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G = C7×D4.C8order 448 = 26·7

Direct product of C7 and D4.C8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C7×D4.C8, D4.C56, Q8.C56, C56.107D4, M5(2)⋊4C14, M4(2).3C28, (C2×C16)⋊2C14, (C2×C112)⋊6C2, C4.3(C2×C56), (C7×D4).3C8, C8.27(C7×D4), (C7×Q8).3C8, C28.32(C2×C8), C4○D4.1C28, C8○D4.2C14, (C7×M5(2))⋊12C2, (C7×M4(2)).7C4, (C2×C14).8M4(2), C14.27(C22⋊C8), (C2×C56).442C22, C28.113(C22⋊C4), C22.1(C7×M4(2)), (C7×C4○D4).5C4, (C7×C8○D4).5C2, C2.8(C7×C22⋊C8), (C2×C8).96(C2×C14), (C2×C4).42(C2×C28), C4.30(C7×C22⋊C4), (C2×C28).263(C2×C4), SmallGroup(448,154)

Series: Derived Chief Lower central Upper central

C1C4 — C7×D4.C8
C1C2C4C8C2×C8C2×C56C2×C112 — C7×D4.C8
C1C2C4 — C7×D4.C8
C1C56C2×C56 — C7×D4.C8

Generators and relations for C7×D4.C8
 G = < a,b,c,d | a7=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=bc >

2C2
4C2
2C4
2C22
2C14
4C14
2C2×C4
2D4
2C8
2C2×C14
2C28
2C2×C8
2C16
2M4(2)
2C16
2C7×D4
2C2×C28
2C56
2C7×M4(2)
2C112
2C112
2C2×C56

Smallest permutation representation of C7×D4.C8
On 224 points
Generators in S224
(1 79 214 40 113 100 52)(2 80 215 41 114 101 53)(3 65 216 42 115 102 54)(4 66 217 43 116 103 55)(5 67 218 44 117 104 56)(6 68 219 45 118 105 57)(7 69 220 46 119 106 58)(8 70 221 47 120 107 59)(9 71 222 48 121 108 60)(10 72 223 33 122 109 61)(11 73 224 34 123 110 62)(12 74 209 35 124 111 63)(13 75 210 36 125 112 64)(14 76 211 37 126 97 49)(15 77 212 38 127 98 50)(16 78 213 39 128 99 51)(17 179 201 132 159 93 165)(18 180 202 133 160 94 166)(19 181 203 134 145 95 167)(20 182 204 135 146 96 168)(21 183 205 136 147 81 169)(22 184 206 137 148 82 170)(23 185 207 138 149 83 171)(24 186 208 139 150 84 172)(25 187 193 140 151 85 173)(26 188 194 141 152 86 174)(27 189 195 142 153 87 175)(28 190 196 143 154 88 176)(29 191 197 144 155 89 161)(30 192 198 129 156 90 162)(31 177 199 130 157 91 163)(32 178 200 131 158 92 164)
(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)(33 45 41 37)(34 46 42 38)(35 47 43 39)(36 48 44 40)(49 61 57 53)(50 62 58 54)(51 63 59 55)(52 64 60 56)(65 77 73 69)(66 78 74 70)(67 79 75 71)(68 80 76 72)(81 85 89 93)(82 86 90 94)(83 87 91 95)(84 88 92 96)(97 109 105 101)(98 110 106 102)(99 111 107 103)(100 112 108 104)(113 125 121 117)(114 126 122 118)(115 127 123 119)(116 128 124 120)(129 133 137 141)(130 134 138 142)(131 135 139 143)(132 136 140 144)(145 149 153 157)(146 150 154 158)(147 151 155 159)(148 152 156 160)(161 165 169 173)(162 166 170 174)(163 167 171 175)(164 168 172 176)(177 181 185 189)(178 182 186 190)(179 183 187 191)(180 184 188 192)(193 197 201 205)(194 198 202 206)(195 199 203 207)(196 200 204 208)(209 221 217 213)(210 222 218 214)(211 223 219 215)(212 224 220 216)
(1 205)(2 202)(3 199)(4 196)(5 193)(6 206)(7 203)(8 200)(9 197)(10 194)(11 207)(12 204)(13 201)(14 198)(15 195)(16 208)(17 112)(18 101)(19 106)(20 111)(21 100)(22 105)(23 110)(24 99)(25 104)(26 109)(27 98)(28 103)(29 108)(30 97)(31 102)(32 107)(33 86)(34 83)(35 96)(36 93)(37 90)(38 87)(39 84)(40 81)(41 94)(42 91)(43 88)(44 85)(45 82)(46 95)(47 92)(48 89)(49 192)(50 189)(51 186)(52 183)(53 180)(54 177)(55 190)(56 187)(57 184)(58 181)(59 178)(60 191)(61 188)(62 185)(63 182)(64 179)(65 130)(66 143)(67 140)(68 137)(69 134)(70 131)(71 144)(72 141)(73 138)(74 135)(75 132)(76 129)(77 142)(78 139)(79 136)(80 133)(113 169)(114 166)(115 163)(116 176)(117 173)(118 170)(119 167)(120 164)(121 161)(122 174)(123 171)(124 168)(125 165)(126 162)(127 175)(128 172)(145 220)(146 209)(147 214)(148 219)(149 224)(150 213)(151 218)(152 223)(153 212)(154 217)(155 222)(156 211)(157 216)(158 221)(159 210)(160 215)
(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)

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

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

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

196 conjugacy classes

class 1 2A2B2C4A4B4C4D7A···7F8A8B8C8D8E8F8G8H14A···14F14G···14L14M···14R16A···16H16I16J16K16L28A···28L28M···28R28S···28X56A···56X56Y···56AJ56AK···56AV112A···112AV112AW···112BT
order122244447···78888888814···1414···1414···1416···161616161628···2828···2828···2856···5656···5656···56112···112112···112
size112411241···1111122441···12···24···42···244441···12···24···41···12···24···42···24···4

196 irreducible representations

dim1111111111111111222222
type+++++
imageC1C2C2C2C4C4C7C8C8C14C14C14C28C28C56C56D4M4(2)C7×D4D4.C8C7×M4(2)C7×D4.C8
kernelC7×D4.C8C2×C112C7×M5(2)C7×C8○D4C7×M4(2)C7×C4○D4D4.C8C7×D4C7×Q8C2×C16M5(2)C8○D4M4(2)C4○D4D4Q8C56C2×C14C8C7C22C1
# reps11112264466612122424221281248

Matrix representation of C7×D4.C8 in GL2(𝔽113) generated by

490
049
,
150
098
,
098
150
,
780
073
G:=sub<GL(2,GF(113))| [49,0,0,49],[15,0,0,98],[0,15,98,0],[78,0,0,73] >;

C7×D4.C8 in GAP, Magma, Sage, TeX

C_7\times D_4.C_8
% in TeX

G:=Group("C7xD4.C8");
// GroupNames label

G:=SmallGroup(448,154);
// by ID

G=gap.SmallGroup(448,154);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3538,248,4911,102,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^4=c^2=1,d^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b*c>;
// generators/relations

Export

Subgroup lattice of C7×D4.C8 in TeX

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