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## G = D7×Q32order 448 = 26·7

### Direct product of D7 and Q32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — D7×Q32
 Chief series C1 — C7 — C14 — C28 — C56 — C8×D7 — D7×Q16 — D7×Q32
 Lower central C7 — C14 — C28 — C56 — D7×Q32
 Upper central C1 — C2 — C4 — C8 — Q32

Generators and relations for D7×Q32
G = < a,b,c,d | a7=b2=c16=1, d2=c8, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 464 in 82 conjugacy classes, 33 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, Q8, D7, C14, C16, C16, C2×C8, Q16, Q16, C2×Q8, Dic7, Dic7, C28, C28, D14, C2×C16, Q32, Q32, C2×Q16, C7⋊C8, C56, Dic14, C4×D7, C4×D7, C7×Q8, C2×Q32, C7⋊C16, C112, C8×D7, Dic28, C7⋊Q16, C7×Q16, Q8×D7, D7×C16, Dic56, C7⋊Q32, C7×Q32, D7×Q16, D7×Q32
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, Q32, C2×D8, C22×D7, C2×Q32, D4×D7, D7×D8, D7×Q32

Smallest permutation representation of D7×Q32
On 224 points
Generators in S224
(1 223 202 113 62 160 186)(2 224 203 114 63 145 187)(3 209 204 115 64 146 188)(4 210 205 116 49 147 189)(5 211 206 117 50 148 190)(6 212 207 118 51 149 191)(7 213 208 119 52 150 192)(8 214 193 120 53 151 177)(9 215 194 121 54 152 178)(10 216 195 122 55 153 179)(11 217 196 123 56 154 180)(12 218 197 124 57 155 181)(13 219 198 125 58 156 182)(14 220 199 126 59 157 183)(15 221 200 127 60 158 184)(16 222 201 128 61 159 185)(17 161 98 141 72 43 85)(18 162 99 142 73 44 86)(19 163 100 143 74 45 87)(20 164 101 144 75 46 88)(21 165 102 129 76 47 89)(22 166 103 130 77 48 90)(23 167 104 131 78 33 91)(24 168 105 132 79 34 92)(25 169 106 133 80 35 93)(26 170 107 134 65 36 94)(27 171 108 135 66 37 95)(28 172 109 136 67 38 96)(29 173 110 137 68 39 81)(30 174 111 138 69 40 82)(31 175 112 139 70 41 83)(32 176 97 140 71 42 84)
(1 186)(2 187)(3 188)(4 189)(5 190)(6 191)(7 192)(8 177)(9 178)(10 179)(11 180)(12 181)(13 182)(14 183)(15 184)(16 185)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 81)(30 82)(31 83)(32 84)(33 167)(34 168)(35 169)(36 170)(37 171)(38 172)(39 173)(40 174)(41 175)(42 176)(43 161)(44 162)(45 163)(46 164)(47 165)(48 166)(49 205)(50 206)(51 207)(52 208)(53 193)(54 194)(55 195)(56 196)(57 197)(58 198)(59 199)(60 200)(61 201)(62 202)(63 203)(64 204)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(145 224)(146 209)(147 210)(148 211)(149 212)(150 213)(151 214)(152 215)(153 216)(154 217)(155 218)(156 219)(157 220)(158 221)(159 222)(160 223)
(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 9 18)(2 25 10 17)(3 24 11 32)(4 23 12 31)(5 22 13 30)(6 21 14 29)(7 20 15 28)(8 19 16 27)(33 155 41 147)(34 154 42 146)(35 153 43 145)(36 152 44 160)(37 151 45 159)(38 150 46 158)(39 149 47 157)(40 148 48 156)(49 78 57 70)(50 77 58 69)(51 76 59 68)(52 75 60 67)(53 74 61 66)(54 73 62 65)(55 72 63 80)(56 71 64 79)(81 191 89 183)(82 190 90 182)(83 189 91 181)(84 188 92 180)(85 187 93 179)(86 186 94 178)(87 185 95 177)(88 184 96 192)(97 204 105 196)(98 203 106 195)(99 202 107 194)(100 201 108 193)(101 200 109 208)(102 199 110 207)(103 198 111 206)(104 197 112 205)(113 134 121 142)(114 133 122 141)(115 132 123 140)(116 131 124 139)(117 130 125 138)(118 129 126 137)(119 144 127 136)(120 143 128 135)(161 224 169 216)(162 223 170 215)(163 222 171 214)(164 221 172 213)(165 220 173 212)(166 219 174 211)(167 218 175 210)(168 217 176 209)

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

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

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

55 conjugacy classes

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

55 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 D4 D4 D7 D8 D8 D14 D14 Q32 D4×D7 D7×D8 D7×Q32 kernel D7×Q32 D7×C16 Dic56 C7⋊Q32 C7×Q32 D7×Q16 C7⋊C8 C4×D7 Q32 Dic7 D14 C16 Q16 D7 C4 C2 C1 # reps 1 1 1 2 1 2 1 1 3 2 2 3 6 8 3 6 12

Matrix representation of D7×Q32 in GL4(𝔽113) generated by

 0 1 0 0 112 9 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 1
,
 1 0 0 0 0 1 0 0 0 0 14 36 0 0 95 91
,
 1 0 0 0 0 1 0 0 0 0 63 3 0 0 108 50
G:=sub<GL(4,GF(113))| [0,112,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,14,95,0,0,36,91],[1,0,0,0,0,1,0,0,0,0,63,108,0,0,3,50] >;

D7×Q32 in GAP, Magma, Sage, TeX

D_7\times Q_{32}
% in TeX

G:=Group("D7xQ32");
// GroupNames label

G:=SmallGroup(448,451);
// by ID

G=gap.SmallGroup(448,451);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,135,184,346,185,192,851,438,102,18822]);
// Polycyclic

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

׿
×
𝔽