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## G = Dic7⋊Q8order 224 = 25·7

### 2nd semidirect product of Dic7 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic7⋊Q8
G = < a,b,c,d | a14=c4=1, b2=a7, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd-1=c-1 >

Subgroups: 222 in 68 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4⋊Q8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, C4×Dic7, Dic7⋊C4, C2×Dic14, Q8×C14, Dic7⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, D14, C4⋊Q8, C7⋊D4, C22×D7, Q8×D7, C2×C7⋊D4, Dic7⋊Q8

Smallest permutation representation of Dic7⋊Q8
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)
(1 149 8 142)(2 148 9 141)(3 147 10 154)(4 146 11 153)(5 145 12 152)(6 144 13 151)(7 143 14 150)(15 180 22 173)(16 179 23 172)(17 178 24 171)(18 177 25 170)(19 176 26 169)(20 175 27 182)(21 174 28 181)(29 138 36 131)(30 137 37 130)(31 136 38 129)(32 135 39 128)(33 134 40 127)(34 133 41 140)(35 132 42 139)(43 87 50 94)(44 86 51 93)(45 85 52 92)(46 98 53 91)(47 97 54 90)(48 96 55 89)(49 95 56 88)(57 156 64 163)(58 155 65 162)(59 168 66 161)(60 167 67 160)(61 166 68 159)(62 165 69 158)(63 164 70 157)(71 210 78 203)(72 209 79 202)(73 208 80 201)(74 207 81 200)(75 206 82 199)(76 205 83 198)(77 204 84 197)(99 217 106 224)(100 216 107 223)(101 215 108 222)(102 214 109 221)(103 213 110 220)(104 212 111 219)(105 211 112 218)(113 191 120 184)(114 190 121 183)(115 189 122 196)(116 188 123 195)(117 187 124 194)(118 186 125 193)(119 185 126 192)
(1 137 181 214)(2 138 182 215)(3 139 169 216)(4 140 170 217)(5 127 171 218)(6 128 172 219)(7 129 173 220)(8 130 174 221)(9 131 175 222)(10 132 176 223)(11 133 177 224)(12 134 178 211)(13 135 179 212)(14 136 180 213)(15 110 143 38)(16 111 144 39)(17 112 145 40)(18 99 146 41)(19 100 147 42)(20 101 148 29)(21 102 149 30)(22 103 150 31)(23 104 151 32)(24 105 152 33)(25 106 153 34)(26 107 154 35)(27 108 141 36)(28 109 142 37)(43 117 167 74)(44 118 168 75)(45 119 155 76)(46 120 156 77)(47 121 157 78)(48 122 158 79)(49 123 159 80)(50 124 160 81)(51 125 161 82)(52 126 162 83)(53 113 163 84)(54 114 164 71)(55 115 165 72)(56 116 166 73)(57 204 91 184)(58 205 92 185)(59 206 93 186)(60 207 94 187)(61 208 95 188)(62 209 96 189)(63 210 97 190)(64 197 98 191)(65 198 85 192)(66 199 86 193)(67 200 87 194)(68 201 88 195)(69 202 89 196)(70 203 90 183)
(1 197 181 191)(2 198 182 192)(3 199 169 193)(4 200 170 194)(5 201 171 195)(6 202 172 196)(7 203 173 183)(8 204 174 184)(9 205 175 185)(10 206 176 186)(11 207 177 187)(12 208 178 188)(13 209 179 189)(14 210 180 190)(15 114 143 71)(16 115 144 72)(17 116 145 73)(18 117 146 74)(19 118 147 75)(20 119 148 76)(21 120 149 77)(22 121 150 78)(23 122 151 79)(24 123 152 80)(25 124 153 81)(26 125 154 82)(27 126 141 83)(28 113 142 84)(29 155 101 45)(30 156 102 46)(31 157 103 47)(32 158 104 48)(33 159 105 49)(34 160 106 50)(35 161 107 51)(36 162 108 52)(37 163 109 53)(38 164 110 54)(39 165 111 55)(40 166 112 56)(41 167 99 43)(42 168 100 44)(57 221 91 130)(58 222 92 131)(59 223 93 132)(60 224 94 133)(61 211 95 134)(62 212 96 135)(63 213 97 136)(64 214 98 137)(65 215 85 138)(66 216 86 139)(67 217 87 140)(68 218 88 127)(69 219 89 128)(70 220 90 129)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 type + + + + + - + + + - image C1 C2 C2 C2 C2 Q8 D4 D7 D14 C7⋊D4 Q8×D7 kernel Dic7⋊Q8 C4×Dic7 Dic7⋊C4 C2×Dic14 Q8×C14 Dic7 C28 C2×Q8 C2×C4 C4 C2 # reps 1 1 4 1 1 4 2 3 9 12 6

Matrix representation of Dic7⋊Q8 in GL4(𝔽29) generated by

 1 28 0 0 5 25 0 0 0 0 1 0 0 0 0 1
,
 12 17 0 0 0 17 0 0 0 0 28 0 0 0 0 28
,
 11 13 0 0 22 18 0 0 0 0 8 28 0 0 7 21
,
 28 0 0 0 0 28 0 0 0 0 25 20 0 0 18 4
G:=sub<GL(4,GF(29))| [1,5,0,0,28,25,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,17,17,0,0,0,0,28,0,0,0,0,28],[11,22,0,0,13,18,0,0,0,0,8,7,0,0,28,21],[28,0,0,0,0,28,0,0,0,0,25,18,0,0,20,4] >;

Dic7⋊Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes Q_8
% in TeX

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

G:=SmallGroup(224,139);
// by ID

G=gap.SmallGroup(224,139);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,55,362,116,50,6917]);
// Polycyclic

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

׿
×
𝔽