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## G = C7×C4.Dic3order 336 = 24·3·7

### Direct product of C7 and C4.Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C7×C4.Dic3
 Chief series C1 — C3 — C6 — C12 — C84 — C7×C3⋊C8 — C7×C4.Dic3
 Lower central C3 — C6 — C7×C4.Dic3
 Upper central C1 — C28 — C2×C28

Generators and relations for C7×C4.Dic3
G = < a,b,c,d | a7=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

Smallest permutation representation of C7×C4.Dic3
On 168 points
Generators in S168
(1 79 67 55 43 31 19)(2 80 68 56 44 32 20)(3 81 69 57 45 33 21)(4 82 70 58 46 34 22)(5 83 71 59 47 35 23)(6 84 72 60 48 36 24)(7 73 61 49 37 25 13)(8 74 62 50 38 26 14)(9 75 63 51 39 27 15)(10 76 64 52 40 28 16)(11 77 65 53 41 29 17)(12 78 66 54 42 30 18)(85 157 145 133 121 109 97)(86 158 146 134 122 110 98)(87 159 147 135 123 111 99)(88 160 148 136 124 112 100)(89 161 149 137 125 113 101)(90 162 150 138 126 114 102)(91 163 151 139 127 115 103)(92 164 152 140 128 116 104)(93 165 153 141 129 117 105)(94 166 154 142 130 118 106)(95 167 155 143 131 119 107)(96 168 156 144 132 120 108)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 46 43 40)(38 47 44 41)(39 48 45 42)(49 58 55 52)(50 59 56 53)(51 60 57 54)(61 70 67 64)(62 71 68 65)(63 72 69 66)(73 82 79 76)(74 83 80 77)(75 84 81 78)(85 88 91 94)(86 89 92 95)(87 90 93 96)(97 100 103 106)(98 101 104 107)(99 102 105 108)(109 112 115 118)(110 113 116 119)(111 114 117 120)(121 124 127 130)(122 125 128 131)(123 126 129 132)(133 136 139 142)(134 137 140 143)(135 138 141 144)(145 148 151 154)(146 149 152 155)(147 150 153 156)(157 160 163 166)(158 161 164 167)(159 162 165 168)
(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 92 10 89 7 86 4 95)(2 85 11 94 8 91 5 88)(3 90 12 87 9 96 6 93)(13 98 22 107 19 104 16 101)(14 103 23 100 20 97 17 106)(15 108 24 105 21 102 18 99)(25 110 34 119 31 116 28 113)(26 115 35 112 32 109 29 118)(27 120 36 117 33 114 30 111)(37 122 46 131 43 128 40 125)(38 127 47 124 44 121 41 130)(39 132 48 129 45 126 42 123)(49 134 58 143 55 140 52 137)(50 139 59 136 56 133 53 142)(51 144 60 141 57 138 54 135)(61 146 70 155 67 152 64 149)(62 151 71 148 68 145 65 154)(63 156 72 153 69 150 66 147)(73 158 82 167 79 164 76 161)(74 163 83 160 80 157 77 166)(75 168 84 165 81 162 78 159)

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

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

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

126 conjugacy classes

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

126 irreducible representations

 dim 1 1 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 C4 C4 C7 C14 C14 C28 C28 S3 Dic3 D6 Dic3 M4(2) S3×C7 C4.Dic3 C7×Dic3 S3×C14 C7×Dic3 C7×M4(2) C7×C4.Dic3 kernel C7×C4.Dic3 C7×C3⋊C8 C2×C84 C84 C2×C42 C4.Dic3 C3⋊C8 C2×C12 C12 C2×C6 C2×C28 C28 C28 C2×C14 C21 C2×C4 C7 C4 C4 C22 C3 C1 # reps 1 2 1 2 2 6 12 6 12 12 1 1 1 1 2 6 4 6 6 6 12 24

Matrix representation of C7×C4.Dic3 in GL2(𝔽337) generated by

 79 0 0 79
,
 189 0 0 148
,
 220 0 0 265
,
 0 1 189 0
G:=sub<GL(2,GF(337))| [79,0,0,79],[189,0,0,148],[220,0,0,265],[0,189,1,0] >;

C7×C4.Dic3 in GAP, Magma, Sage, TeX

C_7\times C_4.{\rm Dic}_3
% in TeX

G:=Group("C7xC4.Dic3");
// GroupNames label

G:=SmallGroup(336,80);
// by ID

G=gap.SmallGroup(336,80);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-3,168,697,69,8069]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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