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## G = SD32⋊3D7order 448 = 26·7

### The semidirect product of SD32 and D7 acting through Inn(SD32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — SD32⋊3D7
 Chief series C1 — C7 — C14 — C28 — C56 — C8×D7 — D8⋊3D7 — SD32⋊3D7
 Lower central C7 — C14 — C28 — C56 — SD32⋊3D7
 Upper central C1 — C2 — C4 — C8 — SD32

Generators and relations for SD323D7
G = < a,b,c,d | a16=b2=c7=d2=1, bab=a7, ac=ca, ad=da, bc=cb, dbd=a8b, dcd=c-1 >

Subgroups: 544 in 84 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, D8, SD16, Q16, Q16, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C16, D16, SD32, SD32, Q32, C4○D8, C7⋊C8, C56, Dic14, C4×D7, C4×D7, D28, C2×Dic7, C7⋊D4, C7×D4, C7×Q8, C4○D16, C7⋊C16, C112, C8×D7, D56, Dic28, D4.D7, Q8⋊D7, C7×D8, C7×Q16, D42D7, Q82D7, D7×C16, C112⋊C2, C7⋊D16, C7⋊Q32, C7×SD32, D83D7, Q8.D14, SD323D7
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C22×D7, C4○D16, D4×D7, D7×D8, SD323D7

Smallest permutation representation of SD323D7
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 26)(2 17)(3 24)(4 31)(5 22)(6 29)(7 20)(8 27)(9 18)(10 25)(11 32)(12 23)(13 30)(14 21)(15 28)(16 19)(33 64)(34 55)(35 62)(36 53)(37 60)(38 51)(39 58)(40 49)(41 56)(42 63)(43 54)(44 61)(45 52)(46 59)(47 50)(48 57)(65 127)(66 118)(67 125)(68 116)(69 123)(70 114)(71 121)(72 128)(73 119)(74 126)(75 117)(76 124)(77 115)(78 122)(79 113)(80 120)(81 140)(82 131)(83 138)(84 129)(85 136)(86 143)(87 134)(88 141)(89 132)(90 139)(91 130)(92 137)(93 144)(94 135)(95 142)(96 133)(97 173)(98 164)(99 171)(100 162)(101 169)(102 176)(103 167)(104 174)(105 165)(106 172)(107 163)(108 170)(109 161)(110 168)(111 175)(112 166)(145 221)(146 212)(147 219)(148 210)(149 217)(150 224)(151 215)(152 222)(153 213)(154 220)(155 211)(156 218)(157 209)(158 216)(159 223)(160 214)(177 205)(178 196)(179 203)(180 194)(181 201)(182 208)(183 199)(184 206)(185 197)(186 204)(187 195)(188 202)(189 193)(190 200)(191 207)(192 198)
(1 72 148 36 86 207 98)(2 73 149 37 87 208 99)(3 74 150 38 88 193 100)(4 75 151 39 89 194 101)(5 76 152 40 90 195 102)(6 77 153 41 91 196 103)(7 78 154 42 92 197 104)(8 79 155 43 93 198 105)(9 80 156 44 94 199 106)(10 65 157 45 95 200 107)(11 66 158 46 96 201 108)(12 67 159 47 81 202 109)(13 68 160 48 82 203 110)(14 69 145 33 83 204 111)(15 70 146 34 84 205 112)(16 71 147 35 85 206 97)(17 119 217 60 134 182 171)(18 120 218 61 135 183 172)(19 121 219 62 136 184 173)(20 122 220 63 137 185 174)(21 123 221 64 138 186 175)(22 124 222 49 139 187 176)(23 125 223 50 140 188 161)(24 126 224 51 141 189 162)(25 127 209 52 142 190 163)(26 128 210 53 143 191 164)(27 113 211 54 144 192 165)(28 114 212 55 129 177 166)(29 115 213 56 130 178 167)(30 116 214 57 131 179 168)(31 117 215 58 132 180 169)(32 118 216 59 133 181 170)
(1 98)(2 99)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 97)(17 163)(18 164)(19 165)(20 166)(21 167)(22 168)(23 169)(24 170)(25 171)(26 172)(27 173)(28 174)(29 175)(30 176)(31 161)(32 162)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(65 200)(66 201)(67 202)(68 203)(69 204)(70 205)(71 206)(72 207)(73 208)(74 193)(75 194)(76 195)(77 196)(78 197)(79 198)(80 199)(81 159)(82 160)(83 145)(84 146)(85 147)(86 148)(87 149)(88 150)(89 151)(90 152)(91 153)(92 154)(93 155)(94 156)(95 157)(96 158)(113 184)(114 185)(115 186)(116 187)(117 188)(118 189)(119 190)(120 191)(121 192)(122 177)(123 178)(124 179)(125 180)(126 181)(127 182)(128 183)(129 220)(130 221)(131 222)(132 223)(133 224)(134 209)(135 210)(136 211)(137 212)(138 213)(139 214)(140 215)(141 216)(142 217)(143 218)(144 219)

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

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

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

55 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 16A 16B 16C 16D 16E 16F 16G 16H 28A 28B 28C 28D 28E 28F 56A ··· 56F 112A ··· 112L order 1 2 2 2 2 4 4 4 4 4 7 7 7 8 8 8 8 14 14 14 14 14 14 16 16 16 16 16 16 16 16 28 28 28 28 28 28 56 ··· 56 112 ··· 112 size 1 1 8 14 56 2 7 7 8 56 2 2 2 2 2 14 14 2 2 2 16 16 16 2 2 2 2 14 14 14 14 4 4 4 16 16 16 4 ··· 4 4 ··· 4

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 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 C4○D16 D4×D7 D7×D8 SD32⋊3D7 kernel SD32⋊3D7 D7×C16 C112⋊C2 C7⋊D16 C7⋊Q32 C7×SD32 D8⋊3D7 Q8.D14 C7⋊C8 C4×D7 SD32 Dic7 D14 C16 D8 Q16 C7 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 3 2 2 3 3 3 8 3 6 12

Matrix representation of SD323D7 in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 0 0 42 0 0 0 0 78
,
 1 0 0 0 0 1 0 0 0 0 0 71 0 0 78 0
,
 0 1 0 0 112 79 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 112
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,42,0,0,0,0,78],[1,0,0,0,0,1,0,0,0,0,0,78,0,0,71,0],[0,112,0,0,1,79,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,112] >;

SD323D7 in GAP, Magma, Sage, TeX

{\rm SD}_{32}\rtimes_3D_7
% in TeX

G:=Group("SD32:3D7");
// GroupNames label

G:=SmallGroup(448,450);
// by ID

G=gap.SmallGroup(448,450);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,758,135,184,346,185,192,851,438,102,18822]);
// 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,a*d=d*a,b*c=c*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽