm Dic}_3 % in TeX
G:=Group("Dic10:Dic3"); // GroupNames label
G:=SmallGroup(480,313);
// by ID
G=gap.SmallGroup(480,313);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,675,346,80,2693,14118,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^6=1,b^2=a^10,d^2=c^3,b*a*b^-1=a^-1,c*a*c^-1=a^9,d*a*d^-1=a^13,c*b*c^-1=a^10*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations