metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.2D28, C28⋊10SD16, C42.52D14, (C4×D4).9D7, C4⋊3(D4.D7), C28⋊C8⋊25C2, (C7×D4).19D4, C4.15(C2×D28), C28.19(C2×D4), (C2×C28).62D4, C4⋊C4.247D14, C28⋊2Q8⋊16C2, (D4×C28).10C2, C7⋊3(D4.D4), (C2×D4).194D14, C4.11(C4○D28), C28.54(C4○D4), C14.Q16⋊31C2, (C4×C28).90C22, C14.53(C2×SD16), C14.66(C4⋊D4), C2.14(C28⋊7D4), (C2×C28).341C23, C2.8(D4.9D14), (D4×C14).236C22, C14.109(C8.C22), (C2×Dic14).100C22, C2.7(C2×D4.D7), (C2×C7⋊C8).96C22, (C2×D4.D7).5C2, (C2×C14).472(C2×D4), (C2×C4).246(C7⋊D4), (C7×C4⋊C4).278C22, (C2×C4).441(C22×D7), C22.151(C2×C7⋊D4), SmallGroup(448,553)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.2D28
G = < a,b,c,d | a4=b2=c28=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Subgroups: 500 in 120 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, C28, C2×C14, C2×C14, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, D4.D4, C2×C7⋊C8, C4⋊Dic7, D4.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C22×C28, D4×C14, C28⋊C8, C14.Q16, C28⋊2Q8, C2×D4.D7, D4×C28, D4.2D28
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, C4○D4, D14, C4⋊D4, C2×SD16, C8.C22, D28, C7⋊D4, C22×D7, D4.D4, D4.D7, C2×D28, C4○D28, C2×C7⋊D4, C28⋊7D4, C2×D4.D7, D4.9D14, D4.2D28
►On 224 points
Generators in S224
(1 171 217 60)(2 172 218 61)(3 173 219 62)(4 174 220 63)(5 175 221 64)(6 176 222 65)(7 177 223 66)(8 178 224 67)(9 179 197 68)(10 180 198 69)(11 181 199 70)(12 182 200 71)(13 183 201 72)(14 184 202 73)(15 185 203 74)(16 186 204 75)(17 187 205 76)(18 188 206 77)(19 189 207 78)(20 190 208 79)(21 191 209 80)(22 192 210 81)(23 193 211 82)(24 194 212 83)(25 195 213 84)(26 196 214 57)(27 169 215 58)(28 170 216 59)(29 99 151 118)(30 100 152 119)(31 101 153 120)(32 102 154 121)(33 103 155 122)(34 104 156 123)(35 105 157 124)(36 106 158 125)(37 107 159 126)(38 108 160 127)(39 109 161 128)(40 110 162 129)(41 111 163 130)(42 112 164 131)(43 85 165 132)(44 86 166 133)(45 87 167 134)(46 88 168 135)(47 89 141 136)(48 90 142 137)(49 91 143 138)(50 92 144 139)(51 93 145 140)(52 94 146 113)(53 95 147 114)(54 96 148 115)(55 97 149 116)(56 98 150 117)
(1 60)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 84)(26 57)(27 58)(28 59)(29 151)(30 152)(31 153)(32 154)(33 155)(34 156)(35 157)(36 158)(37 159)(38 160)(39 161)(40 162)(41 163)(42 164)(43 165)(44 166)(45 167)(46 168)(47 141)(48 142)(49 143)(50 144)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(169 215)(170 216)(171 217)(172 218)(173 219)(174 220)(175 221)(176 222)(177 223)(178 224)(179 197)(180 198)(181 199)(182 200)(183 201)(184 202)(185 203)(186 204)(187 205)(188 206)(189 207)(190 208)(191 209)(192 210)(193 211)(194 212)(195 213)(196 214)
(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 33 217 155)(2 32 218 154)(3 31 219 153)(4 30 220 152)(5 29 221 151)(6 56 222 150)(7 55 223 149)(8 54 224 148)(9 53 197 147)(10 52 198 146)(11 51 199 145)(12 50 200 144)(13 49 201 143)(14 48 202 142)(15 47 203 141)(16 46 204 168)(17 45 205 167)(18 44 206 166)(19 43 207 165)(20 42 208 164)(21 41 209 163)(22 40 210 162)(23 39 211 161)(24 38 212 160)(25 37 213 159)(26 36 214 158)(27 35 215 157)(28 34 216 156)(57 106 196 125)(58 105 169 124)(59 104 170 123)(60 103 171 122)(61 102 172 121)(62 101 173 120)(63 100 174 119)(64 99 175 118)(65 98 176 117)(66 97 177 116)(67 96 178 115)(68 95 179 114)(69 94 180 113)(70 93 181 140)(71 92 182 139)(72 91 183 138)(73 90 184 137)(74 89 185 136)(75 88 186 135)(76 87 187 134)(77 86 188 133)(78 85 189 132)(79 112 190 131)(80 111 191 130)(81 110 192 129)(82 109 193 128)(83 108 194 127)(84 107 195 126)
G:=sub<Sym(224)| (1,171,217,60)(2,172,218,61)(3,173,219,62)(4,174,220,63)(5,175,221,64)(6,176,222,65)(7,177,223,66)(8,178,224,67)(9,179,197,68)(10,180,198,69)(11,181,199,70)(12,182,200,71)(13,183,201,72)(14,184,202,73)(15,185,203,74)(16,186,204,75)(17,187,205,76)(18,188,206,77)(19,189,207,78)(20,190,208,79)(21,191,209,80)(22,192,210,81)(23,193,211,82)(24,194,212,83)(25,195,213,84)(26,196,214,57)(27,169,215,58)(28,170,216,59)(29,99,151,118)(30,100,152,119)(31,101,153,120)(32,102,154,121)(33,103,155,122)(34,104,156,123)(35,105,157,124)(36,106,158,125)(37,107,159,126)(38,108,160,127)(39,109,161,128)(40,110,162,129)(41,111,163,130)(42,112,164,131)(43,85,165,132)(44,86,166,133)(45,87,167,134)(46,88,168,135)(47,89,141,136)(48,90,142,137)(49,91,143,138)(50,92,144,139)(51,93,145,140)(52,94,146,113)(53,95,147,114)(54,96,148,115)(55,97,149,116)(56,98,150,117), (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,57)(27,58)(28,59)(29,151)(30,152)(31,153)(32,154)(33,155)(34,156)(35,157)(36,158)(37,159)(38,160)(39,161)(40,162)(41,163)(42,164)(43,165)(44,166)(45,167)(46,168)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(169,215)(170,216)(171,217)(172,218)(173,219)(174,220)(175,221)(176,222)(177,223)(178,224)(179,197)(180,198)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)(187,205)(188,206)(189,207)(190,208)(191,209)(192,210)(193,211)(194,212)(195,213)(196,214), (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), (1,33,217,155)(2,32,218,154)(3,31,219,153)(4,30,220,152)(5,29,221,151)(6,56,222,150)(7,55,223,149)(8,54,224,148)(9,53,197,147)(10,52,198,146)(11,51,199,145)(12,50,200,144)(13,49,201,143)(14,48,202,142)(15,47,203,141)(16,46,204,168)(17,45,205,167)(18,44,206,166)(19,43,207,165)(20,42,208,164)(21,41,209,163)(22,40,210,162)(23,39,211,161)(24,38,212,160)(25,37,213,159)(26,36,214,158)(27,35,215,157)(28,34,216,156)(57,106,196,125)(58,105,169,124)(59,104,170,123)(60,103,171,122)(61,102,172,121)(62,101,173,120)(63,100,174,119)(64,99,175,118)(65,98,176,117)(66,97,177,116)(67,96,178,115)(68,95,179,114)(69,94,180,113)(70,93,181,140)(71,92,182,139)(72,91,183,138)(73,90,184,137)(74,89,185,136)(75,88,186,135)(76,87,187,134)(77,86,188,133)(78,85,189,132)(79,112,190,131)(80,111,191,130)(81,110,192,129)(82,109,193,128)(83,108,194,127)(84,107,195,126)>;
G:=Group( (1,171,217,60)(2,172,218,61)(3,173,219,62)(4,174,220,63)(5,175,221,64)(6,176,222,65)(7,177,223,66)(8,178,224,67)(9,179,197,68)(10,180,198,69)(11,181,199,70)(12,182,200,71)(13,183,201,72)(14,184,202,73)(15,185,203,74)(16,186,204,75)(17,187,205,76)(18,188,206,77)(19,189,207,78)(20,190,208,79)(21,191,209,80)(22,192,210,81)(23,193,211,82)(24,194,212,83)(25,195,213,84)(26,196,214,57)(27,169,215,58)(28,170,216,59)(29,99,151,118)(30,100,152,119)(31,101,153,120)(32,102,154,121)(33,103,155,122)(34,104,156,123)(35,105,157,124)(36,106,158,125)(37,107,159,126)(38,108,160,127)(39,109,161,128)(40,110,162,129)(41,111,163,130)(42,112,164,131)(43,85,165,132)(44,86,166,133)(45,87,167,134)(46,88,168,135)(47,89,141,136)(48,90,142,137)(49,91,143,138)(50,92,144,139)(51,93,145,140)(52,94,146,113)(53,95,147,114)(54,96,148,115)(55,97,149,116)(56,98,150,117), (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,57)(27,58)(28,59)(29,151)(30,152)(31,153)(32,154)(33,155)(34,156)(35,157)(36,158)(37,159)(38,160)(39,161)(40,162)(41,163)(42,164)(43,165)(44,166)(45,167)(46,168)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(169,215)(170,216)(171,217)(172,218)(173,219)(174,220)(175,221)(176,222)(177,223)(178,224)(179,197)(180,198)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)(187,205)(188,206)(189,207)(190,208)(191,209)(192,210)(193,211)(194,212)(195,213)(196,214), (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), (1,33,217,155)(2,32,218,154)(3,31,219,153)(4,30,220,152)(5,29,221,151)(6,56,222,150)(7,55,223,149)(8,54,224,148)(9,53,197,147)(10,52,198,146)(11,51,199,145)(12,50,200,144)(13,49,201,143)(14,48,202,142)(15,47,203,141)(16,46,204,168)(17,45,205,167)(18,44,206,166)(19,43,207,165)(20,42,208,164)(21,41,209,163)(22,40,210,162)(23,39,211,161)(24,38,212,160)(25,37,213,159)(26,36,214,158)(27,35,215,157)(28,34,216,156)(57,106,196,125)(58,105,169,124)(59,104,170,123)(60,103,171,122)(61,102,172,121)(62,101,173,120)(63,100,174,119)(64,99,175,118)(65,98,176,117)(66,97,177,116)(67,96,178,115)(68,95,179,114)(69,94,180,113)(70,93,181,140)(71,92,182,139)(72,91,183,138)(73,90,184,137)(74,89,185,136)(75,88,186,135)(76,87,187,134)(77,86,188,133)(78,85,189,132)(79,112,190,131)(80,111,191,130)(81,110,192,129)(82,109,193,128)(83,108,194,127)(84,107,195,126) );
G=PermutationGroup([[(1,171,217,60),(2,172,218,61),(3,173,219,62),(4,174,220,63),(5,175,221,64),(6,176,222,65),(7,177,223,66),(8,178,224,67),(9,179,197,68),(10,180,198,69),(11,181,199,70),(12,182,200,71),(13,183,201,72),(14,184,202,73),(15,185,203,74),(16,186,204,75),(17,187,205,76),(18,188,206,77),(19,189,207,78),(20,190,208,79),(21,191,209,80),(22,192,210,81),(23,193,211,82),(24,194,212,83),(25,195,213,84),(26,196,214,57),(27,169,215,58),(28,170,216,59),(29,99,151,118),(30,100,152,119),(31,101,153,120),(32,102,154,121),(33,103,155,122),(34,104,156,123),(35,105,157,124),(36,106,158,125),(37,107,159,126),(38,108,160,127),(39,109,161,128),(40,110,162,129),(41,111,163,130),(42,112,164,131),(43,85,165,132),(44,86,166,133),(45,87,167,134),(46,88,168,135),(47,89,141,136),(48,90,142,137),(49,91,143,138),(50,92,144,139),(51,93,145,140),(52,94,146,113),(53,95,147,114),(54,96,148,115),(55,97,149,116),(56,98,150,117)], [(1,60),(2,61),(3,62),(4,63),(5,64),(6,65),(7,66),(8,67),(9,68),(10,69),(11,70),(12,71),(13,72),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(25,84),(26,57),(27,58),(28,59),(29,151),(30,152),(31,153),(32,154),(33,155),(34,156),(35,157),(36,158),(37,159),(38,160),(39,161),(40,162),(41,163),(42,164),(43,165),(44,166),(45,167),(46,168),(47,141),(48,142),(49,143),(50,144),(51,145),(52,146),(53,147),(54,148),(55,149),(56,150),(169,215),(170,216),(171,217),(172,218),(173,219),(174,220),(175,221),(176,222),(177,223),(178,224),(179,197),(180,198),(181,199),(182,200),(183,201),(184,202),(185,203),(186,204),(187,205),(188,206),(189,207),(190,208),(191,209),(192,210),(193,211),(194,212),(195,213),(196,214)], [(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)], [(1,33,217,155),(2,32,218,154),(3,31,219,153),(4,30,220,152),(5,29,221,151),(6,56,222,150),(7,55,223,149),(8,54,224,148),(9,53,197,147),(10,52,198,146),(11,51,199,145),(12,50,200,144),(13,49,201,143),(14,48,202,142),(15,47,203,141),(16,46,204,168),(17,45,205,167),(18,44,206,166),(19,43,207,165),(20,42,208,164),(21,41,209,163),(22,40,210,162),(23,39,211,161),(24,38,212,160),(25,37,213,159),(26,36,214,158),(27,35,215,157),(28,34,216,156),(57,106,196,125),(58,105,169,124),(59,104,170,123),(60,103,171,122),(61,102,172,121),(62,101,173,120),(63,100,174,119),(64,99,175,118),(65,98,176,117),(66,97,177,116),(67,96,178,115),(68,95,179,114),(69,94,180,113),(70,93,181,140),(71,92,182,139),(72,91,183,138),(73,90,184,137),(74,89,185,136),(75,88,186,135),(76,87,187,134),(77,86,188,133),(78,85,189,132),(79,112,190,131),(80,111,191,130),(81,110,192,129),(82,109,193,128),(83,108,194,127),(84,107,195,126)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 56 | 56 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
79 irreducible representations
| dim | 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 | D4 | D4 | D7 | SD16 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | D28 | C4○D28 | C8.C22 | D4.D7 | D4.9D14 |
| kernel | D4.2D28 | C28⋊C8 | C14.Q16 | C28⋊2Q8 | C2×D4.D7 | D4×C28 | C2×C28 | C7×D4 | C4×D4 | C28 | C28 | C42 | C4⋊C4 | C2×D4 | C2×C4 | D4 | C4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 3 | 3 | 12 | 12 | 12 | 1 | 6 | 6 |
Matrix representation of D4.2D28 ►in GL4(𝔽113) generated by
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 100 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 56 | 0 |
| 0 | 0 | 38 | 111 |
| 13 | 100 | 0 | 0 |
| 100 | 100 | 0 | 0 |
| 0 | 0 | 79 | 40 |
| 0 | 0 | 70 | 34 |
G:=sub<GL(4,GF(113))| [0,112,0,0,1,0,0,0,0,0,112,0,0,0,0,112],[0,1,0,0,1,0,0,0,0,0,112,100,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,56,38,0,0,0,111],[13,100,0,0,100,100,0,0,0,0,79,70,0,0,40,34] >;
D4.2D28 in GAP, Magma, Sage, TeX
D_4._2D_{28} % in TeX
G:=Group("D4.2D28"); // GroupNames label
G:=SmallGroup(448,553);
// by ID
G=gap.SmallGroup(448,553);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,120,254,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^28=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations