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## G = (D4×C14).11C4order 448 = 26·7

### 5th non-split extension by D4×C14 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — (D4×C14).11C4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C7⋊C8 — C22×C7⋊C8 — (D4×C14).11C4
 Lower central C7 — C2×C14 — (D4×C14).11C4
 Upper central C1 — C2×C4 — C2×C4○D4

Generators and relations for (D4×C14).11C4
G = < a,b,c,d | a14=b4=c2=1, d4=b2, ab=ba, ac=ca, dad-1=a-1, cbc=b-1, bd=db, dcd-1=a7b2c >

Subgroups: 404 in 158 conjugacy classes, 71 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C14, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, C22×C8, C2×M4(2), C2×C4○D4, C7⋊C8, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, (C22×C8)⋊C2, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C28.55D4, C22×C7⋊C8, C2×C4.Dic7, C14×C4○D4, (D4×C14).11C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C2×C22⋊C4, C8○D4, C2×Dic7, C7⋊D4, C22×D7, (C22×C8)⋊C2, C23.D7, C22×Dic7, C2×C7⋊D4, Q8.Dic7, C2×C23.D7, (D4×C14).11C4

Smallest permutation representation of (D4×C14).11C4
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)
(1 187 75 130)(2 188 76 131)(3 189 77 132)(4 190 78 133)(5 191 79 134)(6 192 80 135)(7 193 81 136)(8 194 82 137)(9 195 83 138)(10 196 84 139)(11 183 71 140)(12 184 72 127)(13 185 73 128)(14 186 74 129)(15 204 58 96)(16 205 59 97)(17 206 60 98)(18 207 61 85)(19 208 62 86)(20 209 63 87)(21 210 64 88)(22 197 65 89)(23 198 66 90)(24 199 67 91)(25 200 68 92)(26 201 69 93)(27 202 70 94)(28 203 57 95)(29 103 173 147)(30 104 174 148)(31 105 175 149)(32 106 176 150)(33 107 177 151)(34 108 178 152)(35 109 179 153)(36 110 180 154)(37 111 181 141)(38 112 182 142)(39 99 169 143)(40 100 170 144)(41 101 171 145)(42 102 172 146)(43 159 125 221)(44 160 126 222)(45 161 113 223)(46 162 114 224)(47 163 115 211)(48 164 116 212)(49 165 117 213)(50 166 118 214)(51 167 119 215)(52 168 120 216)(53 155 121 217)(54 156 122 218)(55 157 123 219)(56 158 124 220)
(1 157)(2 158)(3 159)(4 160)(5 161)(6 162)(7 163)(8 164)(9 165)(10 166)(11 167)(12 168)(13 155)(14 156)(15 176)(16 177)(17 178)(18 179)(19 180)(20 181)(21 182)(22 169)(23 170)(24 171)(25 172)(26 173)(27 174)(28 175)(29 69)(30 70)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 67)(42 68)(43 189)(44 190)(45 191)(46 192)(47 193)(48 194)(49 195)(50 196)(51 183)(52 184)(53 185)(54 186)(55 187)(56 188)(71 215)(72 216)(73 217)(74 218)(75 219)(76 220)(77 221)(78 222)(79 223)(80 224)(81 211)(82 212)(83 213)(84 214)(85 153)(86 154)(87 141)(88 142)(89 143)(90 144)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 151)(98 152)(99 197)(100 198)(101 199)(102 200)(103 201)(104 202)(105 203)(106 204)(107 205)(108 206)(109 207)(110 208)(111 209)(112 210)(113 134)(114 135)(115 136)(116 137)(117 138)(118 139)(119 140)(120 127)(121 128)(122 129)(123 130)(124 131)(125 132)(126 133)
(1 29 137 154 75 173 194 110)(2 42 138 153 76 172 195 109)(3 41 139 152 77 171 196 108)(4 40 140 151 78 170 183 107)(5 39 127 150 79 169 184 106)(6 38 128 149 80 182 185 105)(7 37 129 148 81 181 186 104)(8 36 130 147 82 180 187 103)(9 35 131 146 83 179 188 102)(10 34 132 145 84 178 189 101)(11 33 133 144 71 177 190 100)(12 32 134 143 72 176 191 99)(13 31 135 142 73 175 192 112)(14 30 136 141 74 174 193 111)(15 120 197 223 58 52 89 161)(16 119 198 222 59 51 90 160)(17 118 199 221 60 50 91 159)(18 117 200 220 61 49 92 158)(19 116 201 219 62 48 93 157)(20 115 202 218 63 47 94 156)(21 114 203 217 64 46 95 155)(22 113 204 216 65 45 96 168)(23 126 205 215 66 44 97 167)(24 125 206 214 67 43 98 166)(25 124 207 213 68 56 85 165)(26 123 208 212 69 55 86 164)(27 122 209 211 70 54 87 163)(28 121 210 224 57 53 88 162)

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

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

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

88 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A ··· 8H 8I 8J 8K 8L 14A ··· 14I 14J ··· 14AA 28A ··· 28L 28M ··· 28AD order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 8 ··· 8 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 4 4 1 1 1 1 2 2 4 4 2 2 2 14 ··· 14 28 28 28 28 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + - - image C1 C2 C2 C2 C2 C4 C4 D4 D7 D14 Dic7 Dic7 C8○D4 C7⋊D4 Q8.Dic7 kernel (D4×C14).11C4 C28.55D4 C22×C7⋊C8 C2×C4.Dic7 C14×C4○D4 D4×C14 Q8×C14 C2×C28 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C14 C2×C4 C2 # reps 1 4 1 1 1 6 2 4 3 9 9 3 8 24 12

Matrix representation of (D4×C14).11C4 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 34 10 0 0 79 0
,
 98 0 0 0 18 15 0 0 0 0 112 0 0 0 0 112
,
 98 88 0 0 18 15 0 0 0 0 91 7 0 0 44 22
,
 95 0 0 0 0 95 0 0 0 0 29 22 0 0 44 84
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,34,79,0,0,10,0],[98,18,0,0,0,15,0,0,0,0,112,0,0,0,0,112],[98,18,0,0,88,15,0,0,0,0,91,44,0,0,7,22],[95,0,0,0,0,95,0,0,0,0,29,44,0,0,22,84] >;

(D4×C14).11C4 in GAP, Magma, Sage, TeX

(D_4\times C_{14})._{11}C_4
% in TeX

G:=Group("(D4xC14).11C4");
// GroupNames label

G:=SmallGroup(448,768);
// by ID

G=gap.SmallGroup(448,768);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,387,136,18822]);
// Polycyclic

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

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