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G = Dic7⋊3Q8order 224 = 25·7

The semidirect product of Dic7 and Q8 acting through Inn(Dic7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — Dic7⋊3Q8
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C4×Dic7 — Dic7⋊3Q8
 Lower central C7 — C14 — Dic7⋊3Q8
 Upper central C1 — C22 — C4⋊C4

Generators and relations for Dic73Q8
G = < a,b,c,d | a14=c4=1, b2=a7, d2=c2, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 214 in 70 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C4, C4, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×Q8, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C4×Dic7, C4×Dic7, Dic7⋊C4, C7×C4⋊C4, C2×Dic14, Dic73Q8
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D7, C22×C4, C2×Q8, C4○D4, D14, C4×Q8, C4×D7, C22×D7, C2×C4×D7, D42D7, Q8×D7, Dic73Q8

Smallest permutation representation of Dic73Q8
Regular action on 224 points
Generators in S224
(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)(169 170 171 172 173 174 175 176 177 178 179 180 181 182)(183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 127 8 134)(2 140 9 133)(3 139 10 132)(4 138 11 131)(5 137 12 130)(6 136 13 129)(7 135 14 128)(15 123 22 116)(16 122 23 115)(17 121 24 114)(18 120 25 113)(19 119 26 126)(20 118 27 125)(21 117 28 124)(29 167 36 160)(30 166 37 159)(31 165 38 158)(32 164 39 157)(33 163 40 156)(34 162 41 155)(35 161 42 168)(43 147 50 154)(44 146 51 153)(45 145 52 152)(46 144 53 151)(47 143 54 150)(48 142 55 149)(49 141 56 148)(57 194 64 187)(58 193 65 186)(59 192 66 185)(60 191 67 184)(61 190 68 183)(62 189 69 196)(63 188 70 195)(71 173 78 180)(72 172 79 179)(73 171 80 178)(74 170 81 177)(75 169 82 176)(76 182 83 175)(77 181 84 174)(85 223 92 216)(86 222 93 215)(87 221 94 214)(88 220 95 213)(89 219 96 212)(90 218 97 211)(91 217 98 224)(99 203 106 210)(100 202 107 209)(101 201 108 208)(102 200 109 207)(103 199 110 206)(104 198 111 205)(105 197 112 204)
(1 141 18 162)(2 154 19 161)(3 153 20 160)(4 152 21 159)(5 151 22 158)(6 150 23 157)(7 149 24 156)(8 148 25 155)(9 147 26 168)(10 146 27 167)(11 145 28 166)(12 144 15 165)(13 143 16 164)(14 142 17 163)(29 139 44 118)(30 138 45 117)(31 137 46 116)(32 136 47 115)(33 135 48 114)(34 134 49 113)(35 133 50 126)(36 132 51 125)(37 131 52 124)(38 130 53 123)(39 129 54 122)(40 128 55 121)(41 127 56 120)(42 140 43 119)(57 222 71 201)(58 221 72 200)(59 220 73 199)(60 219 74 198)(61 218 75 197)(62 217 76 210)(63 216 77 209)(64 215 78 208)(65 214 79 207)(66 213 80 206)(67 212 81 205)(68 211 82 204)(69 224 83 203)(70 223 84 202)(85 181 100 188)(86 180 101 187)(87 179 102 186)(88 178 103 185)(89 177 104 184)(90 176 105 183)(91 175 106 196)(92 174 107 195)(93 173 108 194)(94 172 109 193)(95 171 110 192)(96 170 111 191)(97 169 112 190)(98 182 99 189)
(1 75 18 61)(2 76 19 62)(3 77 20 63)(4 78 21 64)(5 79 22 65)(6 80 23 66)(7 81 24 67)(8 82 25 68)(9 83 26 69)(10 84 27 70)(11 71 28 57)(12 72 15 58)(13 73 16 59)(14 74 17 60)(29 100 44 85)(30 101 45 86)(31 102 46 87)(32 103 47 88)(33 104 48 89)(34 105 49 90)(35 106 50 91)(36 107 51 92)(37 108 52 93)(38 109 53 94)(39 110 54 95)(40 111 55 96)(41 112 56 97)(42 99 43 98)(113 183 134 176)(114 184 135 177)(115 185 136 178)(116 186 137 179)(117 187 138 180)(118 188 139 181)(119 189 140 182)(120 190 127 169)(121 191 128 170)(122 192 129 171)(123 193 130 172)(124 194 131 173)(125 195 132 174)(126 196 133 175)(141 218 162 197)(142 219 163 198)(143 220 164 199)(144 221 165 200)(145 222 166 201)(146 223 167 202)(147 224 168 203)(148 211 155 204)(149 212 156 205)(150 213 157 206)(151 214 158 207)(152 215 159 208)(153 216 160 209)(154 217 161 210)

G:=sub<Sym(224)| (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,127,8,134)(2,140,9,133)(3,139,10,132)(4,138,11,131)(5,137,12,130)(6,136,13,129)(7,135,14,128)(15,123,22,116)(16,122,23,115)(17,121,24,114)(18,120,25,113)(19,119,26,126)(20,118,27,125)(21,117,28,124)(29,167,36,160)(30,166,37,159)(31,165,38,158)(32,164,39,157)(33,163,40,156)(34,162,41,155)(35,161,42,168)(43,147,50,154)(44,146,51,153)(45,145,52,152)(46,144,53,151)(47,143,54,150)(48,142,55,149)(49,141,56,148)(57,194,64,187)(58,193,65,186)(59,192,66,185)(60,191,67,184)(61,190,68,183)(62,189,69,196)(63,188,70,195)(71,173,78,180)(72,172,79,179)(73,171,80,178)(74,170,81,177)(75,169,82,176)(76,182,83,175)(77,181,84,174)(85,223,92,216)(86,222,93,215)(87,221,94,214)(88,220,95,213)(89,219,96,212)(90,218,97,211)(91,217,98,224)(99,203,106,210)(100,202,107,209)(101,201,108,208)(102,200,109,207)(103,199,110,206)(104,198,111,205)(105,197,112,204), (1,141,18,162)(2,154,19,161)(3,153,20,160)(4,152,21,159)(5,151,22,158)(6,150,23,157)(7,149,24,156)(8,148,25,155)(9,147,26,168)(10,146,27,167)(11,145,28,166)(12,144,15,165)(13,143,16,164)(14,142,17,163)(29,139,44,118)(30,138,45,117)(31,137,46,116)(32,136,47,115)(33,135,48,114)(34,134,49,113)(35,133,50,126)(36,132,51,125)(37,131,52,124)(38,130,53,123)(39,129,54,122)(40,128,55,121)(41,127,56,120)(42,140,43,119)(57,222,71,201)(58,221,72,200)(59,220,73,199)(60,219,74,198)(61,218,75,197)(62,217,76,210)(63,216,77,209)(64,215,78,208)(65,214,79,207)(66,213,80,206)(67,212,81,205)(68,211,82,204)(69,224,83,203)(70,223,84,202)(85,181,100,188)(86,180,101,187)(87,179,102,186)(88,178,103,185)(89,177,104,184)(90,176,105,183)(91,175,106,196)(92,174,107,195)(93,173,108,194)(94,172,109,193)(95,171,110,192)(96,170,111,191)(97,169,112,190)(98,182,99,189), (1,75,18,61)(2,76,19,62)(3,77,20,63)(4,78,21,64)(5,79,22,65)(6,80,23,66)(7,81,24,67)(8,82,25,68)(9,83,26,69)(10,84,27,70)(11,71,28,57)(12,72,15,58)(13,73,16,59)(14,74,17,60)(29,100,44,85)(30,101,45,86)(31,102,46,87)(32,103,47,88)(33,104,48,89)(34,105,49,90)(35,106,50,91)(36,107,51,92)(37,108,52,93)(38,109,53,94)(39,110,54,95)(40,111,55,96)(41,112,56,97)(42,99,43,98)(113,183,134,176)(114,184,135,177)(115,185,136,178)(116,186,137,179)(117,187,138,180)(118,188,139,181)(119,189,140,182)(120,190,127,169)(121,191,128,170)(122,192,129,171)(123,193,130,172)(124,194,131,173)(125,195,132,174)(126,196,133,175)(141,218,162,197)(142,219,163,198)(143,220,164,199)(144,221,165,200)(145,222,166,201)(146,223,167,202)(147,224,168,203)(148,211,155,204)(149,212,156,205)(150,213,157,206)(151,214,158,207)(152,215,159,208)(153,216,160,209)(154,217,161,210)>;

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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,127,8,134)(2,140,9,133)(3,139,10,132)(4,138,11,131)(5,137,12,130)(6,136,13,129)(7,135,14,128)(15,123,22,116)(16,122,23,115)(17,121,24,114)(18,120,25,113)(19,119,26,126)(20,118,27,125)(21,117,28,124)(29,167,36,160)(30,166,37,159)(31,165,38,158)(32,164,39,157)(33,163,40,156)(34,162,41,155)(35,161,42,168)(43,147,50,154)(44,146,51,153)(45,145,52,152)(46,144,53,151)(47,143,54,150)(48,142,55,149)(49,141,56,148)(57,194,64,187)(58,193,65,186)(59,192,66,185)(60,191,67,184)(61,190,68,183)(62,189,69,196)(63,188,70,195)(71,173,78,180)(72,172,79,179)(73,171,80,178)(74,170,81,177)(75,169,82,176)(76,182,83,175)(77,181,84,174)(85,223,92,216)(86,222,93,215)(87,221,94,214)(88,220,95,213)(89,219,96,212)(90,218,97,211)(91,217,98,224)(99,203,106,210)(100,202,107,209)(101,201,108,208)(102,200,109,207)(103,199,110,206)(104,198,111,205)(105,197,112,204), (1,141,18,162)(2,154,19,161)(3,153,20,160)(4,152,21,159)(5,151,22,158)(6,150,23,157)(7,149,24,156)(8,148,25,155)(9,147,26,168)(10,146,27,167)(11,145,28,166)(12,144,15,165)(13,143,16,164)(14,142,17,163)(29,139,44,118)(30,138,45,117)(31,137,46,116)(32,136,47,115)(33,135,48,114)(34,134,49,113)(35,133,50,126)(36,132,51,125)(37,131,52,124)(38,130,53,123)(39,129,54,122)(40,128,55,121)(41,127,56,120)(42,140,43,119)(57,222,71,201)(58,221,72,200)(59,220,73,199)(60,219,74,198)(61,218,75,197)(62,217,76,210)(63,216,77,209)(64,215,78,208)(65,214,79,207)(66,213,80,206)(67,212,81,205)(68,211,82,204)(69,224,83,203)(70,223,84,202)(85,181,100,188)(86,180,101,187)(87,179,102,186)(88,178,103,185)(89,177,104,184)(90,176,105,183)(91,175,106,196)(92,174,107,195)(93,173,108,194)(94,172,109,193)(95,171,110,192)(96,170,111,191)(97,169,112,190)(98,182,99,189), (1,75,18,61)(2,76,19,62)(3,77,20,63)(4,78,21,64)(5,79,22,65)(6,80,23,66)(7,81,24,67)(8,82,25,68)(9,83,26,69)(10,84,27,70)(11,71,28,57)(12,72,15,58)(13,73,16,59)(14,74,17,60)(29,100,44,85)(30,101,45,86)(31,102,46,87)(32,103,47,88)(33,104,48,89)(34,105,49,90)(35,106,50,91)(36,107,51,92)(37,108,52,93)(38,109,53,94)(39,110,54,95)(40,111,55,96)(41,112,56,97)(42,99,43,98)(113,183,134,176)(114,184,135,177)(115,185,136,178)(116,186,137,179)(117,187,138,180)(118,188,139,181)(119,189,140,182)(120,190,127,169)(121,191,128,170)(122,192,129,171)(123,193,130,172)(124,194,131,173)(125,195,132,174)(126,196,133,175)(141,218,162,197)(142,219,163,198)(143,220,164,199)(144,221,165,200)(145,222,166,201)(146,223,167,202)(147,224,168,203)(148,211,155,204)(149,212,156,205)(150,213,157,206)(151,214,158,207)(152,215,159,208)(153,216,160,209)(154,217,161,210) );

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),(169,170,171,172,173,174,175,176,177,178,179,180,181,182),(183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210),(211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,127,8,134),(2,140,9,133),(3,139,10,132),(4,138,11,131),(5,137,12,130),(6,136,13,129),(7,135,14,128),(15,123,22,116),(16,122,23,115),(17,121,24,114),(18,120,25,113),(19,119,26,126),(20,118,27,125),(21,117,28,124),(29,167,36,160),(30,166,37,159),(31,165,38,158),(32,164,39,157),(33,163,40,156),(34,162,41,155),(35,161,42,168),(43,147,50,154),(44,146,51,153),(45,145,52,152),(46,144,53,151),(47,143,54,150),(48,142,55,149),(49,141,56,148),(57,194,64,187),(58,193,65,186),(59,192,66,185),(60,191,67,184),(61,190,68,183),(62,189,69,196),(63,188,70,195),(71,173,78,180),(72,172,79,179),(73,171,80,178),(74,170,81,177),(75,169,82,176),(76,182,83,175),(77,181,84,174),(85,223,92,216),(86,222,93,215),(87,221,94,214),(88,220,95,213),(89,219,96,212),(90,218,97,211),(91,217,98,224),(99,203,106,210),(100,202,107,209),(101,201,108,208),(102,200,109,207),(103,199,110,206),(104,198,111,205),(105,197,112,204)], [(1,141,18,162),(2,154,19,161),(3,153,20,160),(4,152,21,159),(5,151,22,158),(6,150,23,157),(7,149,24,156),(8,148,25,155),(9,147,26,168),(10,146,27,167),(11,145,28,166),(12,144,15,165),(13,143,16,164),(14,142,17,163),(29,139,44,118),(30,138,45,117),(31,137,46,116),(32,136,47,115),(33,135,48,114),(34,134,49,113),(35,133,50,126),(36,132,51,125),(37,131,52,124),(38,130,53,123),(39,129,54,122),(40,128,55,121),(41,127,56,120),(42,140,43,119),(57,222,71,201),(58,221,72,200),(59,220,73,199),(60,219,74,198),(61,218,75,197),(62,217,76,210),(63,216,77,209),(64,215,78,208),(65,214,79,207),(66,213,80,206),(67,212,81,205),(68,211,82,204),(69,224,83,203),(70,223,84,202),(85,181,100,188),(86,180,101,187),(87,179,102,186),(88,178,103,185),(89,177,104,184),(90,176,105,183),(91,175,106,196),(92,174,107,195),(93,173,108,194),(94,172,109,193),(95,171,110,192),(96,170,111,191),(97,169,112,190),(98,182,99,189)], [(1,75,18,61),(2,76,19,62),(3,77,20,63),(4,78,21,64),(5,79,22,65),(6,80,23,66),(7,81,24,67),(8,82,25,68),(9,83,26,69),(10,84,27,70),(11,71,28,57),(12,72,15,58),(13,73,16,59),(14,74,17,60),(29,100,44,85),(30,101,45,86),(31,102,46,87),(32,103,47,88),(33,104,48,89),(34,105,49,90),(35,106,50,91),(36,107,51,92),(37,108,52,93),(38,109,53,94),(39,110,54,95),(40,111,55,96),(41,112,56,97),(42,99,43,98),(113,183,134,176),(114,184,135,177),(115,185,136,178),(116,186,137,179),(117,187,138,180),(118,188,139,181),(119,189,140,182),(120,190,127,169),(121,191,128,170),(122,192,129,171),(123,193,130,172),(124,194,131,173),(125,195,132,174),(126,196,133,175),(141,218,162,197),(142,219,163,198),(143,220,164,199),(144,221,165,200),(145,222,166,201),(146,223,167,202),(147,224,168,203),(148,211,155,204),(149,212,156,205),(150,213,157,206),(151,214,158,207),(152,215,159,208),(153,216,160,209),(154,217,161,210)]])

50 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 7A 7B 7C 14A ··· 14I 28A ··· 28R order 1 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 2 ··· 2 7 7 7 7 14 ··· 14 2 2 2 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + - + + - - image C1 C2 C2 C2 C2 C4 Q8 D7 C4○D4 D14 C4×D7 D4⋊2D7 Q8×D7 kernel Dic7⋊3Q8 C4×Dic7 Dic7⋊C4 C7×C4⋊C4 C2×Dic14 Dic14 Dic7 C4⋊C4 C14 C2×C4 C4 C2 C2 # reps 1 3 2 1 1 8 2 3 2 9 12 3 3

Matrix representation of Dic73Q8 in GL4(𝔽29) generated by

 0 28 0 0 1 11 0 0 0 0 28 0 0 0 0 28
,
 2 11 0 0 18 27 0 0 0 0 12 0 0 0 0 12
,
 5 13 0 0 16 24 0 0 0 0 25 11 0 0 9 4
,
 28 0 0 0 0 28 0 0 0 0 19 13 0 0 19 10
G:=sub<GL(4,GF(29))| [0,1,0,0,28,11,0,0,0,0,28,0,0,0,0,28],[2,18,0,0,11,27,0,0,0,0,12,0,0,0,0,12],[5,16,0,0,13,24,0,0,0,0,25,9,0,0,11,4],[28,0,0,0,0,28,0,0,0,0,19,19,0,0,13,10] >;

Dic73Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes_3Q_8
% in TeX

G:=Group("Dic7:3Q8");
// GroupNames label

G:=SmallGroup(224,82);
// by ID

G=gap.SmallGroup(224,82);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,55,116,122,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^14=c^4=1,b^2=a^7,d^2=c^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
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𝔽