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## G = C3×C7⋊C8order 168 = 23·3·7

### Direct product of C3 and C7⋊C8

Aliases: C3×C7⋊C8, C212C8, C73C24, C84.4C2, C42.2C4, C28.6C6, C12.4D7, C14.3C12, C6.2Dic7, C4.2(C3×D7), C2.(C3×Dic7), SmallGroup(168,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C3×C7⋊C8
 Chief series C1 — C7 — C14 — C28 — C84 — C3×C7⋊C8
 Lower central C7 — C3×C7⋊C8
 Upper central C1 — C12

Generators and relations for C3×C7⋊C8
G = < a,b,c | a3=b7=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×C7⋊C8
Regular action on 168 points
Generators in S168
(1 132 165)(2 133 166)(3 134 167)(4 135 168)(5 136 161)(6 129 162)(7 130 163)(8 131 164)(9 116 64)(10 117 57)(11 118 58)(12 119 59)(13 120 60)(14 113 61)(15 114 62)(16 115 63)(17 124 65)(18 125 66)(19 126 67)(20 127 68)(21 128 69)(22 121 70)(23 122 71)(24 123 72)(25 138 88)(26 139 81)(27 140 82)(28 141 83)(29 142 84)(30 143 85)(31 144 86)(32 137 87)(33 73 96)(34 74 89)(35 75 90)(36 76 91)(37 77 92)(38 78 93)(39 79 94)(40 80 95)(41 152 104)(42 145 97)(43 146 98)(44 147 99)(45 148 100)(46 149 101)(47 150 102)(48 151 103)(49 160 112)(50 153 105)(51 154 106)(52 155 107)(53 156 108)(54 157 109)(55 158 110)(56 159 111)
(1 25 113 148 74 65 153)(2 154 66 75 149 114 26)(3 27 115 150 76 67 155)(4 156 68 77 151 116 28)(5 29 117 152 78 69 157)(6 158 70 79 145 118 30)(7 31 119 146 80 71 159)(8 160 72 73 147 120 32)(9 83 168 53 127 37 48)(10 41 38 128 54 161 84)(11 85 162 55 121 39 42)(12 43 40 122 56 163 86)(13 87 164 49 123 33 44)(14 45 34 124 50 165 88)(15 81 166 51 125 35 46)(16 47 36 126 52 167 82)(17 105 132 138 61 100 89)(18 90 101 62 139 133 106)(19 107 134 140 63 102 91)(20 92 103 64 141 135 108)(21 109 136 142 57 104 93)(22 94 97 58 143 129 110)(23 111 130 144 59 98 95)(24 96 99 60 137 131 112)
(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)

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

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

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

C3×C7⋊C8 is a maximal subgroup of
D21⋊C8  D6.Dic7  D42.C4  C7⋊D24  D12.D7  Dic6⋊D7  C7⋊Dic12  D7×C24

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C8 C12 C24 D7 Dic7 C3×D7 C7⋊C8 C3×Dic7 C3×C7⋊C8 kernel C3×C7⋊C8 C84 C7⋊C8 C42 C28 C21 C14 C7 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 4 8 3 3 6 6 6 12

Matrix representation of C3×C7⋊C8 in GL2(𝔽13) generated by

 3 0 0 3
,
 4 7 9 3
,
 0 5 1 0
G:=sub<GL(2,GF(13))| [3,0,0,3],[4,9,7,3],[0,1,5,0] >;

C3×C7⋊C8 in GAP, Magma, Sage, TeX

C_3\times C_7\rtimes C_8
% in TeX

G:=Group("C3xC7:C8");
// GroupNames label

G:=SmallGroup(168,4);
// by ID

G=gap.SmallGroup(168,4);
# by ID

G:=PCGroup([5,-2,-3,-2,-2,-7,30,42,3604]);
// Polycyclic

G:=Group<a,b,c|a^3=b^7=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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