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## G = C56.30C23order 448 = 26·7

### 23rd non-split extension by C56 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — C56.30C23
 Chief series C1 — C7 — C14 — C28 — C56 — D56 — D56⋊7C2 — C56.30C23
 Lower central C7 — C14 — C28 — C56 — C56.30C23
 Upper central C1 — C4 — C2×C4 — C2×C8 — C4○D8

Generators and relations for C56.30C23
G = < a,b,c,d | a56=b2=d2=1, c2=a28, bab=a-1, ac=ca, dad=a15, bc=cb, dbd=a49b, cd=dc >

Subgroups: 436 in 84 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C16, C2×C8, D8, D8, SD16, Q16, Q16, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C16, D16, SD32, Q32, C4○D8, C4○D8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C4○D16, C7⋊C16, C56⋊C2, D56, Dic28, C2×C56, C7×D8, C7×SD16, C7×Q16, C4○D28, C7×C4○D4, C2×C7⋊C16, C7⋊D16, D8.D7, C7⋊SD32, C7⋊Q32, D567C2, C7×C4○D8, C56.30C23
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C7⋊D4, C22×D7, C4○D16, D4⋊D7, C2×C7⋊D4, C2×D4⋊D7, C56.30C23

Smallest permutation representation of C56.30C23
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)
(1 134)(2 133)(3 132)(4 131)(5 130)(6 129)(7 128)(8 127)(9 126)(10 125)(11 124)(12 123)(13 122)(14 121)(15 120)(16 119)(17 118)(18 117)(19 116)(20 115)(21 114)(22 113)(23 168)(24 167)(25 166)(26 165)(27 164)(28 163)(29 162)(30 161)(31 160)(32 159)(33 158)(34 157)(35 156)(36 155)(37 154)(38 153)(39 152)(40 151)(41 150)(42 149)(43 148)(44 147)(45 146)(46 145)(47 144)(48 143)(49 142)(50 141)(51 140)(52 139)(53 138)(54 137)(55 136)(56 135)(57 194)(58 193)(59 192)(60 191)(61 190)(62 189)(63 188)(64 187)(65 186)(66 185)(67 184)(68 183)(69 182)(70 181)(71 180)(72 179)(73 178)(74 177)(75 176)(76 175)(77 174)(78 173)(79 172)(80 171)(81 170)(82 169)(83 224)(84 223)(85 222)(86 221)(87 220)(88 219)(89 218)(90 217)(91 216)(92 215)(93 214)(94 213)(95 212)(96 211)(97 210)(98 209)(99 208)(100 207)(101 206)(102 205)(103 204)(104 203)(105 202)(106 201)(107 200)(108 199)(109 198)(110 197)(111 196)(112 195)
(1 43 29 15)(2 44 30 16)(3 45 31 17)(4 46 32 18)(5 47 33 19)(6 48 34 20)(7 49 35 21)(8 50 36 22)(9 51 37 23)(10 52 38 24)(11 53 39 25)(12 54 40 26)(13 55 41 27)(14 56 42 28)(57 71 85 99)(58 72 86 100)(59 73 87 101)(60 74 88 102)(61 75 89 103)(62 76 90 104)(63 77 91 105)(64 78 92 106)(65 79 93 107)(66 80 94 108)(67 81 95 109)(68 82 96 110)(69 83 97 111)(70 84 98 112)(113 127 141 155)(114 128 142 156)(115 129 143 157)(116 130 144 158)(117 131 145 159)(118 132 146 160)(119 133 147 161)(120 134 148 162)(121 135 149 163)(122 136 150 164)(123 137 151 165)(124 138 152 166)(125 139 153 167)(126 140 154 168)(169 211 197 183)(170 212 198 184)(171 213 199 185)(172 214 200 186)(173 215 201 187)(174 216 202 188)(175 217 203 189)(176 218 204 190)(177 219 205 191)(178 220 206 192)(179 221 207 193)(180 222 208 194)(181 223 209 195)(182 224 210 196)
(1 110)(2 69)(3 84)(4 99)(5 58)(6 73)(7 88)(8 103)(9 62)(10 77)(11 92)(12 107)(13 66)(14 81)(15 96)(16 111)(17 70)(18 85)(19 100)(20 59)(21 74)(22 89)(23 104)(24 63)(25 78)(26 93)(27 108)(28 67)(29 82)(30 97)(31 112)(32 71)(33 86)(34 101)(35 60)(36 75)(37 90)(38 105)(39 64)(40 79)(41 94)(42 109)(43 68)(44 83)(45 98)(46 57)(47 72)(48 87)(49 102)(50 61)(51 76)(52 91)(53 106)(54 65)(55 80)(56 95)(113 169)(114 184)(115 199)(116 214)(117 173)(118 188)(119 203)(120 218)(121 177)(122 192)(123 207)(124 222)(125 181)(126 196)(127 211)(128 170)(129 185)(130 200)(131 215)(132 174)(133 189)(134 204)(135 219)(136 178)(137 193)(138 208)(139 223)(140 182)(141 197)(142 212)(143 171)(144 186)(145 201)(146 216)(147 175)(148 190)(149 205)(150 220)(151 179)(152 194)(153 209)(154 224)(155 183)(156 198)(157 213)(158 172)(159 187)(160 202)(161 217)(162 176)(163 191)(164 206)(165 221)(166 180)(167 195)(168 210)```

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 8D 14A 14B 14C 14D 14E 14F 14G ··· 14L 16A ··· 16H 28A ··· 28F 28G 28H 28I 28J ··· 28O 56A ··· 56L order 1 2 2 2 2 4 4 4 4 4 7 7 7 8 8 8 8 14 14 14 14 14 14 14 ··· 14 16 ··· 16 28 ··· 28 28 28 28 28 ··· 28 56 ··· 56 size 1 1 2 8 56 1 1 2 8 56 2 2 2 2 2 2 2 2 2 2 4 4 4 8 ··· 8 14 ··· 14 2 ··· 2 4 4 4 8 ··· 8 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 D8 D8 D14 D14 D14 C7⋊D4 C7⋊D4 C4○D16 D4⋊D7 D4⋊D7 C56.30C23 kernel C56.30C23 C2×C7⋊C16 C7⋊D16 D8.D7 C7⋊SD32 C7⋊Q32 D56⋊7C2 C7×C4○D8 C56 C2×C28 C4○D8 C28 C2×C14 C2×C8 D8 Q16 C8 C2×C4 C7 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 3 2 2 3 3 3 6 6 8 3 3 12

Matrix representation of C56.30C23 in GL4(𝔽113) generated by

 24 13 0 0 1 10 0 0 0 0 69 0 0 0 5 95
,
 77 109 0 0 13 36 0 0 0 0 100 68 0 0 64 13
,
 1 0 0 0 0 1 0 0 0 0 15 0 0 0 0 15
,
 29 70 0 0 101 84 0 0 0 0 3 106 0 0 98 110
`G:=sub<GL(4,GF(113))| [24,1,0,0,13,10,0,0,0,0,69,5,0,0,0,95],[77,13,0,0,109,36,0,0,0,0,100,64,0,0,68,13],[1,0,0,0,0,1,0,0,0,0,15,0,0,0,0,15],[29,101,0,0,70,84,0,0,0,0,3,98,0,0,106,110] >;`

C56.30C23 in GAP, Magma, Sage, TeX

`C_{56}._{30}C_2^3`
`% in TeX`

`G:=Group("C56.30C2^3");`
`// GroupNames label`

`G:=SmallGroup(448,728);`
`// by ID`

`G=gap.SmallGroup(448,728);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,675,185,192,1684,438,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^56=b^2=d^2=1,c^2=a^28,b*a*b=a^-1,a*c=c*a,d*a*d=a^15,b*c=c*b,d*b*d=a^49*b,c*d=d*c>;`
`// generators/relations`

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