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## G = (C2×C4).20D28order 448 = 26·7

### 13rd non-split extension by C2×C4 of D28 acting via D28/C14=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C14 — (C2×C4).20D28
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C23×D7 — C2×D14⋊C4 — (C2×C4).20D28
 Lower central C7 — C22×C14 — (C2×C4).20D28
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C4).20D28
G = < a,b,c,d | a2=b28=c4=1, d2=b14, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=b14c-1 >

Subgroups: 1148 in 202 conjugacy classes, 61 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C23, D7, C14, C14, C22⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, Dic7, C28, D14, C2×C14, C2×C14, C2.C42, C2.C42, C2×C22⋊C4, C22×Q8, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C23⋊Q8, D14⋊C4, C2×Dic14, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C23×D7, C14.C42, C7×C2.C42, C2×D14⋊C4, C2×D14⋊C4, C22×Dic14, (C2×C4).20D28
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22≀C2, C22⋊Q8, C4.4D4, D28, C22×D7, C23⋊Q8, C2×D28, C4○D28, D4×D7, D42D7, Q8×D7, C4.D28, C22⋊D28, Dic7.D4, D14⋊Q8, D142Q8, (C2×C4).20D28

Smallest permutation representation of (C2×C4).20D28
On 224 points
Generators in S224
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 57)(26 58)(27 59)(28 60)(29 219)(30 220)(31 221)(32 222)(33 223)(34 224)(35 197)(36 198)(37 199)(38 200)(39 201)(40 202)(41 203)(42 204)(43 205)(44 206)(45 207)(46 208)(47 209)(48 210)(49 211)(50 212)(51 213)(52 214)(53 215)(54 216)(55 217)(56 218)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)(93 136)(94 137)(95 138)(96 139)(97 140)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(106 121)(107 122)(108 123)(109 124)(110 125)(111 126)(112 127)(141 186)(142 187)(143 188)(144 189)(145 190)(146 191)(147 192)(148 193)(149 194)(150 195)(151 196)(152 169)(153 170)(154 171)(155 172)(156 173)(157 174)(158 175)(159 176)(160 177)(161 178)(162 179)(163 180)(164 181)(165 182)(166 183)(167 184)(168 185)
(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)
(1 224 129 153)(2 35 130 171)(3 198 131 155)(4 37 132 173)(5 200 133 157)(6 39 134 175)(7 202 135 159)(8 41 136 177)(9 204 137 161)(10 43 138 179)(11 206 139 163)(12 45 140 181)(13 208 113 165)(14 47 114 183)(15 210 115 167)(16 49 116 185)(17 212 117 141)(18 51 118 187)(19 214 119 143)(20 53 120 189)(21 216 121 145)(22 55 122 191)(23 218 123 147)(24 29 124 193)(25 220 125 149)(26 31 126 195)(27 222 127 151)(28 33 128 169)(30 110 194 57)(32 112 196 59)(34 86 170 61)(36 88 172 63)(38 90 174 65)(40 92 176 67)(42 94 178 69)(44 96 180 71)(46 98 182 73)(48 100 184 75)(50 102 186 77)(52 104 188 79)(54 106 190 81)(56 108 192 83)(58 221 111 150)(60 223 85 152)(62 197 87 154)(64 199 89 156)(66 201 91 158)(68 203 93 160)(70 205 95 162)(72 207 97 164)(74 209 99 166)(76 211 101 168)(78 213 103 142)(80 215 105 144)(82 217 107 146)(84 219 109 148)
(1 184 15 170)(2 183 16 169)(3 182 17 196)(4 181 18 195)(5 180 19 194)(6 179 20 193)(7 178 21 192)(8 177 22 191)(9 176 23 190)(10 175 24 189)(11 174 25 188)(12 173 26 187)(13 172 27 186)(14 171 28 185)(29 134 43 120)(30 133 44 119)(31 132 45 118)(32 131 46 117)(33 130 47 116)(34 129 48 115)(35 128 49 114)(36 127 50 113)(37 126 51 140)(38 125 52 139)(39 124 53 138)(40 123 54 137)(41 122 55 136)(42 121 56 135)(57 143 71 157)(58 142 72 156)(59 141 73 155)(60 168 74 154)(61 167 75 153)(62 166 76 152)(63 165 77 151)(64 164 78 150)(65 163 79 149)(66 162 80 148)(67 161 81 147)(68 160 82 146)(69 159 83 145)(70 158 84 144)(85 211 99 197)(86 210 100 224)(87 209 101 223)(88 208 102 222)(89 207 103 221)(90 206 104 220)(91 205 105 219)(92 204 106 218)(93 203 107 217)(94 202 108 216)(95 201 109 215)(96 200 110 214)(97 199 111 213)(98 198 112 212)

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

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

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

82 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4F 4G ··· 4L 7A 7B 7C 14A ··· 14U 28A ··· 28AJ order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 28 28 4 ··· 4 28 ··· 28 2 2 2 2 ··· 2 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - + + + + - - image C1 C2 C2 C2 C2 D4 D4 Q8 D7 C4○D4 D14 D28 C4○D28 D4×D7 D4⋊2D7 Q8×D7 kernel (C2×C4).20D28 C14.C42 C7×C2.C42 C2×D14⋊C4 C22×Dic14 C2×Dic7 C2×C28 C22×D7 C2.C42 C2×C14 C22×C4 C2×C4 C22 C22 C22 C22 # reps 1 2 1 3 1 4 2 2 3 6 9 12 24 6 3 3

Matrix representation of (C2×C4).20D28 in GL6(𝔽29)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 26 11 0 0 0 0 21 0 0 0 0 0 0 0 27 24 0 0 0 0 5 12 0 0 0 0 0 0 17 0 0 0 0 0 2 12
,
 2 24 0 0 0 0 1 27 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 16 18 0 0 0 0 10 13
,
 11 7 0 0 0 0 16 18 0 0 0 0 0 0 12 0 0 0 0 0 7 17 0 0 0 0 0 0 13 11 0 0 0 0 3 16

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[26,21,0,0,0,0,11,0,0,0,0,0,0,0,27,5,0,0,0,0,24,12,0,0,0,0,0,0,17,2,0,0,0,0,0,12],[2,1,0,0,0,0,24,27,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,16,10,0,0,0,0,18,13],[11,16,0,0,0,0,7,18,0,0,0,0,0,0,12,7,0,0,0,0,0,17,0,0,0,0,0,0,13,3,0,0,0,0,11,16] >;

(C2×C4).20D28 in GAP, Magma, Sage, TeX

(C_2\times C_4)._{20}D_{28}
% in TeX

G:=Group("(C2xC4).20D28");
// GroupNames label

G:=SmallGroup(448,207);
// by ID

G=gap.SmallGroup(448,207);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,64,926,387,268,18822]);
// Polycyclic

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

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