direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C22⋊C4×C21, C22⋊2C84, C42.52D4, C23.2C42, (C2×C4)⋊1C42, (C2×C84)⋊4C2, (C2×C6)⋊1C28, (C2×C42)⋊1C4, (C2×C12)⋊2C14, (C2×C14)⋊9C12, (C2×C28)⋊10C6, C2.1(C2×C84), C2.1(D4×C21), C6.12(C7×D4), C42.41(C2×C4), C6.10(C2×C28), C14.28(C3×D4), C14.24(C2×C12), (C22×C6).1C14, C22.2(C2×C42), (C22×C42).1C2, (C22×C14).9C6, (C2×C42).52C22, (C2×C6).13(C2×C14), (C2×C14).30(C2×C6), SmallGroup(336,107)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×C21
G = < a,b,c,d | a21=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 92 in 68 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C7, C2×C4, C23, C12, C2×C6, C2×C6, C2×C6, C14, C14, C14, C22⋊C4, C21, C2×C12, C22×C6, C28, C2×C14, C2×C14, C2×C14, C42, C42, C42, C3×C22⋊C4, C2×C28, C22×C14, C84, C2×C42, C2×C42, C2×C42, C7×C22⋊C4, C2×C84, C22×C42, C22⋊C4×C21
Quotients: C1, C2, C3, C4, C22, C6, C7, C2×C4, D4, C12, C2×C6, C14, C22⋊C4, C21, C2×C12, C3×D4, C28, C2×C14, C42, C3×C22⋊C4, C2×C28, C7×D4, C84, C2×C42, C7×C22⋊C4, C2×C84, D4×C21, C22⋊C4×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)
(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 105)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)
(1 158)(2 159)(3 160)(4 161)(5 162)(6 163)(7 164)(8 165)(9 166)(10 167)(11 168)(12 148)(13 149)(14 150)(15 151)(16 152)(17 153)(18 154)(19 155)(20 156)(21 157)(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 105)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(106 147)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)(113 133)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 140)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)
(1 98 139 52)(2 99 140 53)(3 100 141 54)(4 101 142 55)(5 102 143 56)(6 103 144 57)(7 104 145 58)(8 105 146 59)(9 85 147 60)(10 86 127 61)(11 87 128 62)(12 88 129 63)(13 89 130 43)(14 90 131 44)(15 91 132 45)(16 92 133 46)(17 93 134 47)(18 94 135 48)(19 95 136 49)(20 96 137 50)(21 97 138 51)(22 119 82 158)(23 120 83 159)(24 121 84 160)(25 122 64 161)(26 123 65 162)(27 124 66 163)(28 125 67 164)(29 126 68 165)(30 106 69 166)(31 107 70 167)(32 108 71 168)(33 109 72 148)(34 110 73 149)(35 111 74 150)(36 112 75 151)(37 113 76 152)(38 114 77 153)(39 115 78 154)(40 116 79 155)(41 117 80 156)(42 118 81 157)
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), (22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72), (1,158)(2,159)(3,160)(4,161)(5,162)(6,163)(7,164)(8,165)(9,166)(10,167)(11,168)(12,148)(13,149)(14,150)(15,151)(16,152)(17,153)(18,154)(19,155)(20,156)(21,157)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(106,147)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146), (1,98,139,52)(2,99,140,53)(3,100,141,54)(4,101,142,55)(5,102,143,56)(6,103,144,57)(7,104,145,58)(8,105,146,59)(9,85,147,60)(10,86,127,61)(11,87,128,62)(12,88,129,63)(13,89,130,43)(14,90,131,44)(15,91,132,45)(16,92,133,46)(17,93,134,47)(18,94,135,48)(19,95,136,49)(20,96,137,50)(21,97,138,51)(22,119,82,158)(23,120,83,159)(24,121,84,160)(25,122,64,161)(26,123,65,162)(27,124,66,163)(28,125,67,164)(29,126,68,165)(30,106,69,166)(31,107,70,167)(32,108,71,168)(33,109,72,148)(34,110,73,149)(35,111,74,150)(36,112,75,151)(37,113,76,152)(38,114,77,153)(39,115,78,154)(40,116,79,155)(41,117,80,156)(42,118,81,157)>;
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), (22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72), (1,158)(2,159)(3,160)(4,161)(5,162)(6,163)(7,164)(8,165)(9,166)(10,167)(11,168)(12,148)(13,149)(14,150)(15,151)(16,152)(17,153)(18,154)(19,155)(20,156)(21,157)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(106,147)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146), (1,98,139,52)(2,99,140,53)(3,100,141,54)(4,101,142,55)(5,102,143,56)(6,103,144,57)(7,104,145,58)(8,105,146,59)(9,85,147,60)(10,86,127,61)(11,87,128,62)(12,88,129,63)(13,89,130,43)(14,90,131,44)(15,91,132,45)(16,92,133,46)(17,93,134,47)(18,94,135,48)(19,95,136,49)(20,96,137,50)(21,97,138,51)(22,119,82,158)(23,120,83,159)(24,121,84,160)(25,122,64,161)(26,123,65,162)(27,124,66,163)(28,125,67,164)(29,126,68,165)(30,106,69,166)(31,107,70,167)(32,108,71,168)(33,109,72,148)(34,110,73,149)(35,111,74,150)(36,112,75,151)(37,113,76,152)(38,114,77,153)(39,115,78,154)(40,116,79,155)(41,117,80,156)(42,118,81,157) );
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)], [(22,98),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,105),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72)], [(1,158),(2,159),(3,160),(4,161),(5,162),(6,163),(7,164),(8,165),(9,166),(10,167),(11,168),(12,148),(13,149),(14,150),(15,151),(16,152),(17,153),(18,154),(19,155),(20,156),(21,157),(22,98),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,105),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72),(106,147),(107,127),(108,128),(109,129),(110,130),(111,131),(112,132),(113,133),(114,134),(115,135),(116,136),(117,137),(118,138),(119,139),(120,140),(121,141),(122,142),(123,143),(124,144),(125,145),(126,146)], [(1,98,139,52),(2,99,140,53),(3,100,141,54),(4,101,142,55),(5,102,143,56),(6,103,144,57),(7,104,145,58),(8,105,146,59),(9,85,147,60),(10,86,127,61),(11,87,128,62),(12,88,129,63),(13,89,130,43),(14,90,131,44),(15,91,132,45),(16,92,133,46),(17,93,134,47),(18,94,135,48),(19,95,136,49),(20,96,137,50),(21,97,138,51),(22,119,82,158),(23,120,83,159),(24,121,84,160),(25,122,64,161),(26,123,65,162),(27,124,66,163),(28,125,67,164),(29,126,68,165),(30,106,69,166),(31,107,70,167),(32,108,71,168),(33,109,72,148),(34,110,73,149),(35,111,74,150),(36,112,75,151),(37,113,76,152),(38,114,77,153),(39,115,78,154),(40,116,79,155),(41,117,80,156),(42,118,81,157)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 7A | ··· | 7F | 12A | ··· | 12H | 14A | ··· | 14R | 14S | ··· | 14AD | 21A | ··· | 21L | 28A | ··· | 28X | 42A | ··· | 42AJ | 42AK | ··· | 42BH | 84A | ··· | 84AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 7 | ··· | 7 | 12 | ··· | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 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 | C3 | C4 | C6 | C6 | C7 | C12 | C14 | C14 | C21 | C28 | C42 | C42 | C84 | D4 | C3×D4 | C7×D4 | D4×C21 |
| kernel | C22⋊C4×C21 | C2×C84 | C22×C42 | C7×C22⋊C4 | C2×C42 | C2×C28 | C22×C14 | C3×C22⋊C4 | C2×C14 | C2×C12 | C22×C6 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C22 | C42 | C14 | C6 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 6 | 8 | 12 | 6 | 12 | 24 | 24 | 12 | 48 | 2 | 4 | 12 | 24 |
Matrix representation of C22⋊C4×C21 ►in GL3(𝔽337) generated by
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 336 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 336 |
| 1 | 0 | 0 |
| 0 | 336 | 0 |
| 0 | 0 | 336 |
| 189 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 336 | 0 |
G:=sub<GL(3,GF(337))| [1,0,0,0,4,0,0,0,4],[336,0,0,0,1,0,0,0,336],[1,0,0,0,336,0,0,0,336],[189,0,0,0,0,336,0,1,0] >;
C22⋊C4×C21 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times C_{21} % in TeX
G:=Group("C2^2:C4xC21"); // GroupNames label
G:=SmallGroup(336,107);
// by ID
G=gap.SmallGroup(336,107);
# by ID
G:=PCGroup([6,-2,-2,-3,-7,-2,-2,1008,1033]);
// Polycyclic
G:=Group<a,b,c,d|a^21=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations