direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D10⋊C4, D10⋊3C42, C2.3(C4×D20), (C2×C42)⋊1D5, C10.47(C4×D4), C20⋊8(C22⋊C4), (C2×C4).168D20, (C2×C20).494D4, C2.13(D5×C42), C10.32(C2×C42), C22.38(C2×D20), (C22×C4).459D10, C2.4(C42⋊D5), C22.47(C4○D20), (C23×D5).98C22, C23.271(C22×D5), C10.10C42⋊48C2, C10.33(C42⋊C2), (C22×C20).477C22, (C22×C10).313C23, (C22×Dic5).204C22, (C2×C4×C20)⋊15C2, (C2×C4×D5)⋊15C4, C5⋊4(C4×C22⋊C4), (C2×C4)⋊11(C4×D5), C2.2(C4×C5⋊D4), (C2×C20)⋊40(C2×C4), (C2×C4×Dic5)⋊19C2, C22.53(C2×C4×D5), (D5×C22×C4).18C2, (C2×Dic5)⋊22(C2×C4), C2.1(C2×D10⋊C4), (C2×C10).427(C2×D4), C10.73(C2×C22⋊C4), C22.42(C2×C5⋊D4), (C2×C10).72(C4○D4), (C2×C4).272(C5⋊D4), (C22×D5).77(C2×C4), (C2×D10⋊C4).30C2, (C2×C10).200(C22×C4), SmallGroup(320,565)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D10⋊C4
G = < a,b,c,d | a4=b10=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b5c >
Subgroups: 894 in 258 conjugacy classes, 107 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C10, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C23×C4, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C4×C22⋊C4, C4×Dic5, D10⋊C4, C4×C20, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C23×D5, C10.10C42, C2×C4×Dic5, C2×D10⋊C4, C2×C4×C20, D5×C22×C4, C4×D10⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, D10, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×D5, D20, C5⋊D4, C22×D5, C4×C22⋊C4, D10⋊C4, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, D5×C42, C42⋊D5, C4×D20, C2×D10⋊C4, C4×C5⋊D4, C4×D10⋊C4
►On 160 points
Generators in S160
(1 92 17 82)(2 93 18 83)(3 94 19 84)(4 95 20 85)(5 96 11 86)(6 97 12 87)(7 98 13 88)(8 99 14 89)(9 100 15 90)(10 91 16 81)(21 116 31 106)(22 117 32 107)(23 118 33 108)(24 119 34 109)(25 120 35 110)(26 111 36 101)(27 112 37 102)(28 113 38 103)(29 114 39 104)(30 115 40 105)(41 136 51 126)(42 137 52 127)(43 138 53 128)(44 139 54 129)(45 140 55 130)(46 131 56 121)(47 132 57 122)(48 133 58 123)(49 134 59 124)(50 135 60 125)(61 156 71 146)(62 157 72 147)(63 158 73 148)(64 159 74 149)(65 160 75 150)(66 151 76 141)(67 152 77 142)(68 153 78 143)(69 154 79 144)(70 155 80 145)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 12)(13 20)(14 19)(15 18)(16 17)(21 22)(23 30)(24 29)(25 28)(26 27)(31 32)(33 40)(34 39)(35 38)(36 37)(41 47)(42 46)(43 45)(48 50)(51 57)(52 56)(53 55)(58 60)(61 67)(62 66)(63 65)(68 70)(71 77)(72 76)(73 75)(78 80)(81 82)(83 90)(84 89)(85 88)(86 87)(91 92)(93 100)(94 99)(95 98)(96 97)(101 102)(103 110)(104 109)(105 108)(106 107)(111 112)(113 120)(114 119)(115 118)(116 117)(121 127)(122 126)(123 125)(128 130)(131 137)(132 136)(133 135)(138 140)(141 147)(142 146)(143 145)(148 150)(151 157)(152 156)(153 155)(158 160)
(1 157 37 127)(2 158 38 128)(3 159 39 129)(4 160 40 130)(5 151 31 121)(6 152 32 122)(7 153 33 123)(8 154 34 124)(9 155 35 125)(10 156 36 126)(11 141 21 131)(12 142 22 132)(13 143 23 133)(14 144 24 134)(15 145 25 135)(16 146 26 136)(17 147 27 137)(18 148 28 138)(19 149 29 139)(20 150 30 140)(41 91 71 101)(42 92 72 102)(43 93 73 103)(44 94 74 104)(45 95 75 105)(46 96 76 106)(47 97 77 107)(48 98 78 108)(49 99 79 109)(50 100 80 110)(51 81 61 111)(52 82 62 112)(53 83 63 113)(54 84 64 114)(55 85 65 115)(56 86 66 116)(57 87 67 117)(58 88 68 118)(59 89 69 119)(60 90 70 120)
G:=sub<Sym(160)| (1,92,17,82)(2,93,18,83)(3,94,19,84)(4,95,20,85)(5,96,11,86)(6,97,12,87)(7,98,13,88)(8,99,14,89)(9,100,15,90)(10,91,16,81)(21,116,31,106)(22,117,32,107)(23,118,33,108)(24,119,34,109)(25,120,35,110)(26,111,36,101)(27,112,37,102)(28,113,38,103)(29,114,39,104)(30,115,40,105)(41,136,51,126)(42,137,52,127)(43,138,53,128)(44,139,54,129)(45,140,55,130)(46,131,56,121)(47,132,57,122)(48,133,58,123)(49,134,59,124)(50,135,60,125)(61,156,71,146)(62,157,72,147)(63,158,73,148)(64,159,74,149)(65,160,75,150)(66,151,76,141)(67,152,77,142)(68,153,78,143)(69,154,79,144)(70,155,80,145), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,12)(13,20)(14,19)(15,18)(16,17)(21,22)(23,30)(24,29)(25,28)(26,27)(31,32)(33,40)(34,39)(35,38)(36,37)(41,47)(42,46)(43,45)(48,50)(51,57)(52,56)(53,55)(58,60)(61,67)(62,66)(63,65)(68,70)(71,77)(72,76)(73,75)(78,80)(81,82)(83,90)(84,89)(85,88)(86,87)(91,92)(93,100)(94,99)(95,98)(96,97)(101,102)(103,110)(104,109)(105,108)(106,107)(111,112)(113,120)(114,119)(115,118)(116,117)(121,127)(122,126)(123,125)(128,130)(131,137)(132,136)(133,135)(138,140)(141,147)(142,146)(143,145)(148,150)(151,157)(152,156)(153,155)(158,160), (1,157,37,127)(2,158,38,128)(3,159,39,129)(4,160,40,130)(5,151,31,121)(6,152,32,122)(7,153,33,123)(8,154,34,124)(9,155,35,125)(10,156,36,126)(11,141,21,131)(12,142,22,132)(13,143,23,133)(14,144,24,134)(15,145,25,135)(16,146,26,136)(17,147,27,137)(18,148,28,138)(19,149,29,139)(20,150,30,140)(41,91,71,101)(42,92,72,102)(43,93,73,103)(44,94,74,104)(45,95,75,105)(46,96,76,106)(47,97,77,107)(48,98,78,108)(49,99,79,109)(50,100,80,110)(51,81,61,111)(52,82,62,112)(53,83,63,113)(54,84,64,114)(55,85,65,115)(56,86,66,116)(57,87,67,117)(58,88,68,118)(59,89,69,119)(60,90,70,120)>;
G:=Group( (1,92,17,82)(2,93,18,83)(3,94,19,84)(4,95,20,85)(5,96,11,86)(6,97,12,87)(7,98,13,88)(8,99,14,89)(9,100,15,90)(10,91,16,81)(21,116,31,106)(22,117,32,107)(23,118,33,108)(24,119,34,109)(25,120,35,110)(26,111,36,101)(27,112,37,102)(28,113,38,103)(29,114,39,104)(30,115,40,105)(41,136,51,126)(42,137,52,127)(43,138,53,128)(44,139,54,129)(45,140,55,130)(46,131,56,121)(47,132,57,122)(48,133,58,123)(49,134,59,124)(50,135,60,125)(61,156,71,146)(62,157,72,147)(63,158,73,148)(64,159,74,149)(65,160,75,150)(66,151,76,141)(67,152,77,142)(68,153,78,143)(69,154,79,144)(70,155,80,145), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,12)(13,20)(14,19)(15,18)(16,17)(21,22)(23,30)(24,29)(25,28)(26,27)(31,32)(33,40)(34,39)(35,38)(36,37)(41,47)(42,46)(43,45)(48,50)(51,57)(52,56)(53,55)(58,60)(61,67)(62,66)(63,65)(68,70)(71,77)(72,76)(73,75)(78,80)(81,82)(83,90)(84,89)(85,88)(86,87)(91,92)(93,100)(94,99)(95,98)(96,97)(101,102)(103,110)(104,109)(105,108)(106,107)(111,112)(113,120)(114,119)(115,118)(116,117)(121,127)(122,126)(123,125)(128,130)(131,137)(132,136)(133,135)(138,140)(141,147)(142,146)(143,145)(148,150)(151,157)(152,156)(153,155)(158,160), (1,157,37,127)(2,158,38,128)(3,159,39,129)(4,160,40,130)(5,151,31,121)(6,152,32,122)(7,153,33,123)(8,154,34,124)(9,155,35,125)(10,156,36,126)(11,141,21,131)(12,142,22,132)(13,143,23,133)(14,144,24,134)(15,145,25,135)(16,146,26,136)(17,147,27,137)(18,148,28,138)(19,149,29,139)(20,150,30,140)(41,91,71,101)(42,92,72,102)(43,93,73,103)(44,94,74,104)(45,95,75,105)(46,96,76,106)(47,97,77,107)(48,98,78,108)(49,99,79,109)(50,100,80,110)(51,81,61,111)(52,82,62,112)(53,83,63,113)(54,84,64,114)(55,85,65,115)(56,86,66,116)(57,87,67,117)(58,88,68,118)(59,89,69,119)(60,90,70,120) );
G=PermutationGroup([[(1,92,17,82),(2,93,18,83),(3,94,19,84),(4,95,20,85),(5,96,11,86),(6,97,12,87),(7,98,13,88),(8,99,14,89),(9,100,15,90),(10,91,16,81),(21,116,31,106),(22,117,32,107),(23,118,33,108),(24,119,34,109),(25,120,35,110),(26,111,36,101),(27,112,37,102),(28,113,38,103),(29,114,39,104),(30,115,40,105),(41,136,51,126),(42,137,52,127),(43,138,53,128),(44,139,54,129),(45,140,55,130),(46,131,56,121),(47,132,57,122),(48,133,58,123),(49,134,59,124),(50,135,60,125),(61,156,71,146),(62,157,72,147),(63,158,73,148),(64,159,74,149),(65,160,75,150),(66,151,76,141),(67,152,77,142),(68,153,78,143),(69,154,79,144),(70,155,80,145)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12),(13,20),(14,19),(15,18),(16,17),(21,22),(23,30),(24,29),(25,28),(26,27),(31,32),(33,40),(34,39),(35,38),(36,37),(41,47),(42,46),(43,45),(48,50),(51,57),(52,56),(53,55),(58,60),(61,67),(62,66),(63,65),(68,70),(71,77),(72,76),(73,75),(78,80),(81,82),(83,90),(84,89),(85,88),(86,87),(91,92),(93,100),(94,99),(95,98),(96,97),(101,102),(103,110),(104,109),(105,108),(106,107),(111,112),(113,120),(114,119),(115,118),(116,117),(121,127),(122,126),(123,125),(128,130),(131,137),(132,136),(133,135),(138,140),(141,147),(142,146),(143,145),(148,150),(151,157),(152,156),(153,155),(158,160)], [(1,157,37,127),(2,158,38,128),(3,159,39,129),(4,160,40,130),(5,151,31,121),(6,152,32,122),(7,153,33,123),(8,154,34,124),(9,155,35,125),(10,156,36,126),(11,141,21,131),(12,142,22,132),(13,143,23,133),(14,144,24,134),(15,145,25,135),(16,146,26,136),(17,147,27,137),(18,148,28,138),(19,149,29,139),(20,150,30,140),(41,91,71,101),(42,92,72,102),(43,93,73,103),(44,94,74,104),(45,95,75,105),(46,96,76,106),(47,97,77,107),(48,98,78,108),(49,99,79,109),(50,100,80,110),(51,81,61,111),(52,82,62,112),(53,83,63,113),(54,84,64,114),(55,85,65,115),(56,86,66,116),(57,87,67,117),(58,88,68,118),(59,89,69,119),(60,90,70,120)]])
104 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AB | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D5 | C4○D4 | D10 | C4×D5 | D20 | C5⋊D4 | C4○D20 |
| kernel | C4×D10⋊C4 | C10.10C42 | C2×C4×Dic5 | C2×D10⋊C4 | C2×C4×C20 | D5×C22×C4 | D10⋊C4 | C2×C4×D5 | C2×C20 | C2×C42 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 16 | 8 | 4 | 2 | 4 | 6 | 24 | 8 | 8 | 16 |
Matrix representation of C4×D10⋊C4 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 6 | 0 | 0 |
| 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 0 | 7 | 35 |
| 0 | 0 | 0 | 7 | 0 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 1 |
| 0 | 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 24 | 35 |
| 0 | 0 | 0 | 7 | 17 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,40,35,0,0,0,6,35,0,0,0,0,0,7,7,0,0,0,35,0],[40,0,0,0,0,0,40,0,0,0,0,6,1,0,0,0,0,0,40,0,0,0,0,1,1],[9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,24,7,0,0,0,35,17] >;
C4×D10⋊C4 in GAP, Magma, Sage, TeX
C_4\times D_{10}\rtimes C_4 % in TeX
G:=Group("C4xD10:C4"); // GroupNames label
G:=SmallGroup(320,565);
// by ID
G=gap.SmallGroup(320,565);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,58,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^10=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^5*c>;
// generators/relations