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## G = C6×D28order 336 = 24·3·7

### Direct product of C6 and D28

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C6×D28
 Chief series C1 — C7 — C14 — C42 — C6×D7 — C2×C6×D7 — C6×D28
 Lower central C7 — C14 — C6×D28
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C6×D28
G = < a,b,c | a6=b28=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 464 in 108 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C7, C2×C4, D4, C23, C12, C2×C6, C2×C6, D7, C14, C14, C2×D4, C21, C2×C12, C3×D4, C22×C6, C28, D14, D14, C2×C14, C3×D7, C42, C42, C6×D4, D28, C2×C28, C22×D7, C84, C6×D7, C6×D7, C2×C42, C2×D28, C3×D28, C2×C84, C2×C6×D7, C6×D28
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D7, C2×D4, C3×D4, C22×C6, D14, C3×D7, C6×D4, D28, C22×D7, C6×D7, C2×D28, C3×D28, C2×C6×D7, C6×D28

Smallest permutation representation of C6×D28
On 168 points
Generators in S168
(1 128 162 43 58 90)(2 129 163 44 59 91)(3 130 164 45 60 92)(4 131 165 46 61 93)(5 132 166 47 62 94)(6 133 167 48 63 95)(7 134 168 49 64 96)(8 135 141 50 65 97)(9 136 142 51 66 98)(10 137 143 52 67 99)(11 138 144 53 68 100)(12 139 145 54 69 101)(13 140 146 55 70 102)(14 113 147 56 71 103)(15 114 148 29 72 104)(16 115 149 30 73 105)(17 116 150 31 74 106)(18 117 151 32 75 107)(19 118 152 33 76 108)(20 119 153 34 77 109)(21 120 154 35 78 110)(22 121 155 36 79 111)(23 122 156 37 80 112)(24 123 157 38 81 85)(25 124 158 39 82 86)(26 125 159 40 83 87)(27 126 160 41 84 88)(28 127 161 42 57 89)
(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)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(57 114)(58 113)(59 140)(60 139)(61 138)(62 137)(63 136)(64 135)(65 134)(66 133)(67 132)(68 131)(69 130)(70 129)(71 128)(72 127)(73 126)(74 125)(75 124)(76 123)(77 122)(78 121)(79 120)(80 119)(81 118)(82 117)(83 116)(84 115)(85 152)(86 151)(87 150)(88 149)(89 148)(90 147)(91 146)(92 145)(93 144)(94 143)(95 142)(96 141)(97 168)(98 167)(99 166)(100 165)(101 164)(102 163)(103 162)(104 161)(105 160)(106 159)(107 158)(108 157)(109 156)(110 155)(111 154)(112 153)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A ··· 6F 6G ··· 6N 7A 7B 7C 12A 12B 12C 12D 14A ··· 14I 21A ··· 21F 28A ··· 28L 42A ··· 42R 84A ··· 84X order 1 2 2 2 2 2 2 2 3 3 4 4 6 ··· 6 6 ··· 6 7 7 7 12 12 12 12 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 1 1 14 14 14 14 1 1 2 2 1 ··· 1 14 ··· 14 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D7 C3×D4 D14 D14 C3×D7 D28 C6×D7 C6×D7 C3×D28 kernel C6×D28 C3×D28 C2×C84 C2×C6×D7 C2×D28 D28 C2×C28 C22×D7 C42 C2×C12 C14 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 4 1 2 2 8 2 4 2 3 4 6 3 6 12 12 6 24

Matrix representation of C6×D28 in GL3(𝔽337) generated by

 336 0 0 0 129 0 0 0 129
,
 1 0 0 0 70 101 0 118 281
,
 1 0 0 0 136 169 0 80 201
G:=sub<GL(3,GF(337))| [336,0,0,0,129,0,0,0,129],[1,0,0,0,70,118,0,101,281],[1,0,0,0,136,80,0,169,201] >;

C6×D28 in GAP, Magma, Sage, TeX

C_6\times D_{28}
% in TeX

G:=Group("C6xD28");
// GroupNames label

G:=SmallGroup(336,176);
// by ID

G=gap.SmallGroup(336,176);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,506,122,10373]);
// Polycyclic

G:=Group<a,b,c|a^6=b^28=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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