direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×D4.5D4, C40.100D4, D4.5(C5×D4), C8.20(C5×D4), Q8.5(C5×D4), (C2×Q16)⋊8C10, C8○D4.1C10, (C5×D4).30D4, C4.39(D4×C10), (C5×Q8).30D4, C8.C22.C10, C8.C4⋊7C10, (C10×Q16)⋊22C2, C20.400(C2×D4), C4.10D4⋊4C10, (C2×C40).274C22, (C2×C20).615C23, M4(2).5(C2×C10), C10.156(C4⋊D4), (Q8×C10).168C22, (C5×M4(2)).49C22, (C5×C8○D4).4C2, (C2×C8).26(C2×C10), C2.25(C5×C4⋊D4), (C5×C8.C4)⋊16C2, C22.8(C5×C4○D4), C4○D4.12(C2×C10), (C2×Q8).12(C2×C10), (C5×C8.C22).2C2, (C5×C4.10D4)⋊10C2, (C2×C4).10(C22×C10), (C5×C4○D4).57C22, (C2×C10).117(C4○D4), SmallGroup(320,974)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4.5D4
G = < a,b,c,d,e | a5=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d3 >
Subgroups: 162 in 100 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C10, C10, C2×C8, C2×C8, M4(2), M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C40, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, D4.5D4, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C5×M4(2), C5×SD16, C5×Q16, Q8×C10, C5×C4○D4, C5×C4.10D4, C5×C8.C4, C5×C8○D4, C10×Q16, C5×C8.C22, C5×D4.5D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4⋊D4, C5×D4, C22×C10, D4.5D4, D4×C10, C5×C4○D4, C5×C4⋊D4, C5×D4.5D4
►On 160 points
Generators in S160
(1 17 52 13 61)(2 18 53 14 62)(3 19 54 15 63)(4 20 55 16 64)(5 21 56 9 57)(6 22 49 10 58)(7 23 50 11 59)(8 24 51 12 60)(25 85 73 33 65)(26 86 74 34 66)(27 87 75 35 67)(28 88 76 36 68)(29 81 77 37 69)(30 82 78 38 70)(31 83 79 39 71)(32 84 80 40 72)(41 156 120 148 112)(42 157 113 149 105)(43 158 114 150 106)(44 159 115 151 107)(45 160 116 152 108)(46 153 117 145 109)(47 154 118 146 110)(48 155 119 147 111)(89 126 141 97 133)(90 127 142 98 134)(91 128 143 99 135)(92 121 144 100 136)(93 122 137 101 129)(94 123 138 102 130)(95 124 139 103 131)(96 125 140 104 132)
(1 31 5 27)(2 32 6 28)(3 25 7 29)(4 26 8 30)(9 35 13 39)(10 36 14 40)(11 37 15 33)(12 38 16 34)(17 83 21 87)(18 84 22 88)(19 85 23 81)(20 86 24 82)(41 123 45 127)(42 124 46 128)(43 125 47 121)(44 126 48 122)(49 76 53 80)(50 77 54 73)(51 78 55 74)(52 79 56 75)(57 67 61 71)(58 68 62 72)(59 69 63 65)(60 70 64 66)(89 111 93 107)(90 112 94 108)(91 105 95 109)(92 106 96 110)(97 119 101 115)(98 120 102 116)(99 113 103 117)(100 114 104 118)(129 151 133 147)(130 152 134 148)(131 145 135 149)(132 146 136 150)(137 159 141 155)(138 160 142 156)(139 153 143 157)(140 154 144 158)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 39)(10 40)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 87)(18 88)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(49 80)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 71)(58 72)(59 65)(60 66)(61 67)(62 68)(63 69)(64 70)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)
(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)(121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160)
(1 94 5 90)(2 93 6 89)(3 92 7 96)(4 91 8 95)(9 98 13 102)(10 97 14 101)(11 104 15 100)(12 103 16 99)(17 123 21 127)(18 122 22 126)(19 121 23 125)(20 128 24 124)(25 110 29 106)(26 109 30 105)(27 108 31 112)(28 107 32 111)(33 118 37 114)(34 117 38 113)(35 116 39 120)(36 115 40 119)(41 87 45 83)(42 86 46 82)(43 85 47 81)(44 84 48 88)(49 141 53 137)(50 140 54 144)(51 139 55 143)(52 138 56 142)(57 134 61 130)(58 133 62 129)(59 132 63 136)(60 131 64 135)(65 146 69 150)(66 145 70 149)(67 152 71 148)(68 151 72 147)(73 154 77 158)(74 153 78 157)(75 160 79 156)(76 159 80 155)
G:=sub<Sym(160)| (1,17,52,13,61)(2,18,53,14,62)(3,19,54,15,63)(4,20,55,16,64)(5,21,56,9,57)(6,22,49,10,58)(7,23,50,11,59)(8,24,51,12,60)(25,85,73,33,65)(26,86,74,34,66)(27,87,75,35,67)(28,88,76,36,68)(29,81,77,37,69)(30,82,78,38,70)(31,83,79,39,71)(32,84,80,40,72)(41,156,120,148,112)(42,157,113,149,105)(43,158,114,150,106)(44,159,115,151,107)(45,160,116,152,108)(46,153,117,145,109)(47,154,118,146,110)(48,155,119,147,111)(89,126,141,97,133)(90,127,142,98,134)(91,128,143,99,135)(92,121,144,100,136)(93,122,137,101,129)(94,123,138,102,130)(95,124,139,103,131)(96,125,140,104,132), (1,31,5,27)(2,32,6,28)(3,25,7,29)(4,26,8,30)(9,35,13,39)(10,36,14,40)(11,37,15,33)(12,38,16,34)(17,83,21,87)(18,84,22,88)(19,85,23,81)(20,86,24,82)(41,123,45,127)(42,124,46,128)(43,125,47,121)(44,126,48,122)(49,76,53,80)(50,77,54,73)(51,78,55,74)(52,79,56,75)(57,67,61,71)(58,68,62,72)(59,69,63,65)(60,70,64,66)(89,111,93,107)(90,112,94,108)(91,105,95,109)(92,106,96,110)(97,119,101,115)(98,120,102,116)(99,113,103,117)(100,114,104,118)(129,151,133,147)(130,152,134,148)(131,145,135,149)(132,146,136,150)(137,159,141,155)(138,160,142,156)(139,153,143,157)(140,154,144,158), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,87)(18,88)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(49,80)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,71)(58,72)(59,65)(60,66)(61,67)(62,68)(63,69)(64,70)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144), (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)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160), (1,94,5,90)(2,93,6,89)(3,92,7,96)(4,91,8,95)(9,98,13,102)(10,97,14,101)(11,104,15,100)(12,103,16,99)(17,123,21,127)(18,122,22,126)(19,121,23,125)(20,128,24,124)(25,110,29,106)(26,109,30,105)(27,108,31,112)(28,107,32,111)(33,118,37,114)(34,117,38,113)(35,116,39,120)(36,115,40,119)(41,87,45,83)(42,86,46,82)(43,85,47,81)(44,84,48,88)(49,141,53,137)(50,140,54,144)(51,139,55,143)(52,138,56,142)(57,134,61,130)(58,133,62,129)(59,132,63,136)(60,131,64,135)(65,146,69,150)(66,145,70,149)(67,152,71,148)(68,151,72,147)(73,154,77,158)(74,153,78,157)(75,160,79,156)(76,159,80,155)>;
G:=Group( (1,17,52,13,61)(2,18,53,14,62)(3,19,54,15,63)(4,20,55,16,64)(5,21,56,9,57)(6,22,49,10,58)(7,23,50,11,59)(8,24,51,12,60)(25,85,73,33,65)(26,86,74,34,66)(27,87,75,35,67)(28,88,76,36,68)(29,81,77,37,69)(30,82,78,38,70)(31,83,79,39,71)(32,84,80,40,72)(41,156,120,148,112)(42,157,113,149,105)(43,158,114,150,106)(44,159,115,151,107)(45,160,116,152,108)(46,153,117,145,109)(47,154,118,146,110)(48,155,119,147,111)(89,126,141,97,133)(90,127,142,98,134)(91,128,143,99,135)(92,121,144,100,136)(93,122,137,101,129)(94,123,138,102,130)(95,124,139,103,131)(96,125,140,104,132), (1,31,5,27)(2,32,6,28)(3,25,7,29)(4,26,8,30)(9,35,13,39)(10,36,14,40)(11,37,15,33)(12,38,16,34)(17,83,21,87)(18,84,22,88)(19,85,23,81)(20,86,24,82)(41,123,45,127)(42,124,46,128)(43,125,47,121)(44,126,48,122)(49,76,53,80)(50,77,54,73)(51,78,55,74)(52,79,56,75)(57,67,61,71)(58,68,62,72)(59,69,63,65)(60,70,64,66)(89,111,93,107)(90,112,94,108)(91,105,95,109)(92,106,96,110)(97,119,101,115)(98,120,102,116)(99,113,103,117)(100,114,104,118)(129,151,133,147)(130,152,134,148)(131,145,135,149)(132,146,136,150)(137,159,141,155)(138,160,142,156)(139,153,143,157)(140,154,144,158), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,87)(18,88)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(49,80)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,71)(58,72)(59,65)(60,66)(61,67)(62,68)(63,69)(64,70)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144), (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)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160), (1,94,5,90)(2,93,6,89)(3,92,7,96)(4,91,8,95)(9,98,13,102)(10,97,14,101)(11,104,15,100)(12,103,16,99)(17,123,21,127)(18,122,22,126)(19,121,23,125)(20,128,24,124)(25,110,29,106)(26,109,30,105)(27,108,31,112)(28,107,32,111)(33,118,37,114)(34,117,38,113)(35,116,39,120)(36,115,40,119)(41,87,45,83)(42,86,46,82)(43,85,47,81)(44,84,48,88)(49,141,53,137)(50,140,54,144)(51,139,55,143)(52,138,56,142)(57,134,61,130)(58,133,62,129)(59,132,63,136)(60,131,64,135)(65,146,69,150)(66,145,70,149)(67,152,71,148)(68,151,72,147)(73,154,77,158)(74,153,78,157)(75,160,79,156)(76,159,80,155) );
G=PermutationGroup([[(1,17,52,13,61),(2,18,53,14,62),(3,19,54,15,63),(4,20,55,16,64),(5,21,56,9,57),(6,22,49,10,58),(7,23,50,11,59),(8,24,51,12,60),(25,85,73,33,65),(26,86,74,34,66),(27,87,75,35,67),(28,88,76,36,68),(29,81,77,37,69),(30,82,78,38,70),(31,83,79,39,71),(32,84,80,40,72),(41,156,120,148,112),(42,157,113,149,105),(43,158,114,150,106),(44,159,115,151,107),(45,160,116,152,108),(46,153,117,145,109),(47,154,118,146,110),(48,155,119,147,111),(89,126,141,97,133),(90,127,142,98,134),(91,128,143,99,135),(92,121,144,100,136),(93,122,137,101,129),(94,123,138,102,130),(95,124,139,103,131),(96,125,140,104,132)], [(1,31,5,27),(2,32,6,28),(3,25,7,29),(4,26,8,30),(9,35,13,39),(10,36,14,40),(11,37,15,33),(12,38,16,34),(17,83,21,87),(18,84,22,88),(19,85,23,81),(20,86,24,82),(41,123,45,127),(42,124,46,128),(43,125,47,121),(44,126,48,122),(49,76,53,80),(50,77,54,73),(51,78,55,74),(52,79,56,75),(57,67,61,71),(58,68,62,72),(59,69,63,65),(60,70,64,66),(89,111,93,107),(90,112,94,108),(91,105,95,109),(92,106,96,110),(97,119,101,115),(98,120,102,116),(99,113,103,117),(100,114,104,118),(129,151,133,147),(130,152,134,148),(131,145,135,149),(132,146,136,150),(137,159,141,155),(138,160,142,156),(139,153,143,157),(140,154,144,158)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,39),(10,40),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,87),(18,88),(19,81),(20,82),(21,83),(22,84),(23,85),(24,86),(49,80),(50,73),(51,74),(52,75),(53,76),(54,77),(55,78),(56,79),(57,71),(58,72),(59,65),(60,66),(61,67),(62,68),(63,69),(64,70),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144)], [(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),(121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136),(137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152),(153,154,155,156,157,158,159,160)], [(1,94,5,90),(2,93,6,89),(3,92,7,96),(4,91,8,95),(9,98,13,102),(10,97,14,101),(11,104,15,100),(12,103,16,99),(17,123,21,127),(18,122,22,126),(19,121,23,125),(20,128,24,124),(25,110,29,106),(26,109,30,105),(27,108,31,112),(28,107,32,111),(33,118,37,114),(34,117,38,113),(35,116,39,120),(36,115,40,119),(41,87,45,83),(42,86,46,82),(43,85,47,81),(44,84,48,88),(49,141,53,137),(50,140,54,144),(51,139,55,143),(52,138,56,142),(57,134,61,130),(58,133,62,129),(59,132,63,136),(60,131,64,135),(65,146,69,150),(66,145,70,149),(67,152,71,148),(68,151,72,147),(73,154,77,158),(74,153,78,157),(75,160,79,156),(76,159,80,155)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20T | 40A | ··· | 40H | 40I | ··· | 40T | 40U | ··· | 40AB |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | D4 | C4○D4 | C5×D4 | C5×D4 | C5×D4 | C5×C4○D4 | D4.5D4 | C5×D4.5D4 |
| kernel | C5×D4.5D4 | C5×C4.10D4 | C5×C8.C4 | C5×C8○D4 | C10×Q16 | C5×C8.C22 | D4.5D4 | C4.10D4 | C8.C4 | C8○D4 | C2×Q16 | C8.C22 | C40 | C5×D4 | C5×Q8 | C2×C10 | C8 | D4 | Q8 | C22 | C5 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 8 | 4 | 4 | 4 | 8 | 2 | 1 | 1 | 2 | 8 | 4 | 4 | 8 | 2 | 8 |
Matrix representation of C5×D4.5D4 ►in GL6(𝔽41)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 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 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 22 | 2 | 0 | 0 | 0 | 0 |
| 24 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 0 | 12 | 0 |
| 0 | 0 | 0 | 29 | 0 | 12 |
| 0 | 0 | 29 | 0 | 29 | 0 |
| 0 | 0 | 0 | 29 | 0 | 29 |
| 36 | 13 | 0 | 0 | 0 | 0 |
| 36 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 37 | 2 | 39 |
| 0 | 0 | 37 | 37 | 39 | 39 |
| 0 | 0 | 2 | 39 | 37 | 4 |
| 0 | 0 | 39 | 39 | 4 | 4 |
G:=sub<GL(6,GF(41))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[22,24,0,0,0,0,2,19,0,0,0,0,0,0,29,0,29,0,0,0,0,29,0,29,0,0,12,0,29,0,0,0,0,12,0,29],[36,36,0,0,0,0,13,5,0,0,0,0,0,0,4,37,2,39,0,0,37,37,39,39,0,0,2,39,37,4,0,0,39,39,4,4] >;
C5×D4.5D4 in GAP, Magma, Sage, TeX
C_5\times D_4._5D_4
% in TeX
G:=Group("C5xD4.5D4"); // GroupNames label
G:=SmallGroup(320,974);
// by ID
G=gap.SmallGroup(320,974);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1408,1766,7004,172,10085,2539,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=1,d^4=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d^3>;
// generators/relations