direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C4.Dic3, C60.175C23, C3⋊C8⋊21D10, C3⋊5(D5×M4(2)), (D5×C12).1C4, (C2×C20).77D6, C12.71(C4×D5), C60.114(C2×C4), (C4×D5).101D6, (C3×D5)⋊3M4(2), C15⋊15(C2×M4(2)), (C4×D5).1Dic3, C60.7C4⋊12C2, C4.14(D5×Dic3), C15⋊3C8⋊25C22, (C2×C12).315D10, (C2×C60).46C22, C30.97(C22×C4), (C6×Dic5).10C4, C20.31(C2×Dic3), C20.32D6⋊12C2, C22.6(D5×Dic3), C20.172(C22×S3), D10.19(C2×Dic3), (C2×Dic5).6Dic3, C12.172(C22×D5), (C22×D5).4Dic3, Dic5.11(C2×Dic3), (D5×C12).102C22, C10.15(C22×Dic3), (D5×C3⋊C8)⋊11C2, (C2×C4×D5).1S3, (D5×C2×C6).7C4, C6.78(C2×C4×D5), (D5×C2×C12).1C2, C4.145(C2×S3×D5), C5⋊5(C2×C4.Dic3), C2.4(C2×D5×Dic3), (C5×C3⋊C8)⋊21C22, (C2×C6).51(C4×D5), (C2×C30).94(C2×C4), (C6×D5).48(C2×C4), (C2×C4).142(S3×D5), (C5×C4.Dic3)⋊5C2, (C3×Dic5).56(C2×C4), (C2×C10).24(C2×Dic3), SmallGroup(480,358)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C4.Dic3
G = < a,b,c,d,e | a5=b2=c4=1, d6=c2, e2=c2d3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d5 >
Subgroups: 476 in 136 conjugacy classes, 64 normal (50 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, C6, C6 [×4], C8 [×4], C2×C4, C2×C4 [×5], C23, D5 [×2], D5, C10, C10, C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C3⋊C8 [×2], C3⋊C8 [×2], C2×C12, C2×C12 [×5], C22×C6, C3×D5 [×2], C3×D5, C30, C30, C2×M4(2), C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8 [×2], C4.Dic3, C4.Dic3 [×3], C22×C12, C3×Dic5 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C5×M4(2), C2×C4×D5, C2×C4.Dic3, C5×C3⋊C8 [×2], C15⋊3C8 [×2], D5×C12 [×4], C6×Dic5, C2×C60, D5×C2×C6, D5×M4(2), D5×C3⋊C8 [×2], C20.32D6 [×2], C5×C4.Dic3, C60.7C4, D5×C2×C12, D5×C4.Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, Dic3 [×4], D6 [×3], M4(2) [×2], C22×C4, D10 [×3], C2×Dic3 [×6], C22×S3, C2×M4(2), C4×D5 [×2], C22×D5, C4.Dic3 [×2], C22×Dic3, S3×D5, C2×C4×D5, C2×C4.Dic3, D5×Dic3 [×2], C2×S3×D5, D5×M4(2), C2×D5×Dic3, D5×C4.Dic3
►On 120 points
Generators in S120
(1 73 35 21 70)(2 74 36 22 71)(3 75 25 23 72)(4 76 26 24 61)(5 77 27 13 62)(6 78 28 14 63)(7 79 29 15 64)(8 80 30 16 65)(9 81 31 17 66)(10 82 32 18 67)(11 83 33 19 68)(12 84 34 20 69)(37 96 59 116 102)(38 85 60 117 103)(39 86 49 118 104)(40 87 50 119 105)(41 88 51 120 106)(42 89 52 109 107)(43 90 53 110 108)(44 91 54 111 97)(45 92 55 112 98)(46 93 56 113 99)(47 94 57 114 100)(48 95 58 115 101)
(1 70)(2 71)(3 72)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 77)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 73)(22 74)(23 75)(24 76)(37 96)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 104)(50 105)(51 106)(52 107)(53 108)(54 97)(55 98)(56 99)(57 100)(58 101)(59 102)(60 103)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 52 55 58)(50 53 56 59)(51 54 57 60)(61 70 67 64)(62 71 68 65)(63 72 69 66)(73 82 79 76)(74 83 80 77)(75 84 81 78)(85 88 91 94)(86 89 92 95)(87 90 93 96)(97 100 103 106)(98 101 104 107)(99 102 105 108)(109 112 115 118)(110 113 116 119)(111 114 117 120)
(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 91 10 88 7 85 4 94)(2 96 11 93 8 90 5 87)(3 89 12 86 9 95 6 92)(13 105 22 102 19 99 16 108)(14 98 23 107 20 104 17 101)(15 103 24 100 21 97 18 106)(25 109 34 118 31 115 28 112)(26 114 35 111 32 120 29 117)(27 119 36 116 33 113 30 110)(37 68 46 65 43 62 40 71)(38 61 47 70 44 67 41 64)(39 66 48 63 45 72 42 69)(49 81 58 78 55 75 52 84)(50 74 59 83 56 80 53 77)(51 79 60 76 57 73 54 82)
G:=sub<Sym(120)| (1,73,35,21,70)(2,74,36,22,71)(3,75,25,23,72)(4,76,26,24,61)(5,77,27,13,62)(6,78,28,14,63)(7,79,29,15,64)(8,80,30,16,65)(9,81,31,17,66)(10,82,32,18,67)(11,83,33,19,68)(12,84,34,20,69)(37,96,59,116,102)(38,85,60,117,103)(39,86,49,118,104)(40,87,50,119,105)(41,88,51,120,106)(42,89,52,109,107)(43,90,53,110,108)(44,91,54,111,97)(45,92,55,112,98)(46,93,56,113,99)(47,94,57,114,100)(48,95,58,115,101), (1,70)(2,71)(3,72)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,77)(14,78)(15,79)(16,80)(17,81)(18,82)(19,83)(20,84)(21,73)(22,74)(23,75)(24,76)(37,96)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,104)(50,105)(51,106)(52,107)(53,108)(54,97)(55,98)(56,99)(57,100)(58,101)(59,102)(60,103), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,70,67,64)(62,71,68,65)(63,72,69,66)(73,82,79,76)(74,83,80,77)(75,84,81,78)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,100,103,106)(98,101,104,107)(99,102,105,108)(109,112,115,118)(110,113,116,119)(111,114,117,120), (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), (1,91,10,88,7,85,4,94)(2,96,11,93,8,90,5,87)(3,89,12,86,9,95,6,92)(13,105,22,102,19,99,16,108)(14,98,23,107,20,104,17,101)(15,103,24,100,21,97,18,106)(25,109,34,118,31,115,28,112)(26,114,35,111,32,120,29,117)(27,119,36,116,33,113,30,110)(37,68,46,65,43,62,40,71)(38,61,47,70,44,67,41,64)(39,66,48,63,45,72,42,69)(49,81,58,78,55,75,52,84)(50,74,59,83,56,80,53,77)(51,79,60,76,57,73,54,82)>;
G:=Group( (1,73,35,21,70)(2,74,36,22,71)(3,75,25,23,72)(4,76,26,24,61)(5,77,27,13,62)(6,78,28,14,63)(7,79,29,15,64)(8,80,30,16,65)(9,81,31,17,66)(10,82,32,18,67)(11,83,33,19,68)(12,84,34,20,69)(37,96,59,116,102)(38,85,60,117,103)(39,86,49,118,104)(40,87,50,119,105)(41,88,51,120,106)(42,89,52,109,107)(43,90,53,110,108)(44,91,54,111,97)(45,92,55,112,98)(46,93,56,113,99)(47,94,57,114,100)(48,95,58,115,101), (1,70)(2,71)(3,72)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,77)(14,78)(15,79)(16,80)(17,81)(18,82)(19,83)(20,84)(21,73)(22,74)(23,75)(24,76)(37,96)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,104)(50,105)(51,106)(52,107)(53,108)(54,97)(55,98)(56,99)(57,100)(58,101)(59,102)(60,103), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,70,67,64)(62,71,68,65)(63,72,69,66)(73,82,79,76)(74,83,80,77)(75,84,81,78)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,100,103,106)(98,101,104,107)(99,102,105,108)(109,112,115,118)(110,113,116,119)(111,114,117,120), (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), (1,91,10,88,7,85,4,94)(2,96,11,93,8,90,5,87)(3,89,12,86,9,95,6,92)(13,105,22,102,19,99,16,108)(14,98,23,107,20,104,17,101)(15,103,24,100,21,97,18,106)(25,109,34,118,31,115,28,112)(26,114,35,111,32,120,29,117)(27,119,36,116,33,113,30,110)(37,68,46,65,43,62,40,71)(38,61,47,70,44,67,41,64)(39,66,48,63,45,72,42,69)(49,81,58,78,55,75,52,84)(50,74,59,83,56,80,53,77)(51,79,60,76,57,73,54,82) );
G=PermutationGroup([(1,73,35,21,70),(2,74,36,22,71),(3,75,25,23,72),(4,76,26,24,61),(5,77,27,13,62),(6,78,28,14,63),(7,79,29,15,64),(8,80,30,16,65),(9,81,31,17,66),(10,82,32,18,67),(11,83,33,19,68),(12,84,34,20,69),(37,96,59,116,102),(38,85,60,117,103),(39,86,49,118,104),(40,87,50,119,105),(41,88,51,120,106),(42,89,52,109,107),(43,90,53,110,108),(44,91,54,111,97),(45,92,55,112,98),(46,93,56,113,99),(47,94,57,114,100),(48,95,58,115,101)], [(1,70),(2,71),(3,72),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,68),(12,69),(13,77),(14,78),(15,79),(16,80),(17,81),(18,82),(19,83),(20,84),(21,73),(22,74),(23,75),(24,76),(37,96),(38,85),(39,86),(40,87),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,104),(50,105),(51,106),(52,107),(53,108),(54,97),(55,98),(56,99),(57,100),(58,101),(59,102),(60,103)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,40,43,46),(38,41,44,47),(39,42,45,48),(49,52,55,58),(50,53,56,59),(51,54,57,60),(61,70,67,64),(62,71,68,65),(63,72,69,66),(73,82,79,76),(74,83,80,77),(75,84,81,78),(85,88,91,94),(86,89,92,95),(87,90,93,96),(97,100,103,106),(98,101,104,107),(99,102,105,108),(109,112,115,118),(110,113,116,119),(111,114,117,120)], [(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)], [(1,91,10,88,7,85,4,94),(2,96,11,93,8,90,5,87),(3,89,12,86,9,95,6,92),(13,105,22,102,19,99,16,108),(14,98,23,107,20,104,17,101),(15,103,24,100,21,97,18,106),(25,109,34,118,31,115,28,112),(26,114,35,111,32,120,29,117),(27,119,36,116,33,113,30,110),(37,68,46,65,43,62,40,71),(38,61,47,70,44,67,41,64),(39,66,48,63,45,72,42,69),(49,81,58,78,55,75,52,84),(50,74,59,83,56,80,53,77),(51,79,60,76,57,73,54,82)])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 30A | ··· | 30F | 40A | ··· | 40H | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 2 | 1 | 1 | 2 | 5 | 5 | 10 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 6 | 6 | 6 | 6 | 30 | 30 | 30 | 30 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | + | - | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D5 | Dic3 | D6 | Dic3 | D6 | Dic3 | M4(2) | D10 | D10 | C4×D5 | C4×D5 | C4.Dic3 | S3×D5 | D5×Dic3 | C2×S3×D5 | D5×Dic3 | D5×M4(2) | D5×C4.Dic3 |
| kernel | D5×C4.Dic3 | D5×C3⋊C8 | C20.32D6 | C5×C4.Dic3 | C60.7C4 | D5×C2×C12 | D5×C12 | C6×Dic5 | D5×C2×C6 | C2×C4×D5 | C4.Dic3 | C4×D5 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | C3×D5 | C3⋊C8 | C2×C12 | C12 | C2×C6 | D5 | C2×C4 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of D5×C4.Dic3 ►in GL4(𝔽241) generated by
| 0 | 1 | 0 | 0 |
| 240 | 51 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 118 | 64 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 99 | 4 |
| 177 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 177 | 192 |
| 0 | 0 | 139 | 64 |
G:=sub<GL(4,GF(241))| [0,240,0,0,1,51,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,177,118,0,0,0,64],[240,0,0,0,0,240,0,0,0,0,60,99,0,0,0,4],[177,0,0,0,0,177,0,0,0,0,177,139,0,0,192,64] >;
D5×C4.Dic3 in GAP, Magma, Sage, TeX
D_5\times C_4.{\rm Dic}_3 % in TeX
G:=Group("D5xC4.Dic3"); // GroupNames label
G:=SmallGroup(480,358);
// by ID
G=gap.SmallGroup(480,358);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,219,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^4=1,d^6=c^2,e^2=c^2*d^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^5>;
// generators/relations