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## G = C4○D4×C21order 336 = 24·3·7

### Direct product of C21 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C21
 Chief series C1 — C2 — C14 — C42 — C2×C42 — D4×C21 — C4○D4×C21
 Lower central C1 — C2 — C4○D4×C21
 Upper central C1 — C84 — C4○D4×C21

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

Smallest permutation representation of 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

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