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## G = C7×C4○D12order 336 = 24·3·7

### Direct product of C7 and C4○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C7×C4○D12
 Chief series C1 — C3 — C6 — C42 — S3×C14 — S3×C28 — C7×C4○D12
 Lower central C3 — C6 — C7×C4○D12
 Upper central C1 — C28 — C2×C28

Generators and relations for C7×C4○D12
G = < a,b,c,d | a7=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >

Subgroups: 152 in 80 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6, C6, C7, C2×C4, C2×C4 [×2], D4 [×3], Q8, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C14, C14 [×3], C4○D4, C21, Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, C28 [×2], C28 [×2], C2×C14, C2×C14 [×2], S3×C7 [×2], C42, C42, C4○D12, C2×C28, C2×C28 [×2], C7×D4 [×3], C7×Q8, C7×Dic3 [×2], C84 [×2], S3×C14 [×2], C2×C42, C7×C4○D4, C7×Dic6, S3×C28 [×2], C7×D12, C7×C3⋊D4 [×2], C2×C84, C7×C4○D12
Quotients: C1, C2 [×7], C22 [×7], S3, C7, C23, D6 [×3], C14 [×7], C4○D4, C22×S3, C2×C14 [×7], S3×C7, C4○D12, C22×C14, S3×C14 [×3], C7×C4○D4, S3×C2×C14, C7×C4○D12

Smallest permutation representation of C7×C4○D12
On 168 points
Generators in S168
(1 19 118 90 28 70 44)(2 20 119 91 29 71 45)(3 21 120 92 30 72 46)(4 22 109 93 31 61 47)(5 23 110 94 32 62 48)(6 24 111 95 33 63 37)(7 13 112 96 34 64 38)(8 14 113 85 35 65 39)(9 15 114 86 36 66 40)(10 16 115 87 25 67 41)(11 17 116 88 26 68 42)(12 18 117 89 27 69 43)(49 141 77 98 152 125 163)(50 142 78 99 153 126 164)(51 143 79 100 154 127 165)(52 144 80 101 155 128 166)(53 133 81 102 156 129 167)(54 134 82 103 145 130 168)(55 135 83 104 146 131 157)(56 136 84 105 147 132 158)(57 137 73 106 148 121 159)(58 138 74 107 149 122 160)(59 139 75 108 150 123 161)(60 140 76 97 151 124 162)
(1 138 7 144)(2 139 8 133)(3 140 9 134)(4 141 10 135)(5 142 11 136)(6 143 12 137)(13 80 19 74)(14 81 20 75)(15 82 21 76)(16 83 22 77)(17 84 23 78)(18 73 24 79)(25 131 31 125)(26 132 32 126)(27 121 33 127)(28 122 34 128)(29 123 35 129)(30 124 36 130)(37 51 43 57)(38 52 44 58)(39 53 45 59)(40 54 46 60)(41 55 47 49)(42 56 48 50)(61 163 67 157)(62 164 68 158)(63 165 69 159)(64 166 70 160)(65 167 71 161)(66 168 72 162)(85 156 91 150)(86 145 92 151)(87 146 93 152)(88 147 94 153)(89 148 95 154)(90 149 96 155)(97 114 103 120)(98 115 104 109)(99 116 105 110)(100 117 106 111)(101 118 107 112)(102 119 108 113)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 33)(26 32)(27 31)(28 30)(34 36)(37 41)(38 40)(42 48)(43 47)(44 46)(49 57)(50 56)(51 55)(52 54)(58 60)(61 69)(62 68)(63 67)(64 66)(70 72)(73 77)(74 76)(78 84)(79 83)(80 82)(86 96)(87 95)(88 94)(89 93)(90 92)(97 107)(98 106)(99 105)(100 104)(101 103)(109 117)(110 116)(111 115)(112 114)(118 120)(121 125)(122 124)(126 132)(127 131)(128 130)(134 144)(135 143)(136 142)(137 141)(138 140)(145 155)(146 154)(147 153)(148 152)(149 151)(157 165)(158 164)(159 163)(160 162)(166 168)

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

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

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

126 conjugacy classes

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

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 S3 D6 D6 C4○D4 S3×C7 C4○D12 S3×C14 S3×C14 C7×C4○D4 C7×C4○D12 kernel C7×C4○D12 C7×Dic6 S3×C28 C7×D12 C7×C3⋊D4 C2×C84 C4○D12 Dic6 C4×S3 D12 C3⋊D4 C2×C12 C2×C28 C28 C2×C14 C21 C2×C4 C7 C4 C22 C3 C1 # reps 1 1 2 1 2 1 6 6 12 6 12 6 1 2 1 2 6 4 12 6 12 24

Matrix representation of C7×C4○D12 in GL2(𝔽337) generated by

 64 0 0 64
,
 148 0 0 148
,
 15 15 322 30
,
 0 336 336 0
G:=sub<GL(2,GF(337))| [64,0,0,64],[148,0,0,148],[15,322,15,30],[0,336,336,0] >;

C7×C4○D12 in GAP, Magma, Sage, TeX

C_7\times C_4\circ D_{12}
% in TeX

G:=Group("C7xC4oD12");
// GroupNames label

G:=SmallGroup(336,187);
// by ID

G=gap.SmallGroup(336,187);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-3,343,1082,8069]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^4=d^2=1,c^6=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^5>;
// generators/relations

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