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 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×4], D4 [×8], C23 [×2], D5, C10 [×3], C10, C42, C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, C2×D4, C2×D4 [×3], Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C8⋊C4, D4⋊C4, D4⋊C4 [×3], C42.C2, C4⋊1D4, C5⋊2C8, C40, D20 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], 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 [×2], D4×C10, D20⋊6C4, C40⋊8C4, D20⋊5C4, D4⋊Dic5, C5×D4⋊C4, C4.Dic10, C20⋊D4, C4⋊C4.D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C8⋊C22 [×2], 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 70 123 103)(2 104 124 61)(3 62 125 105)(4 106 126 63)(5 64 127 107)(6 108 128 65)(7 66 129 109)(8 110 130 67)(9 68 121 101)(10 102 122 69)(11 35 81 148)(12 149 82 36)(13 37 83 150)(14 141 84 38)(15 39 85 142)(16 143 86 40)(17 31 87 144)(18 145 88 32)(19 33 89 146)(20 147 90 34)(21 51 134 94)(22 95 135 52)(23 53 136 96)(24 97 137 54)(25 55 138 98)(26 99 139 56)(27 57 140 100)(28 91 131 58)(29 59 132 92)(30 93 133 60)(41 114 154 71)(42 72 155 115)(43 116 156 73)(44 74 157 117)(45 118 158 75)(46 76 159 119)(47 120 160 77)(48 78 151 111)(49 112 152 79)(50 80 153 113)
(1 113 93 40)(2 17 94 41)(3 115 95 32)(4 19 96 43)(5 117 97 34)(6 11 98 45)(7 119 99 36)(8 13 100 47)(9 111 91 38)(10 15 92 49)(12 109 46 26)(14 101 48 28)(16 103 50 30)(18 105 42 22)(20 107 44 24)(21 71 104 144)(23 73 106 146)(25 75 108 148)(27 77 110 150)(29 79 102 142)(31 134 114 61)(33 136 116 63)(35 138 118 65)(37 140 120 67)(39 132 112 69)(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 96)(4 94)(6 92)(7 9)(8 100)(10 98)(11 79)(12 141)(13 77)(14 149)(15 75)(16 147)(17 73)(18 145)(19 71)(20 143)(21 63)(22 135)(23 61)(24 133)(25 69)(26 131)(27 67)(28 139)(29 65)(30 137)(31 156)(32 88)(33 154)(34 86)(35 152)(36 84)(37 160)(38 82)(39 158)(40 90)(41 146)(42 72)(43 144)(44 80)(45 142)(46 78)(47 150)(48 76)(49 148)(50 74)(51 126)(53 124)(54 60)(55 122)(56 58)(57 130)(59 128)(62 105)(64 103)(66 101)(68 109)(70 107)(81 112)(83 120)(85 118)(87 116)(89 114)(91 99)(93 97)(102 138)(104 136)(106 134)(108 132)(110 140)(111 159)(113 157)(115 155)(117 153)(119 151)(121 129)(123 127)
G:=sub<Sym(160)| (1,70,123,103)(2,104,124,61)(3,62,125,105)(4,106,126,63)(5,64,127,107)(6,108,128,65)(7,66,129,109)(8,110,130,67)(9,68,121,101)(10,102,122,69)(11,35,81,148)(12,149,82,36)(13,37,83,150)(14,141,84,38)(15,39,85,142)(16,143,86,40)(17,31,87,144)(18,145,88,32)(19,33,89,146)(20,147,90,34)(21,51,134,94)(22,95,135,52)(23,53,136,96)(24,97,137,54)(25,55,138,98)(26,99,139,56)(27,57,140,100)(28,91,131,58)(29,59,132,92)(30,93,133,60)(41,114,154,71)(42,72,155,115)(43,116,156,73)(44,74,157,117)(45,118,158,75)(46,76,159,119)(47,120,160,77)(48,78,151,111)(49,112,152,79)(50,80,153,113), (1,113,93,40)(2,17,94,41)(3,115,95,32)(4,19,96,43)(5,117,97,34)(6,11,98,45)(7,119,99,36)(8,13,100,47)(9,111,91,38)(10,15,92,49)(12,109,46,26)(14,101,48,28)(16,103,50,30)(18,105,42,22)(20,107,44,24)(21,71,104,144)(23,73,106,146)(25,75,108,148)(27,77,110,150)(29,79,102,142)(31,134,114,61)(33,136,116,63)(35,138,118,65)(37,140,120,67)(39,132,112,69)(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)(2,96)(4,94)(6,92)(7,9)(8,100)(10,98)(11,79)(12,141)(13,77)(14,149)(15,75)(16,147)(17,73)(18,145)(19,71)(20,143)(21,63)(22,135)(23,61)(24,133)(25,69)(26,131)(27,67)(28,139)(29,65)(30,137)(31,156)(32,88)(33,154)(34,86)(35,152)(36,84)(37,160)(38,82)(39,158)(40,90)(41,146)(42,72)(43,144)(44,80)(45,142)(46,78)(47,150)(48,76)(49,148)(50,74)(51,126)(53,124)(54,60)(55,122)(56,58)(57,130)(59,128)(62,105)(64,103)(66,101)(68,109)(70,107)(81,112)(83,120)(85,118)(87,116)(89,114)(91,99)(93,97)(102,138)(104,136)(106,134)(108,132)(110,140)(111,159)(113,157)(115,155)(117,153)(119,151)(121,129)(123,127)>;
G:=Group( (1,70,123,103)(2,104,124,61)(3,62,125,105)(4,106,126,63)(5,64,127,107)(6,108,128,65)(7,66,129,109)(8,110,130,67)(9,68,121,101)(10,102,122,69)(11,35,81,148)(12,149,82,36)(13,37,83,150)(14,141,84,38)(15,39,85,142)(16,143,86,40)(17,31,87,144)(18,145,88,32)(19,33,89,146)(20,147,90,34)(21,51,134,94)(22,95,135,52)(23,53,136,96)(24,97,137,54)(25,55,138,98)(26,99,139,56)(27,57,140,100)(28,91,131,58)(29,59,132,92)(30,93,133,60)(41,114,154,71)(42,72,155,115)(43,116,156,73)(44,74,157,117)(45,118,158,75)(46,76,159,119)(47,120,160,77)(48,78,151,111)(49,112,152,79)(50,80,153,113), (1,113,93,40)(2,17,94,41)(3,115,95,32)(4,19,96,43)(5,117,97,34)(6,11,98,45)(7,119,99,36)(8,13,100,47)(9,111,91,38)(10,15,92,49)(12,109,46,26)(14,101,48,28)(16,103,50,30)(18,105,42,22)(20,107,44,24)(21,71,104,144)(23,73,106,146)(25,75,108,148)(27,77,110,150)(29,79,102,142)(31,134,114,61)(33,136,116,63)(35,138,118,65)(37,140,120,67)(39,132,112,69)(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)(2,96)(4,94)(6,92)(7,9)(8,100)(10,98)(11,79)(12,141)(13,77)(14,149)(15,75)(16,147)(17,73)(18,145)(19,71)(20,143)(21,63)(22,135)(23,61)(24,133)(25,69)(26,131)(27,67)(28,139)(29,65)(30,137)(31,156)(32,88)(33,154)(34,86)(35,152)(36,84)(37,160)(38,82)(39,158)(40,90)(41,146)(42,72)(43,144)(44,80)(45,142)(46,78)(47,150)(48,76)(49,148)(50,74)(51,126)(53,124)(54,60)(55,122)(56,58)(57,130)(59,128)(62,105)(64,103)(66,101)(68,109)(70,107)(81,112)(83,120)(85,118)(87,116)(89,114)(91,99)(93,97)(102,138)(104,136)(106,134)(108,132)(110,140)(111,159)(113,157)(115,155)(117,153)(119,151)(121,129)(123,127) );
G=PermutationGroup([(1,70,123,103),(2,104,124,61),(3,62,125,105),(4,106,126,63),(5,64,127,107),(6,108,128,65),(7,66,129,109),(8,110,130,67),(9,68,121,101),(10,102,122,69),(11,35,81,148),(12,149,82,36),(13,37,83,150),(14,141,84,38),(15,39,85,142),(16,143,86,40),(17,31,87,144),(18,145,88,32),(19,33,89,146),(20,147,90,34),(21,51,134,94),(22,95,135,52),(23,53,136,96),(24,97,137,54),(25,55,138,98),(26,99,139,56),(27,57,140,100),(28,91,131,58),(29,59,132,92),(30,93,133,60),(41,114,154,71),(42,72,155,115),(43,116,156,73),(44,74,157,117),(45,118,158,75),(46,76,159,119),(47,120,160,77),(48,78,151,111),(49,112,152,79),(50,80,153,113)], [(1,113,93,40),(2,17,94,41),(3,115,95,32),(4,19,96,43),(5,117,97,34),(6,11,98,45),(7,119,99,36),(8,13,100,47),(9,111,91,38),(10,15,92,49),(12,109,46,26),(14,101,48,28),(16,103,50,30),(18,105,42,22),(20,107,44,24),(21,71,104,144),(23,73,106,146),(25,75,108,148),(27,77,110,150),(29,79,102,142),(31,134,114,61),(33,136,116,63),(35,138,118,65),(37,140,120,67),(39,132,112,69),(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),(2,96),(4,94),(6,92),(7,9),(8,100),(10,98),(11,79),(12,141),(13,77),(14,149),(15,75),(16,147),(17,73),(18,145),(19,71),(20,143),(21,63),(22,135),(23,61),(24,133),(25,69),(26,131),(27,67),(28,139),(29,65),(30,137),(31,156),(32,88),(33,154),(34,86),(35,152),(36,84),(37,160),(38,82),(39,158),(40,90),(41,146),(42,72),(43,144),(44,80),(45,142),(46,78),(47,150),(48,76),(49,148),(50,74),(51,126),(53,124),(54,60),(55,122),(56,58),(57,130),(59,128),(62,105),(64,103),(66,101),(68,109),(70,107),(81,112),(83,120),(85,118),(87,116),(89,114),(91,99),(93,97),(102,138),(104,136),(106,134),(108,132),(110,140),(111,159),(113,157),(115,155),(117,153),(119,151),(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