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## G = C7×C24⋊C2order 336 = 24·3·7

### Direct product of C7 and C24⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C7×C24⋊C2
 Chief series C1 — C3 — C6 — C12 — C84 — C7×D12 — C7×C24⋊C2
 Lower central C3 — C6 — C12 — C7×C24⋊C2
 Upper central C1 — C14 — C28 — C56

Generators and relations for C7×C24⋊C2
G = < a,b,c | a7=b24=c2=1, ab=ba, ac=ca, cbc=b11 >

Smallest permutation representation of C7×C24⋊C2
On 168 points
Generators in S168
(1 167 111 53 42 90 131)(2 168 112 54 43 91 132)(3 145 113 55 44 92 133)(4 146 114 56 45 93 134)(5 147 115 57 46 94 135)(6 148 116 58 47 95 136)(7 149 117 59 48 96 137)(8 150 118 60 25 73 138)(9 151 119 61 26 74 139)(10 152 120 62 27 75 140)(11 153 97 63 28 76 141)(12 154 98 64 29 77 142)(13 155 99 65 30 78 143)(14 156 100 66 31 79 144)(15 157 101 67 32 80 121)(16 158 102 68 33 81 122)(17 159 103 69 34 82 123)(18 160 104 70 35 83 124)(19 161 105 71 36 84 125)(20 162 106 72 37 85 126)(21 163 107 49 38 86 127)(22 164 108 50 39 87 128)(23 165 109 51 40 88 129)(24 166 110 52 41 89 130)
(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)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 47)(26 34)(27 45)(28 32)(29 43)(31 41)(33 39)(35 37)(36 48)(38 46)(40 44)(49 57)(50 68)(51 55)(52 66)(54 64)(56 62)(58 60)(59 71)(61 69)(63 67)(70 72)(73 95)(74 82)(75 93)(76 80)(77 91)(79 89)(81 87)(83 85)(84 96)(86 94)(88 92)(97 101)(98 112)(100 110)(102 108)(103 119)(104 106)(105 117)(107 115)(109 113)(114 120)(116 118)(121 141)(122 128)(123 139)(124 126)(125 137)(127 135)(129 133)(130 144)(132 142)(134 140)(136 138)(145 165)(146 152)(147 163)(148 150)(149 161)(151 159)(153 157)(154 168)(156 166)(158 164)(160 162)

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

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

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

105 conjugacy classes

 class 1 2A 2B 3 4A 4B 6 7A ··· 7F 8A 8B 12A 12B 14A ··· 14F 14G ··· 14L 21A ··· 21F 24A 24B 24C 24D 28A ··· 28F 28G ··· 28L 42A ··· 42F 56A ··· 56L 84A ··· 84L 168A ··· 168X order 1 2 2 3 4 4 6 7 ··· 7 8 8 12 12 14 ··· 14 14 ··· 14 21 ··· 21 24 24 24 24 28 ··· 28 28 ··· 28 42 ··· 42 56 ··· 56 84 ··· 84 168 ··· 168 size 1 1 12 2 2 12 2 1 ··· 1 2 2 2 2 1 ··· 1 12 ··· 12 2 ··· 2 2 2 2 2 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C7 C14 C14 C14 S3 D4 D6 SD16 D12 S3×C7 C24⋊C2 C7×D4 S3×C14 C7×SD16 C7×D12 C7×C24⋊C2 kernel C7×C24⋊C2 C168 C7×Dic6 C7×D12 C24⋊C2 C24 Dic6 D12 C56 C42 C28 C21 C14 C8 C7 C6 C4 C3 C2 C1 # reps 1 1 1 1 6 6 6 6 1 1 1 2 2 6 4 6 6 12 12 24

Matrix representation of C7×C24⋊C2 in GL4(𝔽337) generated by

 175 0 0 0 0 175 0 0 0 0 8 0 0 0 0 8
,
 1 336 0 0 1 0 0 0 0 0 0 294 0 0 47 141
,
 1 336 0 0 0 336 0 0 0 0 1 0 0 0 224 336
G:=sub<GL(4,GF(337))| [175,0,0,0,0,175,0,0,0,0,8,0,0,0,0,8],[1,1,0,0,336,0,0,0,0,0,0,47,0,0,294,141],[1,0,0,0,336,336,0,0,0,0,1,224,0,0,0,336] >;

C7×C24⋊C2 in GAP, Magma, Sage, TeX

C_7\times C_{24}\rtimes C_2
% in TeX

G:=Group("C7xC24:C2");
// GroupNames label

G:=SmallGroup(336,76);
// by ID

G=gap.SmallGroup(336,76);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-3,361,175,2019,69,8069]);
// Polycyclic

G:=Group<a,b,c|a^7=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^11>;
// generators/relations

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