direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2xDic3:F5, D10.4Dic6, C30:(C4:C4), C6:2(C4:F5), (C2xF5).8D6, (C6xD5).4Q8, D5:(Dic3:C4), C10:(Dic3:C4), (C2xDic3):4F5, (D5xDic3):6C4, Dic3:4(C2xF5), (C6xD5).31D4, (C10xDic3):6C4, Dic15:6(C2xC4), (C2xDic15):7C4, D10.27(C4xS3), D5.2(C2xDic6), (C22xF5).2S3, C6.20(C22xF5), C22.19(S3xF5), C30.20(C22xC4), (C22xD5).76D6, (C6xD5).32C23, (C6xF5).12C22, D10.15(C3:D4), D10.35(C22xS3), (D5xDic3).18C22, C3:3(C2xC4:F5), C15:2(C2xC4:C4), C5:(C2xDic3:C4), (C3xD5):(C4:C4), (C2xC6xF5).3C2, C2.22(C2xS3xF5), C10.20(S3xC2xC4), (C3xD5).5(C2xD4), D5.1(C2xC3:D4), (C2xC6).20(C2xF5), (C3xD5).4(C2xQ8), (C2xC30).15(C2xC4), (C2xC10).17(C4xS3), (C22xC3:F5).3C2, (C5xDic3):6(C2xC4), (C6xD5).25(C2xC4), (C2xD5xDic3).13C2, (D5xC2xC6).69C22, (C2xC3:F5).12C22, SmallGroup(480,1001)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xDic3:F5
G = < a,b,c,d,e | a2=b6=d5=e4=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d3 >
Subgroups: 884 in 184 conjugacy classes, 70 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C6, C6, C2xC4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, C2xC6, C2xC6, C15, C4:C4, C22xC4, Dic5, C20, F5, D10, D10, C2xC10, C2xDic3, C2xDic3, C2xC12, C22xC6, C3xD5, C3xD5, C30, C30, C2xC4:C4, C4xD5, C2xDic5, C2xC20, C2xF5, C2xF5, C22xD5, Dic3:C4, C22xDic3, C22xC12, C5xDic3, Dic15, C3xF5, C3:F5, C6xD5, C6xD5, C2xC30, C4:F5, C2xC4xD5, C22xF5, C22xF5, C2xDic3:C4, D5xDic3, C10xDic3, C2xDic15, C6xF5, C6xF5, C2xC3:F5, C2xC3:F5, D5xC2xC6, C2xC4:F5, Dic3:F5, C2xD5xDic3, C2xC6xF5, C22xC3:F5, C2xDic3:F5
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Q8, C23, D6, C4:C4, C22xC4, C2xD4, C2xQ8, F5, Dic6, C4xS3, C3:D4, C22xS3, C2xC4:C4, C2xF5, Dic3:C4, C2xDic6, S3xC2xC4, C2xC3:D4, C4:F5, C22xF5, C2xDic3:C4, S3xF5, C2xC4:F5, Dic3:F5, C2xS3xF5, C2xDic3:F5
►On 120 points
Generators in S120
(1 59)(2 60)(3 55)(4 56)(5 57)(6 58)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 105)(14 106)(15 107)(16 108)(17 103)(18 104)(19 101)(20 102)(21 97)(22 98)(23 99)(24 100)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 69)(32 70)(33 71)(34 72)(35 67)(36 68)(37 65)(38 66)(39 61)(40 62)(41 63)(42 64)(43 76)(44 77)(45 78)(46 73)(47 74)(48 75)(49 82)(50 83)(51 84)(52 79)(53 80)(54 81)(85 118)(86 119)(87 120)(88 115)(89 116)(90 117)
(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)
(1 115 4 118)(2 120 5 117)(3 119 6 116)(7 54 10 51)(8 53 11 50)(9 52 12 49)(13 48 16 45)(14 47 17 44)(15 46 18 43)(19 31 22 34)(20 36 23 33)(21 35 24 32)(25 41 28 38)(26 40 29 37)(27 39 30 42)(55 86 58 89)(56 85 59 88)(57 90 60 87)(61 96 64 93)(62 95 65 92)(63 94 66 91)(67 100 70 97)(68 99 71 102)(69 98 72 101)(73 104 76 107)(74 103 77 106)(75 108 78 105)(79 114 82 111)(80 113 83 110)(81 112 84 109)
(1 36 39 53 43)(2 31 40 54 44)(3 32 41 49 45)(4 33 42 50 46)(5 34 37 51 47)(6 35 38 52 48)(7 17 117 19 26)(8 18 118 20 27)(9 13 119 21 28)(10 14 120 22 29)(11 15 115 23 30)(12 16 116 24 25)(55 70 63 82 78)(56 71 64 83 73)(57 72 65 84 74)(58 67 66 79 75)(59 68 61 80 76)(60 69 62 81 77)(85 102 93 110 104)(86 97 94 111 105)(87 98 95 112 106)(88 99 96 113 107)(89 100 91 114 108)(90 101 92 109 103)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 55)(7 101 26 103)(8 102 27 104)(9 97 28 105)(10 98 29 106)(11 99 30 107)(12 100 25 108)(13 111 21 94)(14 112 22 95)(15 113 23 96)(16 114 24 91)(17 109 19 92)(18 110 20 93)(31 65 44 84)(32 66 45 79)(33 61 46 80)(34 62 47 81)(35 63 48 82)(36 64 43 83)(37 77 51 69)(38 78 52 70)(39 73 53 71)(40 74 54 72)(41 75 49 67)(42 76 50 68)(85 118)(86 119)(87 120)(88 115)(89 116)(90 117)
G:=sub<Sym(120)| (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,105)(14,106)(15,107)(16,108)(17,103)(18,104)(19,101)(20,102)(21,97)(22,98)(23,99)(24,100)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,69)(32,70)(33,71)(34,72)(35,67)(36,68)(37,65)(38,66)(39,61)(40,62)(41,63)(42,64)(43,76)(44,77)(45,78)(46,73)(47,74)(48,75)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(85,118)(86,119)(87,120)(88,115)(89,116)(90,117), (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), (1,115,4,118)(2,120,5,117)(3,119,6,116)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,48,16,45)(14,47,17,44)(15,46,18,43)(19,31,22,34)(20,36,23,33)(21,35,24,32)(25,41,28,38)(26,40,29,37)(27,39,30,42)(55,86,58,89)(56,85,59,88)(57,90,60,87)(61,96,64,93)(62,95,65,92)(63,94,66,91)(67,100,70,97)(68,99,71,102)(69,98,72,101)(73,104,76,107)(74,103,77,106)(75,108,78,105)(79,114,82,111)(80,113,83,110)(81,112,84,109), (1,36,39,53,43)(2,31,40,54,44)(3,32,41,49,45)(4,33,42,50,46)(5,34,37,51,47)(6,35,38,52,48)(7,17,117,19,26)(8,18,118,20,27)(9,13,119,21,28)(10,14,120,22,29)(11,15,115,23,30)(12,16,116,24,25)(55,70,63,82,78)(56,71,64,83,73)(57,72,65,84,74)(58,67,66,79,75)(59,68,61,80,76)(60,69,62,81,77)(85,102,93,110,104)(86,97,94,111,105)(87,98,95,112,106)(88,99,96,113,107)(89,100,91,114,108)(90,101,92,109,103), (1,56)(2,57)(3,58)(4,59)(5,60)(6,55)(7,101,26,103)(8,102,27,104)(9,97,28,105)(10,98,29,106)(11,99,30,107)(12,100,25,108)(13,111,21,94)(14,112,22,95)(15,113,23,96)(16,114,24,91)(17,109,19,92)(18,110,20,93)(31,65,44,84)(32,66,45,79)(33,61,46,80)(34,62,47,81)(35,63,48,82)(36,64,43,83)(37,77,51,69)(38,78,52,70)(39,73,53,71)(40,74,54,72)(41,75,49,67)(42,76,50,68)(85,118)(86,119)(87,120)(88,115)(89,116)(90,117)>;
G:=Group( (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,105)(14,106)(15,107)(16,108)(17,103)(18,104)(19,101)(20,102)(21,97)(22,98)(23,99)(24,100)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,69)(32,70)(33,71)(34,72)(35,67)(36,68)(37,65)(38,66)(39,61)(40,62)(41,63)(42,64)(43,76)(44,77)(45,78)(46,73)(47,74)(48,75)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(85,118)(86,119)(87,120)(88,115)(89,116)(90,117), (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), (1,115,4,118)(2,120,5,117)(3,119,6,116)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,48,16,45)(14,47,17,44)(15,46,18,43)(19,31,22,34)(20,36,23,33)(21,35,24,32)(25,41,28,38)(26,40,29,37)(27,39,30,42)(55,86,58,89)(56,85,59,88)(57,90,60,87)(61,96,64,93)(62,95,65,92)(63,94,66,91)(67,100,70,97)(68,99,71,102)(69,98,72,101)(73,104,76,107)(74,103,77,106)(75,108,78,105)(79,114,82,111)(80,113,83,110)(81,112,84,109), (1,36,39,53,43)(2,31,40,54,44)(3,32,41,49,45)(4,33,42,50,46)(5,34,37,51,47)(6,35,38,52,48)(7,17,117,19,26)(8,18,118,20,27)(9,13,119,21,28)(10,14,120,22,29)(11,15,115,23,30)(12,16,116,24,25)(55,70,63,82,78)(56,71,64,83,73)(57,72,65,84,74)(58,67,66,79,75)(59,68,61,80,76)(60,69,62,81,77)(85,102,93,110,104)(86,97,94,111,105)(87,98,95,112,106)(88,99,96,113,107)(89,100,91,114,108)(90,101,92,109,103), (1,56)(2,57)(3,58)(4,59)(5,60)(6,55)(7,101,26,103)(8,102,27,104)(9,97,28,105)(10,98,29,106)(11,99,30,107)(12,100,25,108)(13,111,21,94)(14,112,22,95)(15,113,23,96)(16,114,24,91)(17,109,19,92)(18,110,20,93)(31,65,44,84)(32,66,45,79)(33,61,46,80)(34,62,47,81)(35,63,48,82)(36,64,43,83)(37,77,51,69)(38,78,52,70)(39,73,53,71)(40,74,54,72)(41,75,49,67)(42,76,50,68)(85,118)(86,119)(87,120)(88,115)(89,116)(90,117) );
G=PermutationGroup([[(1,59),(2,60),(3,55),(4,56),(5,57),(6,58),(7,109),(8,110),(9,111),(10,112),(11,113),(12,114),(13,105),(14,106),(15,107),(16,108),(17,103),(18,104),(19,101),(20,102),(21,97),(22,98),(23,99),(24,100),(25,91),(26,92),(27,93),(28,94),(29,95),(30,96),(31,69),(32,70),(33,71),(34,72),(35,67),(36,68),(37,65),(38,66),(39,61),(40,62),(41,63),(42,64),(43,76),(44,77),(45,78),(46,73),(47,74),(48,75),(49,82),(50,83),(51,84),(52,79),(53,80),(54,81),(85,118),(86,119),(87,120),(88,115),(89,116),(90,117)], [(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)], [(1,115,4,118),(2,120,5,117),(3,119,6,116),(7,54,10,51),(8,53,11,50),(9,52,12,49),(13,48,16,45),(14,47,17,44),(15,46,18,43),(19,31,22,34),(20,36,23,33),(21,35,24,32),(25,41,28,38),(26,40,29,37),(27,39,30,42),(55,86,58,89),(56,85,59,88),(57,90,60,87),(61,96,64,93),(62,95,65,92),(63,94,66,91),(67,100,70,97),(68,99,71,102),(69,98,72,101),(73,104,76,107),(74,103,77,106),(75,108,78,105),(79,114,82,111),(80,113,83,110),(81,112,84,109)], [(1,36,39,53,43),(2,31,40,54,44),(3,32,41,49,45),(4,33,42,50,46),(5,34,37,51,47),(6,35,38,52,48),(7,17,117,19,26),(8,18,118,20,27),(9,13,119,21,28),(10,14,120,22,29),(11,15,115,23,30),(12,16,116,24,25),(55,70,63,82,78),(56,71,64,83,73),(57,72,65,84,74),(58,67,66,79,75),(59,68,61,80,76),(60,69,62,81,77),(85,102,93,110,104),(86,97,94,111,105),(87,98,95,112,106),(88,99,96,113,107),(89,100,91,114,108),(90,101,92,109,103)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,55),(7,101,26,103),(8,102,27,104),(9,97,28,105),(10,98,29,106),(11,99,30,107),(12,100,25,108),(13,111,21,94),(14,112,22,95),(15,113,23,96),(16,114,24,91),(17,109,19,92),(18,110,20,93),(31,65,44,84),(32,66,45,79),(33,61,46,80),(34,62,47,81),(35,63,48,82),(36,64,43,83),(37,77,51,69),(38,78,52,70),(39,73,53,71),(40,74,54,72),(41,75,49,67),(42,76,50,68),(85,118),(86,119),(87,120),(88,115),(89,116),(90,117)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | 10B | 10C | 12A | ··· | 12H | 15 | 20A | 20B | 20C | 20D | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | ··· | 30 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 10 | ··· | 10 | 8 | 12 | 12 | 12 | 12 | 8 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | + | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | Q8 | D6 | D6 | Dic6 | C4xS3 | C3:D4 | C4xS3 | F5 | C2xF5 | C2xF5 | C4:F5 | S3xF5 | Dic3:F5 | C2xS3xF5 |
| kernel | C2xDic3:F5 | Dic3:F5 | C2xD5xDic3 | C2xC6xF5 | C22xC3:F5 | D5xDic3 | C10xDic3 | C2xDic15 | C22xF5 | C6xD5 | C6xD5 | C2xF5 | C22xD5 | D10 | D10 | D10 | C2xC10 | C2xDic3 | Dic3 | C2xC6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 2 | 1 | 2 | 1 | 4 | 1 | 2 | 1 |
Matrix representation of C2xDic3:F5 ►in GL6(F61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 48 | 0 | 0 | 0 | 0 | 0 |
| 44 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 57 | 53 | 0 | 0 | 0 | 0 |
| 25 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 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 | 1 |
| 0 | 0 | 60 | 60 | 60 | 60 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 50 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[48,44,0,0,0,0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[57,25,0,0,0,0,53,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,1,0,0,60,0,0,0,1,0,60,0,0,0,0,1,60],[11,50,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,1,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,60,0,1] >;
C2xDic3:F5 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_3\rtimes F_5 % in TeX
G:=Group("C2xDic3:F5"); // GroupNames label
G:=SmallGroup(480,1001);
// by ID
G=gap.SmallGroup(480,1001);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,120,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=d^5=e^4=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=d^3>;
// generators/relations