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## G = C2×C3⋊D28order 336 = 24·3·7

### Direct product of C2 and C3⋊D28

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C2×C3⋊D28
 Chief series C1 — C7 — C21 — C42 — C6×D7 — C3⋊D28 — C2×C3⋊D28
 Lower central C21 — C42 — C2×C3⋊D28
 Upper central C1 — C22

Generators and relations for C2×C3⋊D28
G = < a,b,c,d | a2=b3=c28=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 732 in 108 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C7, C2×C4, D4, C23, Dic3, D6, C2×C6, C2×C6, D7, C14, C14, C2×D4, C21, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C28, D14, D14, C2×C14, C3×D7, D21, C42, C42, C2×C3⋊D4, D28, C2×C28, C22×D7, C22×D7, C7×Dic3, C6×D7, C6×D7, D42, D42, C2×C42, C2×D28, C3⋊D28, Dic3×C14, C2×C6×D7, C22×D21, C2×C3⋊D28
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, C3⋊D4, C22×S3, D14, C2×C3⋊D4, D28, C22×D7, S3×D7, C2×D28, C3⋊D28, C2×S3×D7, C2×C3⋊D28

Smallest permutation representation of C2×C3⋊D28
On 168 points
Generators in S168
(1 144)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 151)(9 152)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 160)(18 161)(19 162)(20 163)(21 164)(22 165)(23 166)(24 167)(25 168)(26 141)(27 142)(28 143)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(85 120)(86 121)(87 122)(88 123)(89 124)(90 125)(91 126)(92 127)(93 128)(94 129)(95 130)(96 131)(97 132)(98 133)(99 134)(100 135)(101 136)(102 137)(103 138)(104 139)(105 140)(106 113)(107 114)(108 115)(109 116)(110 117)(111 118)(112 119)
(1 111 81)(2 82 112)(3 85 83)(4 84 86)(5 87 57)(6 58 88)(7 89 59)(8 60 90)(9 91 61)(10 62 92)(11 93 63)(12 64 94)(13 95 65)(14 66 96)(15 97 67)(16 68 98)(17 99 69)(18 70 100)(19 101 71)(20 72 102)(21 103 73)(22 74 104)(23 105 75)(24 76 106)(25 107 77)(26 78 108)(27 109 79)(28 80 110)(29 168 114)(30 115 141)(31 142 116)(32 117 143)(33 144 118)(34 119 145)(35 146 120)(36 121 147)(37 148 122)(38 123 149)(39 150 124)(40 125 151)(41 152 126)(42 127 153)(43 154 128)(44 129 155)(45 156 130)(46 131 157)(47 158 132)(48 133 159)(49 160 134)(50 135 161)(51 162 136)(52 137 163)(53 164 138)(54 139 165)(55 166 140)(56 113 167)
(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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 128)(30 127)(31 126)(32 125)(33 124)(34 123)(35 122)(36 121)(37 120)(38 119)(39 118)(40 117)(41 116)(42 115)(43 114)(44 113)(45 140)(46 139)(47 138)(48 137)(49 136)(50 135)(51 134)(52 133)(53 132)(54 131)(55 130)(56 129)(57 85)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)(141 153)(142 152)(143 151)(144 150)(145 149)(146 148)(154 168)(155 167)(156 166)(157 165)(158 164)(159 163)(160 162)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A 6B 6C 6D 6E 6F 6G 7A 7B 7C 14A ··· 14I 21A 21B 21C 28A ··· 28L 42A ··· 42I order 1 2 2 2 2 2 2 2 3 4 4 6 6 6 6 6 6 6 7 7 7 14 ··· 14 21 21 21 28 ··· 28 42 ··· 42 size 1 1 1 1 14 14 42 42 2 6 6 2 2 2 14 14 14 14 2 2 2 2 ··· 2 4 4 4 6 ··· 6 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 D6 D7 C3⋊D4 D14 D14 D28 S3×D7 C3⋊D28 C2×S3×D7 kernel C2×C3⋊D28 C3⋊D28 Dic3×C14 C2×C6×D7 C22×D21 C22×D7 C42 D14 C2×C14 C2×Dic3 C14 Dic3 C2×C6 C6 C22 C2 C2 # reps 1 4 1 1 1 1 2 2 1 3 4 6 3 12 3 6 3

Matrix representation of C2×C3⋊D28 in GL4(𝔽337) generated by

 1 0 0 0 0 1 0 0 0 0 336 0 0 0 0 336
,
 1 0 0 0 0 1 0 0 0 0 336 1 0 0 336 0
,
 275 219 0 0 228 48 0 0 0 0 0 336 0 0 336 0
,
 336 0 0 0 178 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,336,0,0,0,0,336],[1,0,0,0,0,1,0,0,0,0,336,336,0,0,1,0],[275,228,0,0,219,48,0,0,0,0,0,336,0,0,336,0],[336,178,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C2×C3⋊D28 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes D_{28}
% in TeX

G:=Group("C2xC3:D28");
// GroupNames label

G:=SmallGroup(336,158);
// by ID

G=gap.SmallGroup(336,158);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,121,55,490,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^28=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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