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, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, C2×C12, C2×C12, C22×C6, C3×D5, C3×D5, C30, C30, C2×M4(2), C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C4.Dic3, C4.Dic3, C22×C12, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×C4×D5, C2×C4.Dic3, C5×C3⋊C8, C15⋊3C8, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D5×M4(2), D5×C3⋊C8, C20.32D6, C5×C4.Dic3, C60.7C4, D5×C2×C12, D5×C4.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, Dic3, D6, M4(2), C22×C4, D10, C2×Dic3, C22×S3, C2×M4(2), C4×D5, C22×D5, C4.Dic3, C22×Dic3, S3×D5, C2×C4×D5, C2×C4.Dic3, D5×Dic3, C2×S3×D5, D5×M4(2), C2×D5×Dic3, D5×C4.Dic3
►On 120 points
Generators in S120
(1 47 36 92 62)(2 48 25 93 63)(3 37 26 94 64)(4 38 27 95 65)(5 39 28 96 66)(6 40 29 85 67)(7 41 30 86 68)(8 42 31 87 69)(9 43 32 88 70)(10 44 33 89 71)(11 45 34 90 72)(12 46 35 91 61)(13 75 99 50 116)(14 76 100 51 117)(15 77 101 52 118)(16 78 102 53 119)(17 79 103 54 120)(18 80 104 55 109)(19 81 105 56 110)(20 82 106 57 111)(21 83 107 58 112)(22 84 108 59 113)(23 73 97 60 114)(24 74 98 49 115)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 61)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 73)(24 74)(37 94)(38 95)(39 96)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 109)(105 110)(106 111)(107 112)(108 113)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 46 43 40)(38 47 44 41)(39 48 45 42)(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 76 79 82)(74 77 80 83)(75 78 81 84)(85 94 91 88)(86 95 92 89)(87 96 93 90)(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 76 10 73 7 82 4 79)(2 81 11 78 8 75 5 84)(3 74 12 83 9 80 6 77)(13 66 22 63 19 72 16 69)(14 71 23 68 20 65 17 62)(15 64 24 61 21 70 18 67)(25 56 34 53 31 50 28 59)(26 49 35 58 32 55 29 52)(27 54 36 51 33 60 30 57)(37 98 46 107 43 104 40 101)(38 103 47 100 44 97 41 106)(39 108 48 105 45 102 42 99)(85 118 94 115 91 112 88 109)(86 111 95 120 92 117 89 114)(87 116 96 113 93 110 90 119)
G:=sub<Sym(120)| (1,47,36,92,62)(2,48,25,93,63)(3,37,26,94,64)(4,38,27,95,65)(5,39,28,96,66)(6,40,29,85,67)(7,41,30,86,68)(8,42,31,87,69)(9,43,32,88,70)(10,44,33,89,71)(11,45,34,90,72)(12,46,35,91,61)(13,75,99,50,116)(14,76,100,51,117)(15,77,101,52,118)(16,78,102,53,119)(17,79,103,54,120)(18,80,104,55,109)(19,81,105,56,110)(20,82,106,57,111)(21,83,107,58,112)(22,84,108,59,113)(23,73,97,60,114)(24,74,98,49,115), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,61)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,73)(24,74)(37,94)(38,95)(39,96)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,109)(105,110)(106,111)(107,112)(108,113), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,46,43,40)(38,47,44,41)(39,48,45,42)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,70,67,64)(62,71,68,65)(63,72,69,66)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,94,91,88)(86,95,92,89)(87,96,93,90)(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), (1,76,10,73,7,82,4,79)(2,81,11,78,8,75,5,84)(3,74,12,83,9,80,6,77)(13,66,22,63,19,72,16,69)(14,71,23,68,20,65,17,62)(15,64,24,61,21,70,18,67)(25,56,34,53,31,50,28,59)(26,49,35,58,32,55,29,52)(27,54,36,51,33,60,30,57)(37,98,46,107,43,104,40,101)(38,103,47,100,44,97,41,106)(39,108,48,105,45,102,42,99)(85,118,94,115,91,112,88,109)(86,111,95,120,92,117,89,114)(87,116,96,113,93,110,90,119)>;
G:=Group( (1,47,36,92,62)(2,48,25,93,63)(3,37,26,94,64)(4,38,27,95,65)(5,39,28,96,66)(6,40,29,85,67)(7,41,30,86,68)(8,42,31,87,69)(9,43,32,88,70)(10,44,33,89,71)(11,45,34,90,72)(12,46,35,91,61)(13,75,99,50,116)(14,76,100,51,117)(15,77,101,52,118)(16,78,102,53,119)(17,79,103,54,120)(18,80,104,55,109)(19,81,105,56,110)(20,82,106,57,111)(21,83,107,58,112)(22,84,108,59,113)(23,73,97,60,114)(24,74,98,49,115), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,61)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,73)(24,74)(37,94)(38,95)(39,96)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,109)(105,110)(106,111)(107,112)(108,113), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,46,43,40)(38,47,44,41)(39,48,45,42)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,70,67,64)(62,71,68,65)(63,72,69,66)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,94,91,88)(86,95,92,89)(87,96,93,90)(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), (1,76,10,73,7,82,4,79)(2,81,11,78,8,75,5,84)(3,74,12,83,9,80,6,77)(13,66,22,63,19,72,16,69)(14,71,23,68,20,65,17,62)(15,64,24,61,21,70,18,67)(25,56,34,53,31,50,28,59)(26,49,35,58,32,55,29,52)(27,54,36,51,33,60,30,57)(37,98,46,107,43,104,40,101)(38,103,47,100,44,97,41,106)(39,108,48,105,45,102,42,99)(85,118,94,115,91,112,88,109)(86,111,95,120,92,117,89,114)(87,116,96,113,93,110,90,119) );
G=PermutationGroup([[(1,47,36,92,62),(2,48,25,93,63),(3,37,26,94,64),(4,38,27,95,65),(5,39,28,96,66),(6,40,29,85,67),(7,41,30,86,68),(8,42,31,87,69),(9,43,32,88,70),(10,44,33,89,71),(11,45,34,90,72),(12,46,35,91,61),(13,75,99,50,116),(14,76,100,51,117),(15,77,101,52,118),(16,78,102,53,119),(17,79,103,54,120),(18,80,104,55,109),(19,81,105,56,110),(20,82,106,57,111),(21,83,107,58,112),(22,84,108,59,113),(23,73,97,60,114),(24,74,98,49,115)], [(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,61),(13,75),(14,76),(15,77),(16,78),(17,79),(18,80),(19,81),(20,82),(21,83),(22,84),(23,73),(24,74),(37,94),(38,95),(39,96),(40,85),(41,86),(42,87),(43,88),(44,89),(45,90),(46,91),(47,92),(48,93),(97,114),(98,115),(99,116),(100,117),(101,118),(102,119),(103,120),(104,109),(105,110),(106,111),(107,112),(108,113)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,16,19,22),(14,17,20,23),(15,18,21,24),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,46,43,40),(38,47,44,41),(39,48,45,42),(49,52,55,58),(50,53,56,59),(51,54,57,60),(61,70,67,64),(62,71,68,65),(63,72,69,66),(73,76,79,82),(74,77,80,83),(75,78,81,84),(85,94,91,88),(86,95,92,89),(87,96,93,90),(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)], [(1,76,10,73,7,82,4,79),(2,81,11,78,8,75,5,84),(3,74,12,83,9,80,6,77),(13,66,22,63,19,72,16,69),(14,71,23,68,20,65,17,62),(15,64,24,61,21,70,18,67),(25,56,34,53,31,50,28,59),(26,49,35,58,32,55,29,52),(27,54,36,51,33,60,30,57),(37,98,46,107,43,104,40,101),(38,103,47,100,44,97,41,106),(39,108,48,105,45,102,42,99),(85,118,94,115,91,112,88,109),(86,111,95,120,92,117,89,114),(87,116,96,113,93,110,90,119)]])
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