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## G = Dic7.M4(2)  order 448 = 26·7

### 2nd non-split extension by Dic7 of M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — Dic7.M4(2)
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4×Dic7 — C2×C4×Dic7 — Dic7.M4(2)
 Lower central C7 — C2×C14 — Dic7.M4(2)
 Upper central C1 — C2×C4 — C22⋊C8

Generators and relations for Dic7.M4(2)
G = < a,b,c,d | a14=c8=d2=1, b2=a7, bab-1=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd=a7c5 >

Subgroups: 380 in 110 conjugacy classes, 53 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C42, C2×C8, C2×C8, C22×C4, C22×C4, Dic7, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C2×C42, C7⋊C8, C56, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C42.6C4, C2×C7⋊C8, C4×Dic7, C2×C56, C22×Dic7, C22×C28, Dic7⋊C8, C56⋊C4, C28.55D4, C7×C22⋊C8, C2×C4×Dic7, Dic7.M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, M4(2), C22×C4, C4○D4, D14, C42⋊C2, C2×M4(2), C4×D7, C22×D7, C42.6C4, C8⋊D7, C2×C4×D7, D42D7, C23.11D14, C2×C8⋊D7, D7×M4(2), Dic7.M4(2)

Smallest permutation representation of Dic7.M4(2)
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)
(1 150 8 143)(2 149 9 142)(3 148 10 141)(4 147 11 154)(5 146 12 153)(6 145 13 152)(7 144 14 151)(15 114 22 121)(16 113 23 120)(17 126 24 119)(18 125 25 118)(19 124 26 117)(20 123 27 116)(21 122 28 115)(29 76 36 83)(30 75 37 82)(31 74 38 81)(32 73 39 80)(33 72 40 79)(34 71 41 78)(35 84 42 77)(43 136 50 129)(44 135 51 128)(45 134 52 127)(46 133 53 140)(47 132 54 139)(48 131 55 138)(49 130 56 137)(57 91 64 98)(58 90 65 97)(59 89 66 96)(60 88 67 95)(61 87 68 94)(62 86 69 93)(63 85 70 92)(99 187 106 194)(100 186 107 193)(101 185 108 192)(102 184 109 191)(103 183 110 190)(104 196 111 189)(105 195 112 188)(155 200 162 207)(156 199 163 206)(157 198 164 205)(158 197 165 204)(159 210 166 203)(160 209 167 202)(161 208 168 201)(169 213 176 220)(170 212 177 219)(171 211 178 218)(172 224 179 217)(173 223 180 216)(174 222 181 215)(175 221 182 214)
(1 208 45 171 99 30 23 59)(2 209 46 172 100 31 24 60)(3 210 47 173 101 32 25 61)(4 197 48 174 102 33 26 62)(5 198 49 175 103 34 27 63)(6 199 50 176 104 35 28 64)(7 200 51 177 105 36 15 65)(8 201 52 178 106 37 16 66)(9 202 53 179 107 38 17 67)(10 203 54 180 108 39 18 68)(11 204 55 181 109 40 19 69)(12 205 56 182 110 41 20 70)(13 206 43 169 111 42 21 57)(14 207 44 170 112 29 22 58)(71 123 85 153 164 137 221 190)(72 124 86 154 165 138 222 191)(73 125 87 141 166 139 223 192)(74 126 88 142 167 140 224 193)(75 113 89 143 168 127 211 194)(76 114 90 144 155 128 212 195)(77 115 91 145 156 129 213 196)(78 116 92 146 157 130 214 183)(79 117 93 147 158 131 215 184)(80 118 94 148 159 132 216 185)(81 119 95 149 160 133 217 186)(82 120 96 150 161 134 218 187)(83 121 97 151 162 135 219 188)(84 122 98 152 163 136 220 189)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 105)(8 106)(9 107)(10 108)(11 109)(12 110)(13 111)(14 112)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 192)(142 193)(143 194)(144 195)(145 196)(146 183)(147 184)(148 185)(149 186)(150 187)(151 188)(152 189)(153 190)(154 191)(155 162)(156 163)(157 164)(158 165)(159 166)(160 167)(161 168)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)

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

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

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

88 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G ··· 4N 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 7 7 7 8 8 8 8 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 1 1 1 1 2 2 14 ··· 14 2 2 2 4 4 4 4 28 28 28 28 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 D7 M4(2) C4○D4 M4(2) D14 D14 C4×D7 C4×D7 C8⋊D7 D4⋊2D7 D7×M4(2) kernel Dic7.M4(2) Dic7⋊C8 C56⋊C4 C28.55D4 C7×C22⋊C8 C2×C4×Dic7 C4×Dic7 C22×Dic7 C22⋊C8 Dic7 C28 C2×C14 C2×C8 C22×C4 C2×C4 C23 C22 C4 C2 # reps 1 2 2 1 1 1 4 4 3 4 4 4 6 3 6 6 24 6 6

Matrix representation of Dic7.M4(2) in GL4(𝔽113) generated by

 0 112 0 0 1 89 0 0 0 0 112 0 0 0 0 112
,
 6 88 0 0 6 107 0 0 0 0 15 0 0 0 0 98
,
 80 31 0 0 82 33 0 0 0 0 0 1 0 0 112 0
,
 112 0 0 0 0 112 0 0 0 0 1 0 0 0 0 112
G:=sub<GL(4,GF(113))| [0,1,0,0,112,89,0,0,0,0,112,0,0,0,0,112],[6,6,0,0,88,107,0,0,0,0,15,0,0,0,0,98],[80,82,0,0,31,33,0,0,0,0,0,112,0,0,1,0],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,112] >;

Dic7.M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_7.M_4(2)
% in TeX

G:=Group("Dic7.M4(2)");
// GroupNames label

G:=SmallGroup(448,253);
// by ID

G=gap.SmallGroup(448,253);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,758,219,58,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^14=c^8=d^2=1,b^2=a^7,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=a^7*c^5>;
// generators/relations

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