metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.5D10, C40⋊8C4⋊16C2, D4⋊C4⋊15D5, D20⋊6C4⋊3C2, (C2×D4).19D10, C20.3(C4○D4), (C2×C8).166D10, C20⋊D4.6C2, D4⋊Dic5⋊2C2, C4.Dic10⋊2C2, D20⋊5C4⋊16C2, C4.21(C4○D20), C2.8(D40⋊C2), (C2×Dic5).23D4, C22.166(D4×D5), C2.11(D8⋊D5), C4.47(D4⋊2D5), C10.53(C8⋊C22), (C2×C40).179C22, (C2×C20).204C23, (C2×D20).50C22, (D4×C10).25C22, C4⋊Dic5.63C22, C10.23(C4.4D4), (C4×Dic5).14C22, C5⋊2(C42.29C22), C2.13(Dic5.5D4), (C5×C4⋊C4).9C22, (C5×D4⋊C4)⋊17C2, (C2×C10).217(C2×D4), (C2×C5⋊2C8).10C22, (C2×C4).311(C22×D5), SmallGroup(320,391)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.D10
G = < a,b,c,d | a4=b4=c10=d2=1, bab-1=cac-1=dad=a-1, cbc-1=a-1b-1, dbd=ab, dcd=b2c-1 >
Subgroups: 542 in 110 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C8⋊C4, D4⋊C4, D4⋊C4, C42.C2, C4⋊1D4, C5⋊2C8, C40, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C42.29C22, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×D20, C2×C5⋊D4, D4×C10, D20⋊6C4, C40⋊8C4, D20⋊5C4, D4⋊Dic5, C5×D4⋊C4, C4.Dic10, C20⋊D4, C4⋊C4.D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C8⋊C22, C22×D5, C42.29C22, C4○D20, D4×D5, D4⋊2D5, Dic5.5D4, D8⋊D5, D40⋊C2, C4⋊C4.D10
►On 160 points
Generators in S160
(1 70 123 109)(2 110 124 61)(3 62 125 101)(4 102 126 63)(5 64 127 103)(6 104 128 65)(7 66 129 105)(8 106 130 67)(9 68 121 107)(10 108 122 69)(11 35 85 142)(12 143 86 36)(13 37 87 144)(14 145 88 38)(15 39 89 146)(16 147 90 40)(17 31 81 148)(18 149 82 32)(19 33 83 150)(20 141 84 34)(21 55 138 94)(22 95 139 56)(23 57 140 96)(24 97 131 58)(25 59 132 98)(26 99 133 60)(27 51 134 100)(28 91 135 52)(29 53 136 92)(30 93 137 54)(41 114 158 75)(42 76 159 115)(43 116 160 77)(44 78 151 117)(45 118 152 79)(46 80 153 119)(47 120 154 71)(48 72 155 111)(49 112 156 73)(50 74 157 113)
(1 119 99 36)(2 13 100 47)(3 111 91 38)(4 15 92 49)(5 113 93 40)(6 17 94 41)(7 115 95 32)(8 19 96 43)(9 117 97 34)(10 11 98 45)(12 109 46 26)(14 101 48 28)(16 103 50 30)(18 105 42 22)(20 107 44 24)(21 75 104 148)(23 77 106 150)(25 79 108 142)(27 71 110 144)(29 73 102 146)(31 138 114 65)(33 140 116 67)(35 132 118 69)(37 134 120 61)(39 136 112 63)(51 154 124 87)(52 145 125 72)(53 156 126 89)(54 147 127 74)(55 158 128 81)(56 149 129 76)(57 160 130 83)(58 141 121 78)(59 152 122 85)(60 143 123 80)(62 155 135 88)(64 157 137 90)(66 159 139 82)(68 151 131 84)(70 153 133 86)
(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 5)(2 92)(4 100)(6 98)(7 9)(8 96)(10 94)(11 75)(12 147)(13 73)(14 145)(15 71)(16 143)(17 79)(18 141)(19 77)(20 149)(21 69)(22 131)(23 67)(24 139)(25 65)(26 137)(27 63)(28 135)(29 61)(30 133)(31 152)(32 84)(33 160)(34 82)(35 158)(36 90)(37 156)(38 88)(39 154)(40 86)(41 142)(42 78)(43 150)(44 76)(45 148)(46 74)(47 146)(48 72)(49 144)(50 80)(51 126)(53 124)(54 60)(55 122)(56 58)(57 130)(59 128)(62 101)(64 109)(66 107)(68 105)(70 103)(81 118)(83 116)(85 114)(87 112)(89 120)(93 99)(95 97)(102 134)(104 132)(106 140)(108 138)(110 136)(111 155)(113 153)(115 151)(117 159)(119 157)(121 129)(123 127)
G:=sub<Sym(160)| (1,70,123,109)(2,110,124,61)(3,62,125,101)(4,102,126,63)(5,64,127,103)(6,104,128,65)(7,66,129,105)(8,106,130,67)(9,68,121,107)(10,108,122,69)(11,35,85,142)(12,143,86,36)(13,37,87,144)(14,145,88,38)(15,39,89,146)(16,147,90,40)(17,31,81,148)(18,149,82,32)(19,33,83,150)(20,141,84,34)(21,55,138,94)(22,95,139,56)(23,57,140,96)(24,97,131,58)(25,59,132,98)(26,99,133,60)(27,51,134,100)(28,91,135,52)(29,53,136,92)(30,93,137,54)(41,114,158,75)(42,76,159,115)(43,116,160,77)(44,78,151,117)(45,118,152,79)(46,80,153,119)(47,120,154,71)(48,72,155,111)(49,112,156,73)(50,74,157,113), (1,119,99,36)(2,13,100,47)(3,111,91,38)(4,15,92,49)(5,113,93,40)(6,17,94,41)(7,115,95,32)(8,19,96,43)(9,117,97,34)(10,11,98,45)(12,109,46,26)(14,101,48,28)(16,103,50,30)(18,105,42,22)(20,107,44,24)(21,75,104,148)(23,77,106,150)(25,79,108,142)(27,71,110,144)(29,73,102,146)(31,138,114,65)(33,140,116,67)(35,132,118,69)(37,134,120,61)(39,136,112,63)(51,154,124,87)(52,145,125,72)(53,156,126,89)(54,147,127,74)(55,158,128,81)(56,149,129,76)(57,160,130,83)(58,141,121,78)(59,152,122,85)(60,143,123,80)(62,155,135,88)(64,157,137,90)(66,159,139,82)(68,151,131,84)(70,153,133,86), (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), (1,5)(2,92)(4,100)(6,98)(7,9)(8,96)(10,94)(11,75)(12,147)(13,73)(14,145)(15,71)(16,143)(17,79)(18,141)(19,77)(20,149)(21,69)(22,131)(23,67)(24,139)(25,65)(26,137)(27,63)(28,135)(29,61)(30,133)(31,152)(32,84)(33,160)(34,82)(35,158)(36,90)(37,156)(38,88)(39,154)(40,86)(41,142)(42,78)(43,150)(44,76)(45,148)(46,74)(47,146)(48,72)(49,144)(50,80)(51,126)(53,124)(54,60)(55,122)(56,58)(57,130)(59,128)(62,101)(64,109)(66,107)(68,105)(70,103)(81,118)(83,116)(85,114)(87,112)(89,120)(93,99)(95,97)(102,134)(104,132)(106,140)(108,138)(110,136)(111,155)(113,153)(115,151)(117,159)(119,157)(121,129)(123,127)>;
G:=Group( (1,70,123,109)(2,110,124,61)(3,62,125,101)(4,102,126,63)(5,64,127,103)(6,104,128,65)(7,66,129,105)(8,106,130,67)(9,68,121,107)(10,108,122,69)(11,35,85,142)(12,143,86,36)(13,37,87,144)(14,145,88,38)(15,39,89,146)(16,147,90,40)(17,31,81,148)(18,149,82,32)(19,33,83,150)(20,141,84,34)(21,55,138,94)(22,95,139,56)(23,57,140,96)(24,97,131,58)(25,59,132,98)(26,99,133,60)(27,51,134,100)(28,91,135,52)(29,53,136,92)(30,93,137,54)(41,114,158,75)(42,76,159,115)(43,116,160,77)(44,78,151,117)(45,118,152,79)(46,80,153,119)(47,120,154,71)(48,72,155,111)(49,112,156,73)(50,74,157,113), (1,119,99,36)(2,13,100,47)(3,111,91,38)(4,15,92,49)(5,113,93,40)(6,17,94,41)(7,115,95,32)(8,19,96,43)(9,117,97,34)(10,11,98,45)(12,109,46,26)(14,101,48,28)(16,103,50,30)(18,105,42,22)(20,107,44,24)(21,75,104,148)(23,77,106,150)(25,79,108,142)(27,71,110,144)(29,73,102,146)(31,138,114,65)(33,140,116,67)(35,132,118,69)(37,134,120,61)(39,136,112,63)(51,154,124,87)(52,145,125,72)(53,156,126,89)(54,147,127,74)(55,158,128,81)(56,149,129,76)(57,160,130,83)(58,141,121,78)(59,152,122,85)(60,143,123,80)(62,155,135,88)(64,157,137,90)(66,159,139,82)(68,151,131,84)(70,153,133,86), (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), (1,5)(2,92)(4,100)(6,98)(7,9)(8,96)(10,94)(11,75)(12,147)(13,73)(14,145)(15,71)(16,143)(17,79)(18,141)(19,77)(20,149)(21,69)(22,131)(23,67)(24,139)(25,65)(26,137)(27,63)(28,135)(29,61)(30,133)(31,152)(32,84)(33,160)(34,82)(35,158)(36,90)(37,156)(38,88)(39,154)(40,86)(41,142)(42,78)(43,150)(44,76)(45,148)(46,74)(47,146)(48,72)(49,144)(50,80)(51,126)(53,124)(54,60)(55,122)(56,58)(57,130)(59,128)(62,101)(64,109)(66,107)(68,105)(70,103)(81,118)(83,116)(85,114)(87,112)(89,120)(93,99)(95,97)(102,134)(104,132)(106,140)(108,138)(110,136)(111,155)(113,153)(115,151)(117,159)(119,157)(121,129)(123,127) );
G=PermutationGroup([[(1,70,123,109),(2,110,124,61),(3,62,125,101),(4,102,126,63),(5,64,127,103),(6,104,128,65),(7,66,129,105),(8,106,130,67),(9,68,121,107),(10,108,122,69),(11,35,85,142),(12,143,86,36),(13,37,87,144),(14,145,88,38),(15,39,89,146),(16,147,90,40),(17,31,81,148),(18,149,82,32),(19,33,83,150),(20,141,84,34),(21,55,138,94),(22,95,139,56),(23,57,140,96),(24,97,131,58),(25,59,132,98),(26,99,133,60),(27,51,134,100),(28,91,135,52),(29,53,136,92),(30,93,137,54),(41,114,158,75),(42,76,159,115),(43,116,160,77),(44,78,151,117),(45,118,152,79),(46,80,153,119),(47,120,154,71),(48,72,155,111),(49,112,156,73),(50,74,157,113)], [(1,119,99,36),(2,13,100,47),(3,111,91,38),(4,15,92,49),(5,113,93,40),(6,17,94,41),(7,115,95,32),(8,19,96,43),(9,117,97,34),(10,11,98,45),(12,109,46,26),(14,101,48,28),(16,103,50,30),(18,105,42,22),(20,107,44,24),(21,75,104,148),(23,77,106,150),(25,79,108,142),(27,71,110,144),(29,73,102,146),(31,138,114,65),(33,140,116,67),(35,132,118,69),(37,134,120,61),(39,136,112,63),(51,154,124,87),(52,145,125,72),(53,156,126,89),(54,147,127,74),(55,158,128,81),(56,149,129,76),(57,160,130,83),(58,141,121,78),(59,152,122,85),(60,143,123,80),(62,155,135,88),(64,157,137,90),(66,159,139,82),(68,151,131,84),(70,153,133,86)], [(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)], [(1,5),(2,92),(4,100),(6,98),(7,9),(8,96),(10,94),(11,75),(12,147),(13,73),(14,145),(15,71),(16,143),(17,79),(18,141),(19,77),(20,149),(21,69),(22,131),(23,67),(24,139),(25,65),(26,137),(27,63),(28,135),(29,61),(30,133),(31,152),(32,84),(33,160),(34,82),(35,158),(36,90),(37,156),(38,88),(39,154),(40,86),(41,142),(42,78),(43,150),(44,76),(45,148),(46,74),(47,146),(48,72),(49,144),(50,80),(51,126),(53,124),(54,60),(55,122),(56,58),(57,130),(59,128),(62,101),(64,109),(66,107),(68,105),(70,103),(81,118),(83,116),(85,114),(87,112),(89,120),(93,99),(95,97),(102,134),(104,132),(106,140),(108,138),(110,136),(111,155),(113,153),(115,151),(117,159),(119,157),(121,129),(123,127)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 40 | 2 | 2 | 8 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | C8⋊C22 | D4⋊2D5 | D4×D5 | D8⋊D5 | D40⋊C2 |
| kernel | C4⋊C4.D10 | D20⋊6C4 | C40⋊8C4 | D20⋊5C4 | D4⋊Dic5 | C5×D4⋊C4 | C4.Dic10 | C20⋊D4 | C2×Dic5 | D4⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C4⋊C4.D10 ►in GL8(𝔽41)
| 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 | 39 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 39 |
| 30 | 32 | 39 | 0 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 17 | 38 | 13 | 28 | 0 | 0 | 0 | 0 |
| 3 | 38 | 13 | 22 | 0 | 0 | 0 | 0 |
| 32 | 19 | 24 | 3 | 0 | 0 | 0 | 0 |
| 22 | 22 | 38 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 39 | 27 |
| 0 | 0 | 0 | 0 | 0 | 0 | 14 | 37 |
| 0 | 0 | 0 | 0 | 14 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 26 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 27 | 7 | 7 | 0 | 0 | 0 | 0 |
| 11 | 27 | 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 25 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 16 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,39,13,0,0,0,0,0,0,28,2,0,0,0,0,0,0,0,0,2,28,0,0,0,0,0,0,13,39],[30,9,0,0,0,0,0,0,32,11,0,0,0,0,0,0,39,0,11,32,0,0,0,0,0,39,9,30,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[17,3,32,22,0,0,0,0,38,38,19,22,0,0,0,0,13,13,24,38,0,0,0,0,28,22,3,3,0,0,0,0,0,0,0,0,0,0,14,39,0,0,0,0,0,0,2,26,0,0,0,0,39,14,0,0,0,0,0,0,27,37,0,0],[7,40,14,11,0,0,0,0,7,34,27,27,0,0,0,0,0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,0,0,0,0,35,40,0,0,0,0,0,0,35,6,0,0,0,0,0,0,0,0,25,2,0,0,0,0,0,0,16,16] >;
C4⋊C4.D10 in GAP, Magma, Sage, TeX
C_4\rtimes C_4.D_{10} % in TeX
G:=Group("C4:C4.D10"); // GroupNames label
G:=SmallGroup(320,391);
// by ID
G=gap.SmallGroup(320,391);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,1094,135,100,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b^-1,d*b*d=a*b,d*c*d=b^2*c^-1>;
// generators/relations