direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4○D4×C21, D4⋊2C42, Q8⋊3C42, C84.81C22, C42.60C23, (C2×C4)⋊3C42, (C2×C84)⋊15C2, (C2×C28)⋊15C6, (C2×C12)⋊7C14, (C3×D4)⋊5C14, (C7×D4)⋊11C6, C4.5(C2×C42), (C3×Q8)⋊5C14, (C7×Q8)⋊15C6, C22.(C2×C42), (D4×C21)⋊11C2, C28.44(C2×C6), (Q8×C21)⋊11C2, C12.21(C2×C14), C2.3(C22×C42), C6.13(C22×C14), C14.27(C22×C6), (C2×C42).21C22, (C2×C6).2(C2×C14), (C2×C14).19(C2×C6), SmallGroup(336,207)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D4×C21
G = < a,b,c,d | a21=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Subgroups: 92 in 80 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C7, C2×C4, D4, Q8, C12, C12, C2×C6, C14, C14, C4○D4, C21, C2×C12, C3×D4, C3×Q8, C28, C28, C2×C14, C42, C42, C3×C4○D4, C2×C28, C7×D4, C7×Q8, C84, C84, C2×C42, C7×C4○D4, C2×C84, D4×C21, Q8×C21, C4○D4×C21
Quotients: C1, C2, C3, C22, C6, C7, C23, C2×C6, C14, C4○D4, C21, C22×C6, C2×C14, C42, C3×C4○D4, C22×C14, C2×C42, C7×C4○D4, C22×C42, C4○D4×C21
►On 168 points
Generators in S168
(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)(148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 142 114 73)(2 143 115 74)(3 144 116 75)(4 145 117 76)(5 146 118 77)(6 147 119 78)(7 127 120 79)(8 128 121 80)(9 129 122 81)(10 130 123 82)(11 131 124 83)(12 132 125 84)(13 133 126 64)(14 134 106 65)(15 135 107 66)(16 136 108 67)(17 137 109 68)(18 138 110 69)(19 139 111 70)(20 140 112 71)(21 141 113 72)(22 58 96 152)(23 59 97 153)(24 60 98 154)(25 61 99 155)(26 62 100 156)(27 63 101 157)(28 43 102 158)(29 44 103 159)(30 45 104 160)(31 46 105 161)(32 47 85 162)(33 48 86 163)(34 49 87 164)(35 50 88 165)(36 51 89 166)(37 52 90 167)(38 53 91 168)(39 54 92 148)(40 55 93 149)(41 56 94 150)(42 57 95 151)
(1 73 114 142)(2 74 115 143)(3 75 116 144)(4 76 117 145)(5 77 118 146)(6 78 119 147)(7 79 120 127)(8 80 121 128)(9 81 122 129)(10 82 123 130)(11 83 124 131)(12 84 125 132)(13 64 126 133)(14 65 106 134)(15 66 107 135)(16 67 108 136)(17 68 109 137)(18 69 110 138)(19 70 111 139)(20 71 112 140)(21 72 113 141)(22 58 96 152)(23 59 97 153)(24 60 98 154)(25 61 99 155)(26 62 100 156)(27 63 101 157)(28 43 102 158)(29 44 103 159)(30 45 104 160)(31 46 105 161)(32 47 85 162)(33 48 86 163)(34 49 87 164)(35 50 88 165)(36 51 89 166)(37 52 90 167)(38 53 91 168)(39 54 92 148)(40 55 93 149)(41 56 94 150)(42 57 95 151)
(1 164)(2 165)(3 166)(4 167)(5 168)(6 148)(7 149)(8 150)(9 151)(10 152)(11 153)(12 154)(13 155)(14 156)(15 157)(16 158)(17 159)(18 160)(19 161)(20 162)(21 163)(22 130)(23 131)(24 132)(25 133)(26 134)(27 135)(28 136)(29 137)(30 138)(31 139)(32 140)(33 141)(34 142)(35 143)(36 144)(37 145)(38 146)(39 147)(40 127)(41 128)(42 129)(43 108)(44 109)(45 110)(46 111)(47 112)(48 113)(49 114)(50 115)(51 116)(52 117)(53 118)(54 119)(55 120)(56 121)(57 122)(58 123)(59 124)(60 125)(61 126)(62 106)(63 107)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)
G:=sub<Sym(168)| (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)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,142,114,73)(2,143,115,74)(3,144,116,75)(4,145,117,76)(5,146,118,77)(6,147,119,78)(7,127,120,79)(8,128,121,80)(9,129,122,81)(10,130,123,82)(11,131,124,83)(12,132,125,84)(13,133,126,64)(14,134,106,65)(15,135,107,66)(16,136,108,67)(17,137,109,68)(18,138,110,69)(19,139,111,70)(20,140,112,71)(21,141,113,72)(22,58,96,152)(23,59,97,153)(24,60,98,154)(25,61,99,155)(26,62,100,156)(27,63,101,157)(28,43,102,158)(29,44,103,159)(30,45,104,160)(31,46,105,161)(32,47,85,162)(33,48,86,163)(34,49,87,164)(35,50,88,165)(36,51,89,166)(37,52,90,167)(38,53,91,168)(39,54,92,148)(40,55,93,149)(41,56,94,150)(42,57,95,151), (1,73,114,142)(2,74,115,143)(3,75,116,144)(4,76,117,145)(5,77,118,146)(6,78,119,147)(7,79,120,127)(8,80,121,128)(9,81,122,129)(10,82,123,130)(11,83,124,131)(12,84,125,132)(13,64,126,133)(14,65,106,134)(15,66,107,135)(16,67,108,136)(17,68,109,137)(18,69,110,138)(19,70,111,139)(20,71,112,140)(21,72,113,141)(22,58,96,152)(23,59,97,153)(24,60,98,154)(25,61,99,155)(26,62,100,156)(27,63,101,157)(28,43,102,158)(29,44,103,159)(30,45,104,160)(31,46,105,161)(32,47,85,162)(33,48,86,163)(34,49,87,164)(35,50,88,165)(36,51,89,166)(37,52,90,167)(38,53,91,168)(39,54,92,148)(40,55,93,149)(41,56,94,150)(42,57,95,151), (1,164)(2,165)(3,166)(4,167)(5,168)(6,148)(7,149)(8,150)(9,151)(10,152)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,159)(18,160)(19,161)(20,162)(21,163)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,141)(34,142)(35,143)(36,144)(37,145)(38,146)(39,147)(40,127)(41,128)(42,129)(43,108)(44,109)(45,110)(46,111)(47,112)(48,113)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,106)(63,107)(64,99)(65,100)(66,101)(67,102)(68,103)(69,104)(70,105)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)>;
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)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,142,114,73)(2,143,115,74)(3,144,116,75)(4,145,117,76)(5,146,118,77)(6,147,119,78)(7,127,120,79)(8,128,121,80)(9,129,122,81)(10,130,123,82)(11,131,124,83)(12,132,125,84)(13,133,126,64)(14,134,106,65)(15,135,107,66)(16,136,108,67)(17,137,109,68)(18,138,110,69)(19,139,111,70)(20,140,112,71)(21,141,113,72)(22,58,96,152)(23,59,97,153)(24,60,98,154)(25,61,99,155)(26,62,100,156)(27,63,101,157)(28,43,102,158)(29,44,103,159)(30,45,104,160)(31,46,105,161)(32,47,85,162)(33,48,86,163)(34,49,87,164)(35,50,88,165)(36,51,89,166)(37,52,90,167)(38,53,91,168)(39,54,92,148)(40,55,93,149)(41,56,94,150)(42,57,95,151), (1,73,114,142)(2,74,115,143)(3,75,116,144)(4,76,117,145)(5,77,118,146)(6,78,119,147)(7,79,120,127)(8,80,121,128)(9,81,122,129)(10,82,123,130)(11,83,124,131)(12,84,125,132)(13,64,126,133)(14,65,106,134)(15,66,107,135)(16,67,108,136)(17,68,109,137)(18,69,110,138)(19,70,111,139)(20,71,112,140)(21,72,113,141)(22,58,96,152)(23,59,97,153)(24,60,98,154)(25,61,99,155)(26,62,100,156)(27,63,101,157)(28,43,102,158)(29,44,103,159)(30,45,104,160)(31,46,105,161)(32,47,85,162)(33,48,86,163)(34,49,87,164)(35,50,88,165)(36,51,89,166)(37,52,90,167)(38,53,91,168)(39,54,92,148)(40,55,93,149)(41,56,94,150)(42,57,95,151), (1,164)(2,165)(3,166)(4,167)(5,168)(6,148)(7,149)(8,150)(9,151)(10,152)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,159)(18,160)(19,161)(20,162)(21,163)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,141)(34,142)(35,143)(36,144)(37,145)(38,146)(39,147)(40,127)(41,128)(42,129)(43,108)(44,109)(45,110)(46,111)(47,112)(48,113)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,106)(63,107)(64,99)(65,100)(66,101)(67,102)(68,103)(69,104)(70,105)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98) );
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),(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,142,114,73),(2,143,115,74),(3,144,116,75),(4,145,117,76),(5,146,118,77),(6,147,119,78),(7,127,120,79),(8,128,121,80),(9,129,122,81),(10,130,123,82),(11,131,124,83),(12,132,125,84),(13,133,126,64),(14,134,106,65),(15,135,107,66),(16,136,108,67),(17,137,109,68),(18,138,110,69),(19,139,111,70),(20,140,112,71),(21,141,113,72),(22,58,96,152),(23,59,97,153),(24,60,98,154),(25,61,99,155),(26,62,100,156),(27,63,101,157),(28,43,102,158),(29,44,103,159),(30,45,104,160),(31,46,105,161),(32,47,85,162),(33,48,86,163),(34,49,87,164),(35,50,88,165),(36,51,89,166),(37,52,90,167),(38,53,91,168),(39,54,92,148),(40,55,93,149),(41,56,94,150),(42,57,95,151)], [(1,73,114,142),(2,74,115,143),(3,75,116,144),(4,76,117,145),(5,77,118,146),(6,78,119,147),(7,79,120,127),(8,80,121,128),(9,81,122,129),(10,82,123,130),(11,83,124,131),(12,84,125,132),(13,64,126,133),(14,65,106,134),(15,66,107,135),(16,67,108,136),(17,68,109,137),(18,69,110,138),(19,70,111,139),(20,71,112,140),(21,72,113,141),(22,58,96,152),(23,59,97,153),(24,60,98,154),(25,61,99,155),(26,62,100,156),(27,63,101,157),(28,43,102,158),(29,44,103,159),(30,45,104,160),(31,46,105,161),(32,47,85,162),(33,48,86,163),(34,49,87,164),(35,50,88,165),(36,51,89,166),(37,52,90,167),(38,53,91,168),(39,54,92,148),(40,55,93,149),(41,56,94,150),(42,57,95,151)], [(1,164),(2,165),(3,166),(4,167),(5,168),(6,148),(7,149),(8,150),(9,151),(10,152),(11,153),(12,154),(13,155),(14,156),(15,157),(16,158),(17,159),(18,160),(19,161),(20,162),(21,163),(22,130),(23,131),(24,132),(25,133),(26,134),(27,135),(28,136),(29,137),(30,138),(31,139),(32,140),(33,141),(34,142),(35,143),(36,144),(37,145),(38,146),(39,147),(40,127),(41,128),(42,129),(43,108),(44,109),(45,110),(46,111),(47,112),(48,113),(49,114),(50,115),(51,116),(52,117),(53,118),(54,119),(55,120),(56,121),(57,122),(58,123),(59,124),(60,125),(61,126),(62,106),(63,107),(64,99),(65,100),(66,101),(67,102),(68,103),(69,104),(70,105),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6H | 7A | ··· | 7F | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 14A | ··· | 14F | 14G | ··· | 14X | 21A | ··· | 21L | 28A | ··· | 28L | 28M | ··· | 28AD | 42A | ··· | 42L | 42M | ··· | 42AV | 84A | ··· | 84X | 84Y | ··· | 84BH |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 7 | ··· | 7 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 42 | ··· | 42 | 84 | ··· | 84 | 84 | ··· | 84 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C7 | C14 | C14 | C14 | C21 | C42 | C42 | C42 | C4○D4 | C3×C4○D4 | C7×C4○D4 | C4○D4×C21 |
| kernel | C4○D4×C21 | C2×C84 | D4×C21 | Q8×C21 | C7×C4○D4 | C2×C28 | C7×D4 | C7×Q8 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C21 | C7 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 6 | 18 | 18 | 6 | 12 | 36 | 36 | 12 | 2 | 4 | 12 | 24 |
Matrix representation of C4○D4×C21 ►in GL3(𝔽337) generated by
| 128 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 1 | 0 | 0 |
| 0 | 189 | 0 |
| 0 | 0 | 189 |
| 336 | 0 | 0 |
| 0 | 148 | 0 |
| 0 | 148 | 189 |
| 336 | 0 | 0 |
| 0 | 189 | 296 |
| 0 | 189 | 148 |
G:=sub<GL(3,GF(337))| [128,0,0,0,64,0,0,0,64],[1,0,0,0,189,0,0,0,189],[336,0,0,0,148,148,0,0,189],[336,0,0,0,189,189,0,296,148] >;
C4○D4×C21 in GAP, Magma, Sage, TeX
C_4\circ D_4\times C_{21} % in TeX
G:=Group("C4oD4xC21"); // GroupNames label
G:=SmallGroup(336,207);
// by ID
G=gap.SmallGroup(336,207);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-7,-2,2041,770]);
// Polycyclic
G:=Group<a,b,c,d|a^21=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