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## G = Dic7.D6order 336 = 24·3·7

### 5th non-split extension by Dic7 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — Dic7.D6
 Chief series C1 — C7 — C21 — C42 — C6×D7 — Dic3×D7 — Dic7.D6
 Lower central C21 — C42 — Dic7.D6
 Upper central C1 — C2 — C22

Generators and relations for Dic7.D6
G = < a,b,c,d | a14=c6=1, b2=d2=a7, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=a7b, dcd-1=c-1 >

Subgroups: 444 in 80 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C7, C2×C4, D4, Q8, Dic3, Dic3, C12, D6, C2×C6, C2×C6, D7, C14, C14, C4○D4, C21, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C3×D4, Dic7, Dic7, C28, D14, D14, C2×C14, C3×D7, D21, C42, C42, D42S3, Dic14, C4×D7, D28, C7⋊D4, C7⋊D4, C2×C28, C7×Dic3, C3×Dic7, Dic21, C6×D7, D42, C2×C42, C4○D28, Dic3×D7, D21⋊C4, C3⋊D28, C21⋊Q8, C3×C7⋊D4, Dic3×C14, C217D4, Dic7.D6
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, D42S3, C22×D7, S3×D7, C4○D28, C2×S3×D7, Dic7.D6

Smallest permutation representation of Dic7.D6
On 168 points
Generators in S168
(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)
(1 168 8 161)(2 167 9 160)(3 166 10 159)(4 165 11 158)(5 164 12 157)(6 163 13 156)(7 162 14 155)(15 144 22 151)(16 143 23 150)(17 142 24 149)(18 141 25 148)(19 154 26 147)(20 153 27 146)(21 152 28 145)(29 101 36 108)(30 100 37 107)(31 99 38 106)(32 112 39 105)(33 111 40 104)(34 110 41 103)(35 109 42 102)(43 129 50 136)(44 128 51 135)(45 127 52 134)(46 140 53 133)(47 139 54 132)(48 138 55 131)(49 137 56 130)(57 123 64 116)(58 122 65 115)(59 121 66 114)(60 120 67 113)(61 119 68 126)(62 118 69 125)(63 117 70 124)(71 92 78 85)(72 91 79 98)(73 90 80 97)(74 89 81 96)(75 88 82 95)(76 87 83 94)(77 86 84 93)
(1 141 38 77 134 59)(2 154 39 76 135 58)(3 153 40 75 136 57)(4 152 41 74 137 70)(5 151 42 73 138 69)(6 150 29 72 139 68)(7 149 30 71 140 67)(8 148 31 84 127 66)(9 147 32 83 128 65)(10 146 33 82 129 64)(11 145 34 81 130 63)(12 144 35 80 131 62)(13 143 36 79 132 61)(14 142 37 78 133 60)(15 109 90 48 125 157)(16 108 91 47 126 156)(17 107 92 46 113 155)(18 106 93 45 114 168)(19 105 94 44 115 167)(20 104 95 43 116 166)(21 103 96 56 117 165)(22 102 97 55 118 164)(23 101 98 54 119 163)(24 100 85 53 120 162)(25 99 86 52 121 161)(26 112 87 51 122 160)(27 111 88 50 123 159)(28 110 89 49 124 158)
(1 121 8 114)(2 120 9 113)(3 119 10 126)(4 118 11 125)(5 117 12 124)(6 116 13 123)(7 115 14 122)(15 137 22 130)(16 136 23 129)(17 135 24 128)(18 134 25 127)(19 133 26 140)(20 132 27 139)(21 131 28 138)(29 95 36 88)(30 94 37 87)(31 93 38 86)(32 92 39 85)(33 91 40 98)(34 90 41 97)(35 89 42 96)(43 143 50 150)(44 142 51 149)(45 141 52 148)(46 154 53 147)(47 153 54 146)(48 152 55 145)(49 151 56 144)(57 163 64 156)(58 162 65 155)(59 161 66 168)(60 160 67 167)(61 159 68 166)(62 158 69 165)(63 157 70 164)(71 105 78 112)(72 104 79 111)(73 103 80 110)(74 102 81 109)(75 101 82 108)(76 100 83 107)(77 99 84 106)

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 7A 7B 7C 12 14A ··· 14I 21A 21B 21C 28A ··· 28L 42A ··· 42I order 1 2 2 2 2 3 4 4 4 4 4 6 6 6 7 7 7 12 14 ··· 14 21 21 21 28 ··· 28 42 ··· 42 size 1 1 2 14 42 2 3 3 6 14 42 2 4 28 2 2 2 28 2 ··· 2 4 4 4 6 ··· 6 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D7 C4○D4 D14 D14 C4○D28 D4⋊2S3 S3×D7 C2×S3×D7 Dic7.D6 kernel Dic7.D6 Dic3×D7 D21⋊C4 C3⋊D28 C21⋊Q8 C3×C7⋊D4 Dic3×C14 C21⋊7D4 C7⋊D4 Dic7 D14 C2×C14 C2×Dic3 C21 Dic3 C2×C6 C3 C7 C22 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 3 2 6 3 12 1 3 3 6

Matrix representation of Dic7.D6 in GL4(𝔽337) generated by

 0 336 0 0 1 34 0 0 0 0 1 0 0 0 0 1
,
 189 0 0 0 314 148 0 0 0 0 1 0 0 0 0 1
,
 142 48 0 0 275 195 0 0 0 0 336 336 0 0 1 0
,
 122 27 0 0 260 215 0 0 0 0 336 336 0 0 0 1
G:=sub<GL(4,GF(337))| [0,1,0,0,336,34,0,0,0,0,1,0,0,0,0,1],[189,314,0,0,0,148,0,0,0,0,1,0,0,0,0,1],[142,275,0,0,48,195,0,0,0,0,336,1,0,0,336,0],[122,260,0,0,27,215,0,0,0,0,336,0,0,0,336,1] >;

Dic7.D6 in GAP, Magma, Sage, TeX

{\rm Dic}_7.D_6
% in TeX

G:=Group("Dic7.D6");
// GroupNames label

G:=SmallGroup(336,152);
// by ID

G=gap.SmallGroup(336,152);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,55,218,490,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^14=c^6=1,b^2=d^2=a^7,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations

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