direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C10×D4⋊C4, D4⋊3(C2×C20), (C2×D4)⋊7C20, C2.1(C10×D8), (D4×C10)⋊31C4, (C22×C8)⋊3C10, (C22×C40)⋊7C2, C4.51(D4×C10), C10.73(C2×D8), (C2×C10).53D8, (C2×C40)⋊43C22, C20.458(C2×D4), (C2×C20).414D4, C4.1(C22×C20), C2.1(C10×SD16), C22.12(C5×D8), C23.54(C5×D4), (C2×C10).44SD16, C10.81(C2×SD16), (C22×D4).6C10, C22.41(D4×C10), (C2×C20).890C23, C20.205(C22×C4), (C22×C10).215D4, C22.10(C5×SD16), C20.127(C22⋊C4), (D4×C10).286C22, (C22×C20).581C22, (C2×C4⋊C4)⋊9C10, C4⋊C4⋊7(C2×C10), (C10×C4⋊C4)⋊36C2, (C2×C8)⋊11(C2×C10), (C5×D4)⋊33(C2×C4), (D4×C2×C10).18C2, (C2×C4).68(C5×D4), (C5×C4⋊C4)⋊63C22, (C2×C4).47(C2×C20), C4.12(C5×C22⋊C4), (C2×C20).441(C2×C4), (C2×D4).44(C2×C10), (C2×C10).617(C2×D4), C2.17(C10×C22⋊C4), C10.146(C2×C22⋊C4), (C2×C4).65(C22×C10), C22.33(C5×C22⋊C4), (C22×C4).110(C2×C10), (C2×C10).202(C22⋊C4), SmallGroup(320,915)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×D4⋊C4
G = < a,b,c,d | a10=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >
Subgroups: 402 in 202 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C10, C10, C10, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C20, C20, C20, C2×C10, C2×C10, C2×C10, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, C22×C10, C2×D4⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C22×C20, C22×C20, D4×C10, D4×C10, C23×C10, C5×D4⋊C4, C10×C4⋊C4, C22×C40, D4×C2×C10, C10×D4⋊C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, D8, SD16, C22×C4, C2×D4, C20, C2×C10, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C2×C20, C5×D4, C22×C10, C2×D4⋊C4, C5×C22⋊C4, C5×D8, C5×SD16, C22×C20, D4×C10, C5×D4⋊C4, C10×C22⋊C4, C10×D8, C10×SD16, C10×D4⋊C4
►On 160 points
Generators in S160
(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)
(1 132 81 96)(2 133 82 97)(3 134 83 98)(4 135 84 99)(5 136 85 100)(6 137 86 91)(7 138 87 92)(8 139 88 93)(9 140 89 94)(10 131 90 95)(11 40 51 160)(12 31 52 151)(13 32 53 152)(14 33 54 153)(15 34 55 154)(16 35 56 155)(17 36 57 156)(18 37 58 157)(19 38 59 158)(20 39 60 159)(21 75 61 46)(22 76 62 47)(23 77 63 48)(24 78 64 49)(25 79 65 50)(26 80 66 41)(27 71 67 42)(28 72 68 43)(29 73 69 44)(30 74 70 45)(101 123 147 116)(102 124 148 117)(103 125 149 118)(104 126 150 119)(105 127 141 120)(106 128 142 111)(107 129 143 112)(108 130 144 113)(109 121 145 114)(110 122 146 115)
(1 145)(2 146)(3 147)(4 148)(5 149)(6 150)(7 141)(8 142)(9 143)(10 144)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 61)(19 62)(20 63)(21 58)(22 59)(23 60)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 79)(32 80)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(41 152)(42 153)(43 154)(44 155)(45 156)(46 157)(47 158)(48 159)(49 160)(50 151)(81 109)(82 110)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(91 119)(92 120)(93 111)(94 112)(95 113)(96 114)(97 115)(98 116)(99 117)(100 118)(121 132)(122 133)(123 134)(124 135)(125 136)(126 137)(127 138)(128 139)(129 140)(130 131)
(1 157 114 75)(2 158 115 76)(3 159 116 77)(4 160 117 78)(5 151 118 79)(6 152 119 80)(7 153 120 71)(8 154 111 72)(9 155 112 73)(10 156 113 74)(11 148 64 99)(12 149 65 100)(13 150 66 91)(14 141 67 92)(15 142 68 93)(16 143 69 94)(17 144 70 95)(18 145 61 96)(19 146 62 97)(20 147 63 98)(21 132 58 109)(22 133 59 110)(23 134 60 101)(24 135 51 102)(25 136 52 103)(26 137 53 104)(27 138 54 105)(28 139 55 106)(29 140 56 107)(30 131 57 108)(31 125 50 85)(32 126 41 86)(33 127 42 87)(34 128 43 88)(35 129 44 89)(36 130 45 90)(37 121 46 81)(38 122 47 82)(39 123 48 83)(40 124 49 84)
G:=sub<Sym(160)| (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), (1,132,81,96)(2,133,82,97)(3,134,83,98)(4,135,84,99)(5,136,85,100)(6,137,86,91)(7,138,87,92)(8,139,88,93)(9,140,89,94)(10,131,90,95)(11,40,51,160)(12,31,52,151)(13,32,53,152)(14,33,54,153)(15,34,55,154)(16,35,56,155)(17,36,57,156)(18,37,58,157)(19,38,59,158)(20,39,60,159)(21,75,61,46)(22,76,62,47)(23,77,63,48)(24,78,64,49)(25,79,65,50)(26,80,66,41)(27,71,67,42)(28,72,68,43)(29,73,69,44)(30,74,70,45)(101,123,147,116)(102,124,148,117)(103,125,149,118)(104,126,150,119)(105,127,141,120)(106,128,142,111)(107,129,143,112)(108,130,144,113)(109,121,145,114)(110,122,146,115), (1,145)(2,146)(3,147)(4,148)(5,149)(6,150)(7,141)(8,142)(9,143)(10,144)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,61)(19,62)(20,63)(21,58)(22,59)(23,60)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,79)(32,80)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,152)(42,153)(43,154)(44,155)(45,156)(46,157)(47,158)(48,159)(49,160)(50,151)(81,109)(82,110)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,119)(92,120)(93,111)(94,112)(95,113)(96,114)(97,115)(98,116)(99,117)(100,118)(121,132)(122,133)(123,134)(124,135)(125,136)(126,137)(127,138)(128,139)(129,140)(130,131), (1,157,114,75)(2,158,115,76)(3,159,116,77)(4,160,117,78)(5,151,118,79)(6,152,119,80)(7,153,120,71)(8,154,111,72)(9,155,112,73)(10,156,113,74)(11,148,64,99)(12,149,65,100)(13,150,66,91)(14,141,67,92)(15,142,68,93)(16,143,69,94)(17,144,70,95)(18,145,61,96)(19,146,62,97)(20,147,63,98)(21,132,58,109)(22,133,59,110)(23,134,60,101)(24,135,51,102)(25,136,52,103)(26,137,53,104)(27,138,54,105)(28,139,55,106)(29,140,56,107)(30,131,57,108)(31,125,50,85)(32,126,41,86)(33,127,42,87)(34,128,43,88)(35,129,44,89)(36,130,45,90)(37,121,46,81)(38,122,47,82)(39,123,48,83)(40,124,49,84)>;
G:=Group( (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), (1,132,81,96)(2,133,82,97)(3,134,83,98)(4,135,84,99)(5,136,85,100)(6,137,86,91)(7,138,87,92)(8,139,88,93)(9,140,89,94)(10,131,90,95)(11,40,51,160)(12,31,52,151)(13,32,53,152)(14,33,54,153)(15,34,55,154)(16,35,56,155)(17,36,57,156)(18,37,58,157)(19,38,59,158)(20,39,60,159)(21,75,61,46)(22,76,62,47)(23,77,63,48)(24,78,64,49)(25,79,65,50)(26,80,66,41)(27,71,67,42)(28,72,68,43)(29,73,69,44)(30,74,70,45)(101,123,147,116)(102,124,148,117)(103,125,149,118)(104,126,150,119)(105,127,141,120)(106,128,142,111)(107,129,143,112)(108,130,144,113)(109,121,145,114)(110,122,146,115), (1,145)(2,146)(3,147)(4,148)(5,149)(6,150)(7,141)(8,142)(9,143)(10,144)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,61)(19,62)(20,63)(21,58)(22,59)(23,60)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,79)(32,80)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,152)(42,153)(43,154)(44,155)(45,156)(46,157)(47,158)(48,159)(49,160)(50,151)(81,109)(82,110)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,119)(92,120)(93,111)(94,112)(95,113)(96,114)(97,115)(98,116)(99,117)(100,118)(121,132)(122,133)(123,134)(124,135)(125,136)(126,137)(127,138)(128,139)(129,140)(130,131), (1,157,114,75)(2,158,115,76)(3,159,116,77)(4,160,117,78)(5,151,118,79)(6,152,119,80)(7,153,120,71)(8,154,111,72)(9,155,112,73)(10,156,113,74)(11,148,64,99)(12,149,65,100)(13,150,66,91)(14,141,67,92)(15,142,68,93)(16,143,69,94)(17,144,70,95)(18,145,61,96)(19,146,62,97)(20,147,63,98)(21,132,58,109)(22,133,59,110)(23,134,60,101)(24,135,51,102)(25,136,52,103)(26,137,53,104)(27,138,54,105)(28,139,55,106)(29,140,56,107)(30,131,57,108)(31,125,50,85)(32,126,41,86)(33,127,42,87)(34,128,43,88)(35,129,44,89)(36,130,45,90)(37,121,46,81)(38,122,47,82)(39,123,48,83)(40,124,49,84) );
G=PermutationGroup([[(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)], [(1,132,81,96),(2,133,82,97),(3,134,83,98),(4,135,84,99),(5,136,85,100),(6,137,86,91),(7,138,87,92),(8,139,88,93),(9,140,89,94),(10,131,90,95),(11,40,51,160),(12,31,52,151),(13,32,53,152),(14,33,54,153),(15,34,55,154),(16,35,56,155),(17,36,57,156),(18,37,58,157),(19,38,59,158),(20,39,60,159),(21,75,61,46),(22,76,62,47),(23,77,63,48),(24,78,64,49),(25,79,65,50),(26,80,66,41),(27,71,67,42),(28,72,68,43),(29,73,69,44),(30,74,70,45),(101,123,147,116),(102,124,148,117),(103,125,149,118),(104,126,150,119),(105,127,141,120),(106,128,142,111),(107,129,143,112),(108,130,144,113),(109,121,145,114),(110,122,146,115)], [(1,145),(2,146),(3,147),(4,148),(5,149),(6,150),(7,141),(8,142),(9,143),(10,144),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,61),(19,62),(20,63),(21,58),(22,59),(23,60),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,79),(32,80),(33,71),(34,72),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78),(41,152),(42,153),(43,154),(44,155),(45,156),(46,157),(47,158),(48,159),(49,160),(50,151),(81,109),(82,110),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(91,119),(92,120),(93,111),(94,112),(95,113),(96,114),(97,115),(98,116),(99,117),(100,118),(121,132),(122,133),(123,134),(124,135),(125,136),(126,137),(127,138),(128,139),(129,140),(130,131)], [(1,157,114,75),(2,158,115,76),(3,159,116,77),(4,160,117,78),(5,151,118,79),(6,152,119,80),(7,153,120,71),(8,154,111,72),(9,155,112,73),(10,156,113,74),(11,148,64,99),(12,149,65,100),(13,150,66,91),(14,141,67,92),(15,142,68,93),(16,143,69,94),(17,144,70,95),(18,145,61,96),(19,146,62,97),(20,147,63,98),(21,132,58,109),(22,133,59,110),(23,134,60,101),(24,135,51,102),(25,136,52,103),(26,137,53,104),(27,138,54,105),(28,139,55,106),(29,140,56,107),(30,131,57,108),(31,125,50,85),(32,126,41,86),(33,127,42,87),(34,128,43,88),(35,129,44,89),(36,130,45,90),(37,121,46,81),(38,122,47,82),(39,123,48,83),(40,124,49,84)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | D4 | D4 | D8 | SD16 | C5×D4 | C5×D4 | C5×D8 | C5×SD16 |
| kernel | C10×D4⋊C4 | C5×D4⋊C4 | C10×C4⋊C4 | C22×C40 | D4×C2×C10 | D4×C10 | C2×D4⋊C4 | D4⋊C4 | C2×C4⋊C4 | C22×C8 | C22×D4 | C2×D4 | C2×C20 | C22×C10 | C2×C10 | C2×C10 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 16 | 4 | 4 | 4 | 32 | 3 | 1 | 4 | 4 | 12 | 4 | 16 | 16 |
Matrix representation of C10×D4⋊C4 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 0 | 29 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,25,0,0,0,0,25,0,0,0,0,25],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,39,40],[1,0,0,0,0,32,0,0,0,0,0,29,0,0,17,0] >;
C10×D4⋊C4 in GAP, Magma, Sage, TeX
C_{10}\times D_4\rtimes C_4 % in TeX
G:=Group("C10xD4:C4"); // GroupNames label
G:=SmallGroup(320,915);
// by ID
G=gap.SmallGroup(320,915);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations