metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6:5F5, Dic30:3C4, C3:F5:Q8, C3:2(Q8xF5), C15:Q8:2C4, C15:2(C4xQ8), C4:F5.2S3, C60.5(C2xC4), (C2xF5).2D6, C4.12(S3xF5), D5.1(S3xQ8), C20.10(C4xS3), C5:(Dic6:C4), (C5xDic6):3C4, (C4xD5).32D6, C12.14(C2xF5), C6.3(C22xF5), Dic3:F5.2C2, C30.3(C22xC4), (D5xDic6).4C2, Dic5.2(C4xS3), Dic3.2(C2xF5), (Dic3xF5).2C2, (C6xF5).2C22, Dic15.2(C2xC4), (C6xD5).23C23, D5.1(D4:2S3), D10.26(C22xS3), (D5xC12).34C22, (D5xDic3).6C22, C2.8(C2xS3xF5), C10.3(S3xC2xC4), (C4xC3:F5).1C2, (C3xC4:F5).2C2, (C3xD5).2(C2xQ8), (C2xC3:F5).7C22, (C3xD5).3(C4oD4), (C5xDic3).2(C2xC4), (C3xDic5).21(C2xC4), SmallGroup(480,984)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6:5F5
G = < a,b,c,d | a12=c5=d4=1, b2=a6, bab-1=a-1, ac=ca, dad-1=a7, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 660 in 140 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2xC4, Q8, D5, C10, Dic3, Dic3, C12, C12, C2xC6, C15, C42, C4:C4, C2xQ8, Dic5, Dic5, C20, C20, F5, D10, Dic6, Dic6, C2xDic3, C2xC12, C3xD5, C30, C4xQ8, Dic10, C4xD5, C4xD5, C5xQ8, C2xF5, C2xF5, C4xDic3, Dic3:C4, C3xC4:C4, C2xDic6, C5xDic3, C3xDic5, Dic15, C60, C3xF5, C3:F5, C3:F5, C6xD5, C4xF5, C4:F5, C4:F5, Q8xD5, Dic6:C4, D5xDic3, C15:Q8, D5xC12, C5xDic6, Dic30, C6xF5, C2xC3:F5, Q8xF5, Dic3xF5, Dic3:F5, C3xC4:F5, C4xC3:F5, D5xDic6, Dic6:5F5
Quotients: C1, C2, C4, C22, S3, C2xC4, Q8, C23, D6, C22xC4, C2xQ8, C4oD4, F5, C4xS3, C22xS3, C4xQ8, C2xF5, S3xC2xC4, D4:2S3, S3xQ8, C22xF5, Dic6:C4, S3xF5, Q8xF5, C2xS3xF5, Dic6:5F5
►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 15 7 21)(2 14 8 20)(3 13 9 19)(4 24 10 18)(5 23 11 17)(6 22 12 16)(25 44 31 38)(26 43 32 37)(27 42 33 48)(28 41 34 47)(29 40 35 46)(30 39 36 45)(49 71 55 65)(50 70 56 64)(51 69 57 63)(52 68 58 62)(53 67 59 61)(54 66 60 72)(73 96 79 90)(74 95 80 89)(75 94 81 88)(76 93 82 87)(77 92 83 86)(78 91 84 85)(97 119 103 113)(98 118 104 112)(99 117 105 111)(100 116 106 110)(101 115 107 109)(102 114 108 120)
(1 54 36 102 85)(2 55 25 103 86)(3 56 26 104 87)(4 57 27 105 88)(5 58 28 106 89)(6 59 29 107 90)(7 60 30 108 91)(8 49 31 97 92)(9 50 32 98 93)(10 51 33 99 94)(11 52 34 100 95)(12 53 35 101 96)(13 64 43 112 76)(14 65 44 113 77)(15 66 45 114 78)(16 67 46 115 79)(17 68 47 116 80)(18 69 48 117 81)(19 70 37 118 82)(20 71 38 119 83)(21 72 39 120 84)(22 61 40 109 73)(23 62 41 110 74)(24 63 42 111 75)
(1 21 7 15)(2 16 8 22)(3 23 9 17)(4 18 10 24)(5 13 11 19)(6 20 12 14)(25 79 97 61)(26 74 98 68)(27 81 99 63)(28 76 100 70)(29 83 101 65)(30 78 102 72)(31 73 103 67)(32 80 104 62)(33 75 105 69)(34 82 106 64)(35 77 107 71)(36 84 108 66)(37 89 112 52)(38 96 113 59)(39 91 114 54)(40 86 115 49)(41 93 116 56)(42 88 117 51)(43 95 118 58)(44 90 119 53)(45 85 120 60)(46 92 109 55)(47 87 110 50)(48 94 111 57)
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), (1,15,7,21)(2,14,8,20)(3,13,9,19)(4,24,10,18)(5,23,11,17)(6,22,12,16)(25,44,31,38)(26,43,32,37)(27,42,33,48)(28,41,34,47)(29,40,35,46)(30,39,36,45)(49,71,55,65)(50,70,56,64)(51,69,57,63)(52,68,58,62)(53,67,59,61)(54,66,60,72)(73,96,79,90)(74,95,80,89)(75,94,81,88)(76,93,82,87)(77,92,83,86)(78,91,84,85)(97,119,103,113)(98,118,104,112)(99,117,105,111)(100,116,106,110)(101,115,107,109)(102,114,108,120), (1,54,36,102,85)(2,55,25,103,86)(3,56,26,104,87)(4,57,27,105,88)(5,58,28,106,89)(6,59,29,107,90)(7,60,30,108,91)(8,49,31,97,92)(9,50,32,98,93)(10,51,33,99,94)(11,52,34,100,95)(12,53,35,101,96)(13,64,43,112,76)(14,65,44,113,77)(15,66,45,114,78)(16,67,46,115,79)(17,68,47,116,80)(18,69,48,117,81)(19,70,37,118,82)(20,71,38,119,83)(21,72,39,120,84)(22,61,40,109,73)(23,62,41,110,74)(24,63,42,111,75), (1,21,7,15)(2,16,8,22)(3,23,9,17)(4,18,10,24)(5,13,11,19)(6,20,12,14)(25,79,97,61)(26,74,98,68)(27,81,99,63)(28,76,100,70)(29,83,101,65)(30,78,102,72)(31,73,103,67)(32,80,104,62)(33,75,105,69)(34,82,106,64)(35,77,107,71)(36,84,108,66)(37,89,112,52)(38,96,113,59)(39,91,114,54)(40,86,115,49)(41,93,116,56)(42,88,117,51)(43,95,118,58)(44,90,119,53)(45,85,120,60)(46,92,109,55)(47,87,110,50)(48,94,111,57)>;
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), (1,15,7,21)(2,14,8,20)(3,13,9,19)(4,24,10,18)(5,23,11,17)(6,22,12,16)(25,44,31,38)(26,43,32,37)(27,42,33,48)(28,41,34,47)(29,40,35,46)(30,39,36,45)(49,71,55,65)(50,70,56,64)(51,69,57,63)(52,68,58,62)(53,67,59,61)(54,66,60,72)(73,96,79,90)(74,95,80,89)(75,94,81,88)(76,93,82,87)(77,92,83,86)(78,91,84,85)(97,119,103,113)(98,118,104,112)(99,117,105,111)(100,116,106,110)(101,115,107,109)(102,114,108,120), (1,54,36,102,85)(2,55,25,103,86)(3,56,26,104,87)(4,57,27,105,88)(5,58,28,106,89)(6,59,29,107,90)(7,60,30,108,91)(8,49,31,97,92)(9,50,32,98,93)(10,51,33,99,94)(11,52,34,100,95)(12,53,35,101,96)(13,64,43,112,76)(14,65,44,113,77)(15,66,45,114,78)(16,67,46,115,79)(17,68,47,116,80)(18,69,48,117,81)(19,70,37,118,82)(20,71,38,119,83)(21,72,39,120,84)(22,61,40,109,73)(23,62,41,110,74)(24,63,42,111,75), (1,21,7,15)(2,16,8,22)(3,23,9,17)(4,18,10,24)(5,13,11,19)(6,20,12,14)(25,79,97,61)(26,74,98,68)(27,81,99,63)(28,76,100,70)(29,83,101,65)(30,78,102,72)(31,73,103,67)(32,80,104,62)(33,75,105,69)(34,82,106,64)(35,77,107,71)(36,84,108,66)(37,89,112,52)(38,96,113,59)(39,91,114,54)(40,86,115,49)(41,93,116,56)(42,88,117,51)(43,95,118,58)(44,90,119,53)(45,85,120,60)(46,92,109,55)(47,87,110,50)(48,94,111,57) );
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)], [(1,15,7,21),(2,14,8,20),(3,13,9,19),(4,24,10,18),(5,23,11,17),(6,22,12,16),(25,44,31,38),(26,43,32,37),(27,42,33,48),(28,41,34,47),(29,40,35,46),(30,39,36,45),(49,71,55,65),(50,70,56,64),(51,69,57,63),(52,68,58,62),(53,67,59,61),(54,66,60,72),(73,96,79,90),(74,95,80,89),(75,94,81,88),(76,93,82,87),(77,92,83,86),(78,91,84,85),(97,119,103,113),(98,118,104,112),(99,117,105,111),(100,116,106,110),(101,115,107,109),(102,114,108,120)], [(1,54,36,102,85),(2,55,25,103,86),(3,56,26,104,87),(4,57,27,105,88),(5,58,28,106,89),(6,59,29,107,90),(7,60,30,108,91),(8,49,31,97,92),(9,50,32,98,93),(10,51,33,99,94),(11,52,34,100,95),(12,53,35,101,96),(13,64,43,112,76),(14,65,44,113,77),(15,66,45,114,78),(16,67,46,115,79),(17,68,47,116,80),(18,69,48,117,81),(19,70,37,118,82),(20,71,38,119,83),(21,72,39,120,84),(22,61,40,109,73),(23,62,41,110,74),(24,63,42,111,75)], [(1,21,7,15),(2,16,8,22),(3,23,9,17),(4,18,10,24),(5,13,11,19),(6,20,12,14),(25,79,97,61),(26,74,98,68),(27,81,99,63),(28,76,100,70),(29,83,101,65),(30,78,102,72),(31,73,103,67),(32,80,104,62),(33,75,105,69),(34,82,106,64),(35,77,107,71),(36,84,108,66),(37,89,112,52),(38,96,113,59),(39,91,114,54),(40,86,115,49),(41,93,116,56),(42,88,117,51),(43,95,118,58),(44,90,119,53),(45,85,120,60),(46,92,109,55),(47,87,110,50),(48,94,111,57)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5 | 6A | 6B | 6C | 10 | 12A | 12B | ··· | 12F | 15 | 20A | 20B | 20C | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 10 | 12 | 12 | ··· | 12 | 15 | 20 | 20 | 20 | 30 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 6 | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | 30 | 30 | 30 | 4 | 2 | 10 | 10 | 4 | 4 | 20 | ··· | 20 | 8 | 8 | 24 | 24 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | Q8 | D6 | D6 | C4oD4 | C4xS3 | C4xS3 | F5 | C2xF5 | C2xF5 | D4:2S3 | S3xQ8 | S3xF5 | Q8xF5 | C2xS3xF5 | Dic6:5F5 |
| kernel | Dic6:5F5 | Dic3xF5 | Dic3:F5 | C3xC4:F5 | C4xC3:F5 | D5xDic6 | C15:Q8 | C5xDic6 | Dic30 | C4:F5 | C3:F5 | C4xD5 | C2xF5 | C3xD5 | Dic5 | C20 | Dic6 | Dic3 | C12 | D5 | D5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
Matrix representation of Dic6:5F5 ►in GL8(F61)
| 60 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 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 |
| 0 | 0 | 0 | 0 | 60 | 60 | 60 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 50 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
G:=sub<GL(8,GF(61))| [60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,43,38,0,0,0,0,0,0,38,18,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,60,0,1,0,0,0,0,0,60,0,0,1,0,0,0,0,60,0,0,0],[50,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,15,9,0,0,0,0,0,0,9,46,0,0,0,0,0,0,0,0,60,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,1,0,0,0,0,0,60,0,1] >;
Dic6:5F5 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_5F_5 % in TeX
G:=Group("Dic6:5F5"); // GroupNames label
G:=SmallGroup(480,984);
// by ID
G=gap.SmallGroup(480,984);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,219,100,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^5=d^4=1,b^2=a^6,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations