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## G = C2×C21⋊7D4order 336 = 24·3·7

### Direct product of C2 and C21⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C2×C21⋊7D4
 Chief series C1 — C7 — C21 — C42 — D42 — C22×D21 — C2×C21⋊7D4
 Lower central C21 — C42 — C2×C21⋊7D4
 Upper central C1 — C22 — C23

Generators and relations for C2×C217D4
G = < a,b,c,d | a2=b21=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 672 in 108 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C7, C2×C4, D4, C23, C23, Dic3, D6, C2×C6, C2×C6, C2×C6, D7, C14, C14, C14, C2×D4, C21, C2×Dic3, C3⋊D4, C22×S3, C22×C6, Dic7, D14, C2×C14, C2×C14, C2×C14, D21, C42, C42, C42, C2×C3⋊D4, C2×Dic7, C7⋊D4, C22×D7, C22×C14, Dic21, D42, D42, C2×C42, C2×C42, C2×C42, C2×C7⋊D4, C2×Dic21, C217D4, C22×D21, C22×C42, C2×C217D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, C3⋊D4, C22×S3, D14, D21, C2×C3⋊D4, C7⋊D4, C22×D7, D42, C2×C7⋊D4, C217D4, C22×D21, C2×C217D4

Smallest permutation representation of C2×C217D4
On 168 points
Generators in S168
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 101)(9 102)(10 103)(11 104)(12 105)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 113)(43 144)(44 145)(45 146)(46 147)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 154)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 161)(72 162)(73 163)(74 164)(75 165)(76 166)(77 167)(78 168)(79 148)(80 149)(81 150)(82 151)(83 152)(84 153)
(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 56 35 75)(2 55 36 74)(3 54 37 73)(4 53 38 72)(5 52 39 71)(6 51 40 70)(7 50 41 69)(8 49 42 68)(9 48 22 67)(10 47 23 66)(11 46 24 65)(12 45 25 64)(13 44 26 84)(14 43 27 83)(15 63 28 82)(16 62 29 81)(17 61 30 80)(18 60 31 79)(19 59 32 78)(20 58 33 77)(21 57 34 76)(85 145 118 153)(86 144 119 152)(87 143 120 151)(88 142 121 150)(89 141 122 149)(90 140 123 148)(91 139 124 168)(92 138 125 167)(93 137 126 166)(94 136 106 165)(95 135 107 164)(96 134 108 163)(97 133 109 162)(98 132 110 161)(99 131 111 160)(100 130 112 159)(101 129 113 158)(102 128 114 157)(103 127 115 156)(104 147 116 155)(105 146 117 154)
(1 94)(2 93)(3 92)(4 91)(5 90)(6 89)(7 88)(8 87)(9 86)(10 85)(11 105)(12 104)(13 103)(14 102)(15 101)(16 100)(17 99)(18 98)(19 97)(20 96)(21 95)(22 119)(23 118)(24 117)(25 116)(26 115)(27 114)(28 113)(29 112)(30 111)(31 110)(32 109)(33 108)(34 107)(35 106)(36 126)(37 125)(38 124)(39 123)(40 122)(41 121)(42 120)(43 157)(44 156)(45 155)(46 154)(47 153)(48 152)(49 151)(50 150)(51 149)(52 148)(53 168)(54 167)(55 166)(56 165)(57 164)(58 163)(59 162)(60 161)(61 160)(62 159)(63 158)(64 147)(65 146)(66 145)(67 144)(68 143)(69 142)(70 141)(71 140)(72 139)(73 138)(74 137)(75 136)(76 135)(77 134)(78 133)(79 132)(80 131)(81 130)(82 129)(83 128)(84 127)

G:=sub<Sym(168)| (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,113)(43,144)(44,145)(45,146)(46,147)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,154)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,161)(72,162)(73,163)(74,164)(75,165)(76,166)(77,167)(78,168)(79,148)(80,149)(81,150)(82,151)(83,152)(84,153), (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), (1,56,35,75)(2,55,36,74)(3,54,37,73)(4,53,38,72)(5,52,39,71)(6,51,40,70)(7,50,41,69)(8,49,42,68)(9,48,22,67)(10,47,23,66)(11,46,24,65)(12,45,25,64)(13,44,26,84)(14,43,27,83)(15,63,28,82)(16,62,29,81)(17,61,30,80)(18,60,31,79)(19,59,32,78)(20,58,33,77)(21,57,34,76)(85,145,118,153)(86,144,119,152)(87,143,120,151)(88,142,121,150)(89,141,122,149)(90,140,123,148)(91,139,124,168)(92,138,125,167)(93,137,126,166)(94,136,106,165)(95,135,107,164)(96,134,108,163)(97,133,109,162)(98,132,110,161)(99,131,111,160)(100,130,112,159)(101,129,113,158)(102,128,114,157)(103,127,115,156)(104,147,116,155)(105,146,117,154), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,105)(12,104)(13,103)(14,102)(15,101)(16,100)(17,99)(18,98)(19,97)(20,96)(21,95)(22,119)(23,118)(24,117)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,126)(37,125)(38,124)(39,123)(40,122)(41,121)(42,120)(43,157)(44,156)(45,155)(46,154)(47,153)(48,152)(49,151)(50,150)(51,149)(52,148)(53,168)(54,167)(55,166)(56,165)(57,164)(58,163)(59,162)(60,161)(61,160)(62,159)(63,158)(64,147)(65,146)(66,145)(67,144)(68,143)(69,142)(70,141)(71,140)(72,139)(73,138)(74,137)(75,136)(76,135)(77,134)(78,133)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127)>;

G:=Group( (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,113)(43,144)(44,145)(45,146)(46,147)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,154)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,161)(72,162)(73,163)(74,164)(75,165)(76,166)(77,167)(78,168)(79,148)(80,149)(81,150)(82,151)(83,152)(84,153), (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), (1,56,35,75)(2,55,36,74)(3,54,37,73)(4,53,38,72)(5,52,39,71)(6,51,40,70)(7,50,41,69)(8,49,42,68)(9,48,22,67)(10,47,23,66)(11,46,24,65)(12,45,25,64)(13,44,26,84)(14,43,27,83)(15,63,28,82)(16,62,29,81)(17,61,30,80)(18,60,31,79)(19,59,32,78)(20,58,33,77)(21,57,34,76)(85,145,118,153)(86,144,119,152)(87,143,120,151)(88,142,121,150)(89,141,122,149)(90,140,123,148)(91,139,124,168)(92,138,125,167)(93,137,126,166)(94,136,106,165)(95,135,107,164)(96,134,108,163)(97,133,109,162)(98,132,110,161)(99,131,111,160)(100,130,112,159)(101,129,113,158)(102,128,114,157)(103,127,115,156)(104,147,116,155)(105,146,117,154), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,105)(12,104)(13,103)(14,102)(15,101)(16,100)(17,99)(18,98)(19,97)(20,96)(21,95)(22,119)(23,118)(24,117)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,126)(37,125)(38,124)(39,123)(40,122)(41,121)(42,120)(43,157)(44,156)(45,155)(46,154)(47,153)(48,152)(49,151)(50,150)(51,149)(52,148)(53,168)(54,167)(55,166)(56,165)(57,164)(58,163)(59,162)(60,161)(61,160)(62,159)(63,158)(64,147)(65,146)(66,145)(67,144)(68,143)(69,142)(70,141)(71,140)(72,139)(73,138)(74,137)(75,136)(76,135)(77,134)(78,133)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127) );

G=PermutationGroup([[(1,94),(2,95),(3,96),(4,97),(5,98),(6,99),(7,100),(8,101),(9,102),(10,103),(11,104),(12,105),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,114),(23,115),(24,116),(25,117),(26,118),(27,119),(28,120),(29,121),(30,122),(31,123),(32,124),(33,125),(34,126),(35,106),(36,107),(37,108),(38,109),(39,110),(40,111),(41,112),(42,113),(43,144),(44,145),(45,146),(46,147),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,154),(65,155),(66,156),(67,157),(68,158),(69,159),(70,160),(71,161),(72,162),(73,163),(74,164),(75,165),(76,166),(77,167),(78,168),(79,148),(80,149),(81,150),(82,151),(83,152),(84,153)], [(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)], [(1,56,35,75),(2,55,36,74),(3,54,37,73),(4,53,38,72),(5,52,39,71),(6,51,40,70),(7,50,41,69),(8,49,42,68),(9,48,22,67),(10,47,23,66),(11,46,24,65),(12,45,25,64),(13,44,26,84),(14,43,27,83),(15,63,28,82),(16,62,29,81),(17,61,30,80),(18,60,31,79),(19,59,32,78),(20,58,33,77),(21,57,34,76),(85,145,118,153),(86,144,119,152),(87,143,120,151),(88,142,121,150),(89,141,122,149),(90,140,123,148),(91,139,124,168),(92,138,125,167),(93,137,126,166),(94,136,106,165),(95,135,107,164),(96,134,108,163),(97,133,109,162),(98,132,110,161),(99,131,111,160),(100,130,112,159),(101,129,113,158),(102,128,114,157),(103,127,115,156),(104,147,116,155),(105,146,117,154)], [(1,94),(2,93),(3,92),(4,91),(5,90),(6,89),(7,88),(8,87),(9,86),(10,85),(11,105),(12,104),(13,103),(14,102),(15,101),(16,100),(17,99),(18,98),(19,97),(20,96),(21,95),(22,119),(23,118),(24,117),(25,116),(26,115),(27,114),(28,113),(29,112),(30,111),(31,110),(32,109),(33,108),(34,107),(35,106),(36,126),(37,125),(38,124),(39,123),(40,122),(41,121),(42,120),(43,157),(44,156),(45,155),(46,154),(47,153),(48,152),(49,151),(50,150),(51,149),(52,148),(53,168),(54,167),(55,166),(56,165),(57,164),(58,163),(59,162),(60,161),(61,160),(62,159),(63,158),(64,147),(65,146),(66,145),(67,144),(68,143),(69,142),(70,141),(71,140),(72,139),(73,138),(74,137),(75,136),(76,135),(77,134),(78,133),(79,132),(80,131),(81,130),(82,129),(83,128),(84,127)]])

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A ··· 6G 7A 7B 7C 14A ··· 14U 21A ··· 21F 42A ··· 42AP order 1 2 2 2 2 2 2 2 3 4 4 6 ··· 6 7 7 7 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 1 1 2 2 42 42 2 42 42 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 D7 C3⋊D4 D14 D21 C7⋊D4 D42 C21⋊7D4 kernel C2×C21⋊7D4 C2×Dic21 C21⋊7D4 C22×D21 C22×C42 C22×C14 C42 C2×C14 C22×C6 C14 C2×C6 C23 C6 C22 C2 # reps 1 1 4 1 1 1 2 3 3 4 9 6 12 18 24

Matrix representation of C2×C217D4 in GL5(𝔽337)

 336 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 336 336 0 0 0 1 0 0 0 0 0 0 193 143 0 0 0 159 34
,
 1 0 0 0 0 0 198 59 0 0 0 198 139 0 0 0 0 0 144 336 0 0 0 178 193
,
 1 0 0 0 0 0 336 0 0 0 0 1 1 0 0 0 0 0 193 1 0 0 0 159 144

G:=sub<GL(5,GF(337))| [336,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,336,1,0,0,0,336,0,0,0,0,0,0,193,159,0,0,0,143,34],[1,0,0,0,0,0,198,198,0,0,0,59,139,0,0,0,0,0,144,178,0,0,0,336,193],[1,0,0,0,0,0,336,1,0,0,0,0,1,0,0,0,0,0,193,159,0,0,0,1,144] >;

C2×C217D4 in GAP, Magma, Sage, TeX

C_2\times C_{21}\rtimes_7D_4
% in TeX

G:=Group("C2xC21:7D4");
// GroupNames label

G:=SmallGroup(336,203);
// by ID

G=gap.SmallGroup(336,203);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,218,964,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^21=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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