direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4○D4×C20, D4⋊6(C2×C20), Q8⋊6(C2×C20), (D4×C20)⋊52C2, (C4×D4)⋊23C10, (C2×C42)⋊8C10, (C4×Q8)⋊18C10, (Q8×C20)⋊38C2, C2.6(C23×C20), C4.18(C22×C20), C42.87(C2×C10), C10.79(C23×C4), C42⋊C2⋊19C10, (C4×C20).371C22, (C2×C10).337C24, (C2×C20).959C23, C20.222(C22×C4), C22.1(C22×C20), (D4×C10).331C22, C22.10(C23×C10), C23.29(C22×C10), (Q8×C10).283C22, (C22×C10).253C23, (C22×C20).595C22, (C2×C4×C20)⋊21C2, (C2×C4)⋊8(C2×C20), (C2×C20)⋊46(C2×C4), (C5×D4)⋊36(C2×C4), (C5×Q8)⋊33(C2×C4), C2.4(C10×C4○D4), C4⋊C4.81(C2×C10), (C2×C4○D4).13C10, (C10×C4○D4).27C2, (C2×D4).77(C2×C10), C10.223(C2×C4○D4), (C2×Q8).71(C2×C10), (C5×C42⋊C2)⋊40C2, C22⋊C4.28(C2×C10), (C5×C4⋊C4).406C22, (C22×C4).99(C2×C10), (C2×C4).134(C22×C10), (C2×C10).133(C22×C4), (C5×C22⋊C4).159C22, SmallGroup(320,1519)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D4×C20
G = < a,b,c,d | a20=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Subgroups: 370 in 310 conjugacy classes, 250 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C42, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C4×C4○D4, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C2×C4×C20, C5×C42⋊C2, D4×C20, Q8×C20, C10×C4○D4, C4○D4×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C4○D4, C24, C20, C2×C10, C23×C4, C2×C4○D4, C2×C20, C22×C10, C4×C4○D4, C22×C20, C5×C4○D4, C23×C10, C23×C20, C10×C4○D4, C4○D4×C20
►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 28 158 71)(2 29 159 72)(3 30 160 73)(4 31 141 74)(5 32 142 75)(6 33 143 76)(7 34 144 77)(8 35 145 78)(9 36 146 79)(10 37 147 80)(11 38 148 61)(12 39 149 62)(13 40 150 63)(14 21 151 64)(15 22 152 65)(16 23 153 66)(17 24 154 67)(18 25 155 68)(19 26 156 69)(20 27 157 70)(41 100 120 133)(42 81 101 134)(43 82 102 135)(44 83 103 136)(45 84 104 137)(46 85 105 138)(47 86 106 139)(48 87 107 140)(49 88 108 121)(50 89 109 122)(51 90 110 123)(52 91 111 124)(53 92 112 125)(54 93 113 126)(55 94 114 127)(56 95 115 128)(57 96 116 129)(58 97 117 130)(59 98 118 131)(60 99 119 132)
(1 132 158 99)(2 133 159 100)(3 134 160 81)(4 135 141 82)(5 136 142 83)(6 137 143 84)(7 138 144 85)(8 139 145 86)(9 140 146 87)(10 121 147 88)(11 122 148 89)(12 123 149 90)(13 124 150 91)(14 125 151 92)(15 126 152 93)(16 127 153 94)(17 128 154 95)(18 129 155 96)(19 130 156 97)(20 131 157 98)(21 53 64 112)(22 54 65 113)(23 55 66 114)(24 56 67 115)(25 57 68 116)(26 58 69 117)(27 59 70 118)(28 60 71 119)(29 41 72 120)(30 42 73 101)(31 43 74 102)(32 44 75 103)(33 45 76 104)(34 46 77 105)(35 47 78 106)(36 48 79 107)(37 49 80 108)(38 50 61 109)(39 51 62 110)(40 52 63 111)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 110)(42 111)(43 112)(44 113)(45 114)(46 115)(47 116)(48 117)(49 118)(50 119)(51 120)(52 101)(53 102)(54 103)(55 104)(56 105)(57 106)(58 107)(59 108)(60 109)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 124)(82 125)(83 126)(84 127)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)(93 136)(94 137)(95 138)(96 139)(97 140)(98 121)(99 122)(100 123)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
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), (1,28,158,71)(2,29,159,72)(3,30,160,73)(4,31,141,74)(5,32,142,75)(6,33,143,76)(7,34,144,77)(8,35,145,78)(9,36,146,79)(10,37,147,80)(11,38,148,61)(12,39,149,62)(13,40,150,63)(14,21,151,64)(15,22,152,65)(16,23,153,66)(17,24,154,67)(18,25,155,68)(19,26,156,69)(20,27,157,70)(41,100,120,133)(42,81,101,134)(43,82,102,135)(44,83,103,136)(45,84,104,137)(46,85,105,138)(47,86,106,139)(48,87,107,140)(49,88,108,121)(50,89,109,122)(51,90,110,123)(52,91,111,124)(53,92,112,125)(54,93,113,126)(55,94,114,127)(56,95,115,128)(57,96,116,129)(58,97,117,130)(59,98,118,131)(60,99,119,132), (1,132,158,99)(2,133,159,100)(3,134,160,81)(4,135,141,82)(5,136,142,83)(6,137,143,84)(7,138,144,85)(8,139,145,86)(9,140,146,87)(10,121,147,88)(11,122,148,89)(12,123,149,90)(13,124,150,91)(14,125,151,92)(15,126,152,93)(16,127,153,94)(17,128,154,95)(18,129,155,96)(19,130,156,97)(20,131,157,98)(21,53,64,112)(22,54,65,113)(23,55,66,114)(24,56,67,115)(25,57,68,116)(26,58,69,117)(27,59,70,118)(28,60,71,119)(29,41,72,120)(30,42,73,101)(31,43,74,102)(32,44,75,103)(33,45,76,104)(34,46,77,105)(35,47,78,106)(36,48,79,107)(37,49,80,108)(38,50,61,109)(39,51,62,110)(40,52,63,111), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,109)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139)(97,140)(98,121)(99,122)(100,123)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)>;
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), (1,28,158,71)(2,29,159,72)(3,30,160,73)(4,31,141,74)(5,32,142,75)(6,33,143,76)(7,34,144,77)(8,35,145,78)(9,36,146,79)(10,37,147,80)(11,38,148,61)(12,39,149,62)(13,40,150,63)(14,21,151,64)(15,22,152,65)(16,23,153,66)(17,24,154,67)(18,25,155,68)(19,26,156,69)(20,27,157,70)(41,100,120,133)(42,81,101,134)(43,82,102,135)(44,83,103,136)(45,84,104,137)(46,85,105,138)(47,86,106,139)(48,87,107,140)(49,88,108,121)(50,89,109,122)(51,90,110,123)(52,91,111,124)(53,92,112,125)(54,93,113,126)(55,94,114,127)(56,95,115,128)(57,96,116,129)(58,97,117,130)(59,98,118,131)(60,99,119,132), (1,132,158,99)(2,133,159,100)(3,134,160,81)(4,135,141,82)(5,136,142,83)(6,137,143,84)(7,138,144,85)(8,139,145,86)(9,140,146,87)(10,121,147,88)(11,122,148,89)(12,123,149,90)(13,124,150,91)(14,125,151,92)(15,126,152,93)(16,127,153,94)(17,128,154,95)(18,129,155,96)(19,130,156,97)(20,131,157,98)(21,53,64,112)(22,54,65,113)(23,55,66,114)(24,56,67,115)(25,57,68,116)(26,58,69,117)(27,59,70,118)(28,60,71,119)(29,41,72,120)(30,42,73,101)(31,43,74,102)(32,44,75,103)(33,45,76,104)(34,46,77,105)(35,47,78,106)(36,48,79,107)(37,49,80,108)(38,50,61,109)(39,51,62,110)(40,52,63,111), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,109)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139)(97,140)(98,121)(99,122)(100,123)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160) );
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)], [(1,28,158,71),(2,29,159,72),(3,30,160,73),(4,31,141,74),(5,32,142,75),(6,33,143,76),(7,34,144,77),(8,35,145,78),(9,36,146,79),(10,37,147,80),(11,38,148,61),(12,39,149,62),(13,40,150,63),(14,21,151,64),(15,22,152,65),(16,23,153,66),(17,24,154,67),(18,25,155,68),(19,26,156,69),(20,27,157,70),(41,100,120,133),(42,81,101,134),(43,82,102,135),(44,83,103,136),(45,84,104,137),(46,85,105,138),(47,86,106,139),(48,87,107,140),(49,88,108,121),(50,89,109,122),(51,90,110,123),(52,91,111,124),(53,92,112,125),(54,93,113,126),(55,94,114,127),(56,95,115,128),(57,96,116,129),(58,97,117,130),(59,98,118,131),(60,99,119,132)], [(1,132,158,99),(2,133,159,100),(3,134,160,81),(4,135,141,82),(5,136,142,83),(6,137,143,84),(7,138,144,85),(8,139,145,86),(9,140,146,87),(10,121,147,88),(11,122,148,89),(12,123,149,90),(13,124,150,91),(14,125,151,92),(15,126,152,93),(16,127,153,94),(17,128,154,95),(18,129,155,96),(19,130,156,97),(20,131,157,98),(21,53,64,112),(22,54,65,113),(23,55,66,114),(24,56,67,115),(25,57,68,116),(26,58,69,117),(27,59,70,118),(28,60,71,119),(29,41,72,120),(30,42,73,101),(31,43,74,102),(32,44,75,103),(33,45,76,104),(34,46,77,105),(35,47,78,106),(36,48,79,107),(37,49,80,108),(38,50,61,109),(39,51,62,110),(40,52,63,111)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,110),(42,111),(43,112),(44,113),(45,114),(46,115),(47,116),(48,117),(49,118),(50,119),(51,120),(52,101),(53,102),(54,103),(55,104),(56,105),(57,106),(58,107),(59,108),(60,109),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,124),(82,125),(83,126),(84,127),(85,128),(86,129),(87,130),(88,131),(89,132),(90,133),(91,134),(92,135),(93,136),(94,137),(95,138),(96,139),(97,140),(98,121),(99,122),(100,123),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160)]])
200 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4L | 4M | ··· | 4AD | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AJ | 20A | ··· | 20AV | 20AW | ··· | 20DP |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | C4○D4 | C5×C4○D4 |
| kernel | C4○D4×C20 | C2×C4×C20 | C5×C42⋊C2 | D4×C20 | Q8×C20 | C10×C4○D4 | C5×C4○D4 | C4×C4○D4 | C2×C42 | C42⋊C2 | C4×D4 | C4×Q8 | C2×C4○D4 | C4○D4 | C20 | C4 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 16 | 4 | 12 | 12 | 24 | 8 | 4 | 64 | 8 | 32 |
Matrix representation of C4○D4×C20 ►in GL3(𝔽41) generated by
| 32 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 1 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 40 | 0 | 0 |
| 0 | 40 | 39 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 40 | 40 |
G:=sub<GL(3,GF(41))| [32,0,0,0,10,0,0,0,10],[1,0,0,0,32,0,0,0,32],[40,0,0,0,40,1,0,39,1],[1,0,0,0,1,40,0,0,40] >;
C4○D4×C20 in GAP, Magma, Sage, TeX
C_4\circ D_4\times C_{20} % in TeX
G:=Group("C4oD4xC20"); // GroupNames label
G:=SmallGroup(320,1519);
// by ID
G=gap.SmallGroup(320,1519);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,856,304]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations