direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C4○D20, C42.275D10, (C2×C42)⋊8D5, (C4×D20)⋊53C2, D20⋊35(C2×C4), C20⋊12(C4○D4), (D5×C42)⋊15C2, (C4×Dic10)⋊55C2, Dic10⋊33(C2×C4), C42⋊D5⋊38C2, C10.26(C23×C4), (C2×C10).18C24, (C2×C20).876C23, (C4×C20).333C22, C20.177(C22×C4), D10.10(C22×C4), (C22×C4).437D10, C22.15(C23×D5), (C2×D20).293C22, C4⋊Dic5.395C22, Dic5.10(C22×C4), C23.217(C22×D5), C23.21D10⋊40C2, (C22×C20).564C22, (C22×C10).380C23, (C2×Dic5).184C23, (C4×Dic5).330C22, (C22×D5).157C23, C23.D5.139C22, D10⋊C4.162C22, (C2×Dic10).322C22, C10.D4.174C22, C5⋊2(C4×C4○D4), (C2×C4×C20)⋊12C2, (C2×C4)⋊13(C4×D5), C4.117(C2×C4×D5), (C2×C20)⋊43(C2×C4), (C4×D5)⋊13(C2×C4), (C4×C5⋊D4)⋊62C2, C5⋊D4⋊12(C2×C4), C22.9(C2×C4×D5), C2.7(D5×C22×C4), C10.6(C2×C4○D4), C2.4(C2×C4○D20), (C2×C4○D20).28C2, (C2×C4×D5).370C22, (C2×C4).818(C22×D5), (C2×C10).249(C22×C4), (C2×C5⋊D4).157C22, SmallGroup(320,1146)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C4○D20
G = < a,b,c,d | a4=b4=d2=1, c10=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c9 >
Subgroups: 894 in 310 conjugacy classes, 159 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C42, C2×C42, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×C10, C4×C4○D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C4×C20, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C22×C20, C4×Dic10, D5×C42, C42⋊D5, C4×D20, C23.21D10, C4×C5⋊D4, C2×C4×C20, C2×C4○D20, C4×C4○D20
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C4○D4, C24, D10, C23×C4, C2×C4○D4, C4×D5, C22×D5, C4×C4○D4, C2×C4×D5, C4○D20, C23×D5, D5×C22×C4, C2×C4○D20, C4×C4○D20
►On 160 points
Generators in S160
(1 52 71 136)(2 53 72 137)(3 54 73 138)(4 55 74 139)(5 56 75 140)(6 57 76 121)(7 58 77 122)(8 59 78 123)(9 60 79 124)(10 41 80 125)(11 42 61 126)(12 43 62 127)(13 44 63 128)(14 45 64 129)(15 46 65 130)(16 47 66 131)(17 48 67 132)(18 49 68 133)(19 50 69 134)(20 51 70 135)(21 85 110 151)(22 86 111 152)(23 87 112 153)(24 88 113 154)(25 89 114 155)(26 90 115 156)(27 91 116 157)(28 92 117 158)(29 93 118 159)(30 94 119 160)(31 95 120 141)(32 96 101 142)(33 97 102 143)(34 98 103 144)(35 99 104 145)(36 100 105 146)(37 81 106 147)(38 82 107 148)(39 83 108 149)(40 84 109 150)
(1 149 11 159)(2 150 12 160)(3 151 13 141)(4 152 14 142)(5 153 15 143)(6 154 16 144)(7 155 17 145)(8 156 18 146)(9 157 19 147)(10 158 20 148)(21 44 31 54)(22 45 32 55)(23 46 33 56)(24 47 34 57)(25 48 35 58)(26 49 36 59)(27 50 37 60)(28 51 38 41)(29 52 39 42)(30 53 40 43)(61 93 71 83)(62 94 72 84)(63 95 73 85)(64 96 74 86)(65 97 75 87)(66 98 76 88)(67 99 77 89)(68 100 78 90)(69 81 79 91)(70 82 80 92)(101 139 111 129)(102 140 112 130)(103 121 113 131)(104 122 114 132)(105 123 115 133)(106 124 116 134)(107 125 117 135)(108 126 118 136)(109 127 119 137)(110 128 120 138)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 31)(22 30)(23 29)(24 28)(25 27)(32 40)(33 39)(34 38)(35 37)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 60)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 80)(77 79)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(101 109)(102 108)(103 107)(104 106)(110 120)(111 119)(112 118)(113 117)(114 116)(121 125)(122 124)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(141 151)(142 150)(143 149)(144 148)(145 147)(152 160)(153 159)(154 158)(155 157)
G:=sub<Sym(160)| (1,52,71,136)(2,53,72,137)(3,54,73,138)(4,55,74,139)(5,56,75,140)(6,57,76,121)(7,58,77,122)(8,59,78,123)(9,60,79,124)(10,41,80,125)(11,42,61,126)(12,43,62,127)(13,44,63,128)(14,45,64,129)(15,46,65,130)(16,47,66,131)(17,48,67,132)(18,49,68,133)(19,50,69,134)(20,51,70,135)(21,85,110,151)(22,86,111,152)(23,87,112,153)(24,88,113,154)(25,89,114,155)(26,90,115,156)(27,91,116,157)(28,92,117,158)(29,93,118,159)(30,94,119,160)(31,95,120,141)(32,96,101,142)(33,97,102,143)(34,98,103,144)(35,99,104,145)(36,100,105,146)(37,81,106,147)(38,82,107,148)(39,83,108,149)(40,84,109,150), (1,149,11,159)(2,150,12,160)(3,151,13,141)(4,152,14,142)(5,153,15,143)(6,154,16,144)(7,155,17,145)(8,156,18,146)(9,157,19,147)(10,158,20,148)(21,44,31,54)(22,45,32,55)(23,46,33,56)(24,47,34,57)(25,48,35,58)(26,49,36,59)(27,50,37,60)(28,51,38,41)(29,52,39,42)(30,53,40,43)(61,93,71,83)(62,94,72,84)(63,95,73,85)(64,96,74,86)(65,97,75,87)(66,98,76,88)(67,99,77,89)(68,100,78,90)(69,81,79,91)(70,82,80,92)(101,139,111,129)(102,140,112,130)(103,121,113,131)(104,122,114,132)(105,123,115,133)(106,124,116,134)(107,125,117,135)(108,126,118,136)(109,127,119,137)(110,128,120,138), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,31)(22,30)(23,29)(24,28)(25,27)(32,40)(33,39)(34,38)(35,37)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,80)(77,79)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(101,109)(102,108)(103,107)(104,106)(110,120)(111,119)(112,118)(113,117)(114,116)(121,125)(122,124)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(141,151)(142,150)(143,149)(144,148)(145,147)(152,160)(153,159)(154,158)(155,157)>;
G:=Group( (1,52,71,136)(2,53,72,137)(3,54,73,138)(4,55,74,139)(5,56,75,140)(6,57,76,121)(7,58,77,122)(8,59,78,123)(9,60,79,124)(10,41,80,125)(11,42,61,126)(12,43,62,127)(13,44,63,128)(14,45,64,129)(15,46,65,130)(16,47,66,131)(17,48,67,132)(18,49,68,133)(19,50,69,134)(20,51,70,135)(21,85,110,151)(22,86,111,152)(23,87,112,153)(24,88,113,154)(25,89,114,155)(26,90,115,156)(27,91,116,157)(28,92,117,158)(29,93,118,159)(30,94,119,160)(31,95,120,141)(32,96,101,142)(33,97,102,143)(34,98,103,144)(35,99,104,145)(36,100,105,146)(37,81,106,147)(38,82,107,148)(39,83,108,149)(40,84,109,150), (1,149,11,159)(2,150,12,160)(3,151,13,141)(4,152,14,142)(5,153,15,143)(6,154,16,144)(7,155,17,145)(8,156,18,146)(9,157,19,147)(10,158,20,148)(21,44,31,54)(22,45,32,55)(23,46,33,56)(24,47,34,57)(25,48,35,58)(26,49,36,59)(27,50,37,60)(28,51,38,41)(29,52,39,42)(30,53,40,43)(61,93,71,83)(62,94,72,84)(63,95,73,85)(64,96,74,86)(65,97,75,87)(66,98,76,88)(67,99,77,89)(68,100,78,90)(69,81,79,91)(70,82,80,92)(101,139,111,129)(102,140,112,130)(103,121,113,131)(104,122,114,132)(105,123,115,133)(106,124,116,134)(107,125,117,135)(108,126,118,136)(109,127,119,137)(110,128,120,138), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,31)(22,30)(23,29)(24,28)(25,27)(32,40)(33,39)(34,38)(35,37)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,80)(77,79)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(101,109)(102,108)(103,107)(104,106)(110,120)(111,119)(112,118)(113,117)(114,116)(121,125)(122,124)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(141,151)(142,150)(143,149)(144,148)(145,147)(152,160)(153,159)(154,158)(155,157) );
G=PermutationGroup([[(1,52,71,136),(2,53,72,137),(3,54,73,138),(4,55,74,139),(5,56,75,140),(6,57,76,121),(7,58,77,122),(8,59,78,123),(9,60,79,124),(10,41,80,125),(11,42,61,126),(12,43,62,127),(13,44,63,128),(14,45,64,129),(15,46,65,130),(16,47,66,131),(17,48,67,132),(18,49,68,133),(19,50,69,134),(20,51,70,135),(21,85,110,151),(22,86,111,152),(23,87,112,153),(24,88,113,154),(25,89,114,155),(26,90,115,156),(27,91,116,157),(28,92,117,158),(29,93,118,159),(30,94,119,160),(31,95,120,141),(32,96,101,142),(33,97,102,143),(34,98,103,144),(35,99,104,145),(36,100,105,146),(37,81,106,147),(38,82,107,148),(39,83,108,149),(40,84,109,150)], [(1,149,11,159),(2,150,12,160),(3,151,13,141),(4,152,14,142),(5,153,15,143),(6,154,16,144),(7,155,17,145),(8,156,18,146),(9,157,19,147),(10,158,20,148),(21,44,31,54),(22,45,32,55),(23,46,33,56),(24,47,34,57),(25,48,35,58),(26,49,36,59),(27,50,37,60),(28,51,38,41),(29,52,39,42),(30,53,40,43),(61,93,71,83),(62,94,72,84),(63,95,73,85),(64,96,74,86),(65,97,75,87),(66,98,76,88),(67,99,77,89),(68,100,78,90),(69,81,79,91),(70,82,80,92),(101,139,111,129),(102,140,112,130),(103,121,113,131),(104,122,114,132),(105,123,115,133),(106,124,116,134),(107,125,117,135),(108,126,118,136),(109,127,119,137),(110,128,120,138)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,31),(22,30),(23,29),(24,28),(25,27),(32,40),(33,39),(34,38),(35,37),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(58,60),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69),(76,80),(77,79),(81,99),(82,98),(83,97),(84,96),(85,95),(86,94),(87,93),(88,92),(89,91),(101,109),(102,108),(103,107),(104,106),(110,120),(111,119),(112,118),(113,117),(114,116),(121,125),(122,124),(126,140),(127,139),(128,138),(129,137),(130,136),(131,135),(132,134),(141,151),(142,150),(143,149),(144,148),(145,147),(152,160),(153,159),(154,158),(155,157)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4L | 4M | ··· | 4R | 4S | ··· | 4AD | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | C4○D4 | D10 | D10 | C4×D5 | C4○D20 |
| kernel | C4×C4○D20 | C4×Dic10 | D5×C42 | C42⋊D5 | C4×D20 | C23.21D10 | C4×C5⋊D4 | C2×C4×C20 | C2×C4○D20 | C4○D20 | C2×C42 | C20 | C42 | C22×C4 | C2×C4 | C4 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 4 | 1 | 1 | 16 | 2 | 8 | 8 | 6 | 16 | 32 |
Matrix representation of C4×C4○D20 ►in GL3(𝔽41) generated by
| 9 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 1 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 40 | 0 | 0 |
| 0 | 30 | 39 |
| 0 | 16 | 14 |
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 8 | 40 |
G:=sub<GL(3,GF(41))| [9,0,0,0,40,0,0,0,40],[1,0,0,0,32,0,0,0,32],[40,0,0,0,30,16,0,39,14],[40,0,0,0,1,8,0,0,40] >;
C4×C4○D20 in GAP, Magma, Sage, TeX
C_4\times C_4\circ D_{20} % in TeX
G:=Group("C4xC4oD20"); // GroupNames label
G:=SmallGroup(320,1146);
// by ID
G=gap.SmallGroup(320,1146);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^10=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^9>;
// generators/relations