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## G = C28⋊Q8⋊C2order 448 = 26·7

### 4th semidirect product of C28⋊Q8 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C28⋊Q8⋊C2
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4×Dic7 — C28⋊Q8 — C28⋊Q8⋊C2
 Lower central C7 — C14 — C2×C28 — C28⋊Q8⋊C2
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for C28⋊Q8⋊C2
G = < a,b,c,d | a28=b4=d2=1, c2=b2, bab-1=dad=a15, cac-1=a13, cbc-1=b-1, dbd=a7b-1, dcd=a14b2c >

Subgroups: 500 in 100 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C42.28C22, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C23.D7, C7×C4⋊C4, C2×C56, C2×Dic14, C2×Dic14, D4×C14, C14.Q16, C56⋊C4, C28.44D4, D4⋊Dic7, C7×D4⋊C4, C28⋊Q8, C28.17D4, C28⋊Q8⋊C2
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, C8⋊C22, C8.C22, C22×D7, C42.28C22, C4○D28, D4×D7, D42D7, Dic7.D4, D8⋊D7, SD16⋊D7, C28⋊Q8⋊C2

Smallest permutation representation of C28⋊Q8⋊C2
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 54 218 185)(2 41 219 172)(3 56 220 187)(4 43 221 174)(5 30 222 189)(6 45 223 176)(7 32 224 191)(8 47 197 178)(9 34 198 193)(10 49 199 180)(11 36 200 195)(12 51 201 182)(13 38 202 169)(14 53 203 184)(15 40 204 171)(16 55 205 186)(17 42 206 173)(18 29 207 188)(19 44 208 175)(20 31 209 190)(21 46 210 177)(22 33 211 192)(23 48 212 179)(24 35 213 194)(25 50 214 181)(26 37 215 196)(27 52 216 183)(28 39 217 170)(57 98 120 157)(58 85 121 144)(59 100 122 159)(60 87 123 146)(61 102 124 161)(62 89 125 148)(63 104 126 163)(64 91 127 150)(65 106 128 165)(66 93 129 152)(67 108 130 167)(68 95 131 154)(69 110 132 141)(70 97 133 156)(71 112 134 143)(72 99 135 158)(73 86 136 145)(74 101 137 160)(75 88 138 147)(76 103 139 162)(77 90 140 149)(78 105 113 164)(79 92 114 151)(80 107 115 166)(81 94 116 153)(82 109 117 168)(83 96 118 155)(84 111 119 142)
(1 106 218 165)(2 91 219 150)(3 104 220 163)(4 89 221 148)(5 102 222 161)(6 87 223 146)(7 100 224 159)(8 85 197 144)(9 98 198 157)(10 111 199 142)(11 96 200 155)(12 109 201 168)(13 94 202 153)(14 107 203 166)(15 92 204 151)(16 105 205 164)(17 90 206 149)(18 103 207 162)(19 88 208 147)(20 101 209 160)(21 86 210 145)(22 99 211 158)(23 112 212 143)(24 97 213 156)(25 110 214 141)(26 95 215 154)(27 108 216 167)(28 93 217 152)(29 76 188 139)(30 61 189 124)(31 74 190 137)(32 59 191 122)(33 72 192 135)(34 57 193 120)(35 70 194 133)(36 83 195 118)(37 68 196 131)(38 81 169 116)(39 66 170 129)(40 79 171 114)(41 64 172 127)(42 77 173 140)(43 62 174 125)(44 75 175 138)(45 60 176 123)(46 73 177 136)(47 58 178 121)(48 71 179 134)(49 84 180 119)(50 69 181 132)(51 82 182 117)(52 67 183 130)(53 80 184 115)(54 65 185 128)(55 78 186 113)(56 63 187 126)
(2 16)(4 18)(6 20)(8 22)(10 24)(12 26)(14 28)(29 195)(30 182)(31 169)(32 184)(33 171)(34 186)(35 173)(36 188)(37 175)(38 190)(39 177)(40 192)(41 179)(42 194)(43 181)(44 196)(45 183)(46 170)(47 185)(48 172)(49 187)(50 174)(51 189)(52 176)(53 191)(54 178)(55 193)(56 180)(57 64)(58 79)(59 66)(60 81)(61 68)(62 83)(63 70)(65 72)(67 74)(69 76)(71 78)(73 80)(75 82)(77 84)(85 144)(86 159)(87 146)(88 161)(89 148)(90 163)(91 150)(92 165)(93 152)(94 167)(95 154)(96 141)(97 156)(98 143)(99 158)(100 145)(101 160)(102 147)(103 162)(104 149)(105 164)(106 151)(107 166)(108 153)(109 168)(110 155)(111 142)(112 157)(113 134)(114 121)(115 136)(116 123)(117 138)(118 125)(119 140)(120 127)(122 129)(124 131)(126 133)(128 135)(130 137)(132 139)(197 211)(199 213)(201 215)(203 217)(205 219)(207 221)(209 223)

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

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

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

58 conjugacy classes

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

58 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D7 C4○D4 D14 D14 D14 C4○D28 C8⋊C22 C8.C22 D4⋊2D7 D4×D7 D8⋊D7 SD16⋊D7 kernel C28⋊Q8⋊C2 C14.Q16 C56⋊C4 C28.44D4 D4⋊Dic7 C7×D4⋊C4 C28⋊Q8 C28.17D4 C2×Dic7 D4⋊C4 C28 C4⋊C4 C2×C8 C2×D4 C4 C14 C14 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 3 4 3 3 3 12 1 1 3 3 6 6

Matrix representation of C28⋊Q8⋊C2 in GL6(𝔽113)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 105 46 4 0 0 97 14 91 75 0 0 31 42 41 8 0 0 19 53 38 57
,
 34 31 0 0 0 0 21 79 0 0 0 0 0 0 15 76 57 68 0 0 17 56 101 98 0 0 70 97 43 37 0 0 32 67 8 112
,
 55 100 0 0 0 0 85 58 0 0 0 0 0 0 112 76 0 0 0 0 110 1 0 0 0 0 78 3 92 12 0 0 59 76 95 21
,
 1 0 0 0 0 0 78 112 0 0 0 0 0 0 106 80 57 26 0 0 22 7 74 76 0 0 0 0 8 33 0 0 0 0 70 105

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,97,31,19,0,0,105,14,42,53,0,0,46,91,41,38,0,0,4,75,8,57],[34,21,0,0,0,0,31,79,0,0,0,0,0,0,15,17,70,32,0,0,76,56,97,67,0,0,57,101,43,8,0,0,68,98,37,112],[55,85,0,0,0,0,100,58,0,0,0,0,0,0,112,110,78,59,0,0,76,1,3,76,0,0,0,0,92,95,0,0,0,0,12,21],[1,78,0,0,0,0,0,112,0,0,0,0,0,0,106,22,0,0,0,0,80,7,0,0,0,0,57,74,8,70,0,0,26,76,33,105] >;

C28⋊Q8⋊C2 in GAP, Magma, Sage, TeX

C_{28}\rtimes Q_8\rtimes C_2
% in TeX

G:=Group("C28:Q8:C2");
// GroupNames label

G:=SmallGroup(448,299);
// by ID

G=gap.SmallGroup(448,299);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,1094,135,570,297,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^28=b^4=d^2=1,c^2=b^2,b*a*b^-1=d*a*d=a^15,c*a*c^-1=a^13,c*b*c^-1=b^-1,d*b*d=a^7*b^-1,d*c*d=a^14*b^2*c>;
// generators/relations

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