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G = C7×C8.4Q8order 448 = 26·7

Direct product of C7 and C8.4Q8

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C7×C8.4Q8, C112.5C4, C16.1C28, C28.70D8, C56.20Q8, C8.5(C7×Q8), C4.19(C7×D8), C56.83(C2×C4), (C2×C16).5C14, C8.15(C2×C28), C28.58(C4⋊C4), (C2×C14).6Q16, (C2×C112).11C2, (C2×C28).410D4, C8.C4.3C14, C22.1(C7×Q16), C14.15(C2.D8), (C2×C56).431C22, C4.9(C7×C4⋊C4), C2.5(C7×C2.D8), (C2×C4).64(C7×D4), (C2×C8).89(C2×C14), (C7×C8.C4).6C2, SmallGroup(448,172)

Series: Derived Chief Lower central Upper central

C1C8 — C7×C8.4Q8
C1C2C4C2×C4C2×C8C2×C56C7×C8.C4 — C7×C8.4Q8
C1C2C4C8 — C7×C8.4Q8
C1C28C2×C28C2×C56 — C7×C8.4Q8

Generators and relations for C7×C8.4Q8
 G = < a,b,c,d | a7=b8=1, c4=b2, d2=bc2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b6c3 >

2C2
2C14
4C8
4C8
2M4(2)
2M4(2)
4C56
4C56
2C7×M4(2)
2C7×M4(2)

Smallest permutation representation of C7×C8.4Q8
On 224 points
Generators in S224
(1 60 165 205 192 34 120)(2 61 166 206 177 35 121)(3 62 167 207 178 36 122)(4 63 168 208 179 37 123)(5 64 169 193 180 38 124)(6 49 170 194 181 39 125)(7 50 171 195 182 40 126)(8 51 172 196 183 41 127)(9 52 173 197 184 42 128)(10 53 174 198 185 43 113)(11 54 175 199 186 44 114)(12 55 176 200 187 45 115)(13 56 161 201 188 46 116)(14 57 162 202 189 47 117)(15 58 163 203 190 48 118)(16 59 164 204 191 33 119)(17 154 108 92 217 74 138)(18 155 109 93 218 75 139)(19 156 110 94 219 76 140)(20 157 111 95 220 77 141)(21 158 112 96 221 78 142)(22 159 97 81 222 79 143)(23 160 98 82 223 80 144)(24 145 99 83 224 65 129)(25 146 100 84 209 66 130)(26 147 101 85 210 67 131)(27 148 102 86 211 68 132)(28 149 103 87 212 69 133)(29 150 104 88 213 70 134)(30 151 105 89 214 71 135)(31 152 106 90 215 72 136)(32 153 107 91 216 73 137)
(1 11 5 15 9 3 13 7)(2 12 6 16 10 4 14 8)(17 19 21 23 25 27 29 31)(18 20 22 24 26 28 30 32)(33 43 37 47 41 35 45 39)(34 44 38 48 42 36 46 40)(49 59 53 63 57 51 61 55)(50 60 54 64 58 52 62 56)(65 67 69 71 73 75 77 79)(66 68 70 72 74 76 78 80)(81 83 85 87 89 91 93 95)(82 84 86 88 90 92 94 96)(97 99 101 103 105 107 109 111)(98 100 102 104 106 108 110 112)(113 123 117 127 121 115 125 119)(114 124 118 128 122 116 126 120)(129 131 133 135 137 139 141 143)(130 132 134 136 138 140 142 144)(145 147 149 151 153 155 157 159)(146 148 150 152 154 156 158 160)(161 171 165 175 169 163 173 167)(162 172 166 176 170 164 174 168)(177 187 181 191 185 179 189 183)(178 188 182 192 186 180 190 184)(193 203 197 207 201 195 205 199)(194 204 198 208 202 196 206 200)(209 211 213 215 217 219 221 223)(210 212 214 216 218 220 222 224)
(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 71 13 75 9 79 5 67)(2 70 14 74 10 78 6 66)(3 69 15 73 11 77 7 65)(4 68 16 72 12 76 8 80)(17 174 21 170 25 166 29 162)(18 173 22 169 26 165 30 161)(19 172 23 168 27 164 31 176)(20 171 24 167 28 163 32 175)(33 90 45 94 41 82 37 86)(34 89 46 93 42 81 38 85)(35 88 47 92 43 96 39 84)(36 87 48 91 44 95 40 83)(49 130 61 134 57 138 53 142)(50 129 62 133 58 137 54 141)(51 144 63 132 59 136 55 140)(52 143 64 131 60 135 56 139)(97 180 101 192 105 188 109 184)(98 179 102 191 106 187 110 183)(99 178 103 190 107 186 111 182)(100 177 104 189 108 185 112 181)(113 221 125 209 121 213 117 217)(114 220 126 224 122 212 118 216)(115 219 127 223 123 211 119 215)(116 218 128 222 124 210 120 214)(145 207 149 203 153 199 157 195)(146 206 150 202 154 198 158 194)(147 205 151 201 155 197 159 193)(148 204 152 200 156 196 160 208)

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

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

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

154 conjugacy classes

class 1 2A2B4A4B4C7A···7F8A8B8C8D8E8F8G8H14A···14F14G···14L16A···16H28A···28L28M···28R56A···56X56Y···56AV112A···112AV
order1224447···78888888814···1414···1416···1628···2828···2856···5656···56112···112
size1121121···1222288881···12···22···21···12···22···28···82···2

154 irreducible representations

dim111111112222222222
type+++-++-
imageC1C2C2C4C7C14C14C28Q8D4D8Q16C7×Q8C7×D4C8.4Q8C7×D8C7×Q16C7×C8.4Q8
kernelC7×C8.4Q8C7×C8.C4C2×C112C112C8.4Q8C8.C4C2×C16C16C56C2×C28C28C2×C14C8C2×C4C7C4C22C1
# reps12146126241122668121248

Matrix representation of C7×C8.4Q8 in GL2(𝔽113) generated by

1090
0109
,
18107
069
,
7854
071
,
7031
8343
G:=sub<GL(2,GF(113))| [109,0,0,109],[18,0,107,69],[78,0,54,71],[70,83,31,43] >;

C7×C8.4Q8 in GAP, Magma, Sage, TeX

C_7\times C_8._4Q_8
% in TeX

G:=Group("C7xC8.4Q8");
// GroupNames label

G:=SmallGroup(448,172);
// by ID

G=gap.SmallGroup(448,172);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,988,3923,360,172,14117,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^8=1,c^4=b^2,d^2=b*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^6*c^3>;
// generators/relations

Export

Subgroup lattice of C7×C8.4Q8 in TeX

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