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G = SD32⋊D7order 448 = 26·7

2nd semidirect product of SD32 and D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: SD322D7, D14.8D8, D8.4D14, C16.2D14, Dic566C2, Q16.1D14, C56.18C23, C112.9C22, Dic7.10D8, Dic28.3C22, C7⋊C8.4D4, C4.6(D4×D7), D8.D74C2, D83D7.C2, (D7×Q16)⋊4C2, (C4×D7).9D4, C7⋊Q321C2, C2.21(D7×D8), C16⋊D72C2, (C7×SD32)⋊2C2, C14.37(C2×D8), C28.12(C2×D4), C72(Q32⋊C2), C7⋊C16.1C22, (C8×D7).5C22, (C7×D8).4C22, C8.24(C22×D7), (C7×Q16).2C22, SmallGroup(448,449)

Series: Derived Chief Lower central Upper central

C1C56 — SD32⋊D7
C1C7C14C28C56C8×D7D7×Q16 — SD32⋊D7
C7C14C28C56 — SD32⋊D7
C1C2C4C8SD32

Generators and relations for SD32⋊D7
 G = < a,b,c,d | a16=b2=c7=d2=1, bab=a7, ac=ca, dad=a9, bc=cb, dbd=a8b, dcd=c-1 >

Subgroups: 480 in 82 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, Q8, D7, C14, C14, C16, C16, C2×C8, D8, SD16, Q16, Q16, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, C2×C14, M5(2), SD32, SD32, Q32, C2×Q16, C4○D8, C7⋊C8, C56, Dic14, C4×D7, C4×D7, C2×Dic7, C7⋊D4, C7×D4, C7×Q8, Q32⋊C2, C7⋊C16, C112, C8×D7, Dic28, D4.D7, C7⋊Q16, C7×D8, C7×Q16, D42D7, Q8×D7, C16⋊D7, Dic56, D8.D7, C7⋊Q32, C7×SD32, D83D7, D7×Q16, SD32⋊D7
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C22×D7, Q32⋊C2, D4×D7, D7×D8, SD32⋊D7

Smallest permutation representation of SD32⋊D7
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)
(1 181)(2 188)(3 179)(4 186)(5 177)(6 184)(7 191)(8 182)(9 189)(10 180)(11 187)(12 178)(13 185)(14 192)(15 183)(16 190)(17 196)(18 203)(19 194)(20 201)(21 208)(22 199)(23 206)(24 197)(25 204)(26 195)(27 202)(28 193)(29 200)(30 207)(31 198)(32 205)(33 120)(34 127)(35 118)(36 125)(37 116)(38 123)(39 114)(40 121)(41 128)(42 119)(43 126)(44 117)(45 124)(46 115)(47 122)(48 113)(49 216)(50 223)(51 214)(52 221)(53 212)(54 219)(55 210)(56 217)(57 224)(58 215)(59 222)(60 213)(61 220)(62 211)(63 218)(64 209)(65 130)(66 137)(67 144)(68 135)(69 142)(70 133)(71 140)(72 131)(73 138)(74 129)(75 136)(76 143)(77 134)(78 141)(79 132)(80 139)(81 102)(82 109)(83 100)(84 107)(85 98)(86 105)(87 112)(88 103)(89 110)(90 101)(91 108)(92 99)(93 106)(94 97)(95 104)(96 111)(145 175)(146 166)(147 173)(148 164)(149 171)(150 162)(151 169)(152 176)(153 167)(154 174)(155 165)(156 172)(157 163)(158 170)(159 161)(160 168)
(1 94 156 39 21 137 221)(2 95 157 40 22 138 222)(3 96 158 41 23 139 223)(4 81 159 42 24 140 224)(5 82 160 43 25 141 209)(6 83 145 44 26 142 210)(7 84 146 45 27 143 211)(8 85 147 46 28 144 212)(9 86 148 47 29 129 213)(10 87 149 48 30 130 214)(11 88 150 33 31 131 215)(12 89 151 34 32 132 216)(13 90 152 35 17 133 217)(14 91 153 36 18 134 218)(15 92 154 37 19 135 219)(16 93 155 38 20 136 220)(49 178 110 169 127 205 79)(50 179 111 170 128 206 80)(51 180 112 171 113 207 65)(52 181 97 172 114 208 66)(53 182 98 173 115 193 67)(54 183 99 174 116 194 68)(55 184 100 175 117 195 69)(56 185 101 176 118 196 70)(57 186 102 161 119 197 71)(58 187 103 162 120 198 72)(59 188 104 163 121 199 73)(60 189 105 164 122 200 74)(61 190 106 165 123 201 75)(62 191 107 166 124 202 76)(63 192 108 167 125 203 77)(64 177 109 168 126 204 78)
(1 221)(2 214)(3 223)(4 216)(5 209)(6 218)(7 211)(8 220)(9 213)(10 222)(11 215)(12 224)(13 217)(14 210)(15 219)(16 212)(17 152)(18 145)(19 154)(20 147)(21 156)(22 149)(23 158)(24 151)(25 160)(26 153)(27 146)(28 155)(29 148)(30 157)(31 150)(32 159)(34 42)(36 44)(38 46)(40 48)(49 178)(50 187)(51 180)(52 189)(53 182)(54 191)(55 184)(56 177)(57 186)(58 179)(59 188)(60 181)(61 190)(62 183)(63 192)(64 185)(65 112)(66 105)(67 98)(68 107)(69 100)(70 109)(71 102)(72 111)(73 104)(74 97)(75 106)(76 99)(77 108)(78 101)(79 110)(80 103)(81 132)(82 141)(83 134)(84 143)(85 136)(86 129)(87 138)(88 131)(89 140)(90 133)(91 142)(92 135)(93 144)(94 137)(95 130)(96 139)(114 122)(116 124)(118 126)(120 128)(161 197)(162 206)(163 199)(164 208)(165 201)(166 194)(167 203)(168 196)(169 205)(170 198)(171 207)(172 200)(173 193)(174 202)(175 195)(176 204)

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

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

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

49 conjugacy classes

class 1 2A2B2C4A4B4C4D4E7A7B7C8A8B8C14A14B14C14D14E14F16A16B16C16D28A28B28C28D28E28F56A···56F112A···112L
order1222444447778881414141414141616161628282828282856···56112···112
size118142814565622222282221616164428284441616164···44···4

49 irreducible representations

dim11111111222222224444
type++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2D4D4D7D8D8D14D14D14Q32⋊C2D4×D7D7×D8SD32⋊D7
kernelSD32⋊D7C16⋊D7Dic56D8.D7C7⋊Q32C7×SD32D83D7D7×Q16C7⋊C8C4×D7SD32Dic7D14C16D8Q16C7C4C2C1
# reps111111111132233323612

Matrix representation of SD32⋊D7 in GL8(𝔽113)

7054340000
077900000
04810600000
657001060000
000022951587
0000188569104
000083673
0000747695112
,
1003270000
0108600000
07910300000
345401030000
0000691360
0000340112
000078894422
000010618108109
,
91000000
1120000000
008810000
0053340000
00001000
00000100
00000010
00000001
,
8033000000
10433000000
003490000
0060790000
00001000
00000100
000015691120
0000680112

G:=sub<GL(8,GF(113))| [7,0,0,65,0,0,0,0,0,7,48,70,0,0,0,0,54,79,106,0,0,0,0,0,34,0,0,106,0,0,0,0,0,0,0,0,22,18,8,74,0,0,0,0,95,85,36,76,0,0,0,0,15,69,7,95,0,0,0,0,87,104,3,112],[10,0,0,34,0,0,0,0,0,10,79,54,0,0,0,0,3,86,103,0,0,0,0,0,27,0,0,103,0,0,0,0,0,0,0,0,69,3,78,106,0,0,0,0,1,4,89,18,0,0,0,0,36,0,44,108,0,0,0,0,0,112,22,109],[9,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,88,53,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[80,104,0,0,0,0,0,0,33,33,0,0,0,0,0,0,0,0,34,60,0,0,0,0,0,0,9,79,0,0,0,0,0,0,0,0,1,0,15,6,0,0,0,0,0,1,69,8,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112] >;

SD32⋊D7 in GAP, Magma, Sage, TeX

{\rm SD}_{32}\rtimes D_7
% in TeX

G:=Group("SD32:D7");
// GroupNames label

G:=SmallGroup(448,449);
// by ID

G=gap.SmallGroup(448,449);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,135,346,185,192,851,438,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^16=b^2=c^7=d^2=1,b*a*b=a^7,a*c=c*a,d*a*d=a^9,b*c=c*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations

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