metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7⋊C8⋊1D4, C7⋊1(C8⋊D4), C4⋊C4.11D14, C4.160(D4×D7), D4⋊C4⋊18D7, D14⋊2Q8⋊3C2, (C2×D4).28D14, C28.9(C4○D4), C28.Q8⋊8C2, C28⋊2D4.5C2, C28.110(C2×D4), (C2×C8).169D14, C4.26(C4○D28), (C2×Dic7).22D4, (C22×D7).12D4, C22.178(D4×D7), C2.16(D8⋊D7), C28.44D4⋊21C2, C14.17(C4⋊D4), C14.34(C8⋊C22), (C2×C56).186C22, (C2×C28).220C23, (D4×C14).41C22, C4⋊Dic7.74C22, C2.20(D14⋊D4), C2.12(SD16⋊D7), C14.30(C8.C22), (C2×Dic14).58C22, (C2×D4.D7)⋊5C2, (C2×C8⋊D7)⋊17C2, (C2×C7⋊C8).18C22, (C7×D4⋊C4)⋊24C2, (C2×C4×D7).12C22, (C2×C14).233(C2×D4), (C7×C4⋊C4).21C22, (C2×C4).327(C22×D7), SmallGroup(448,314)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊C8⋊1D4
G = < a,b,c,d | a7=b8=c4=d2=1, bab-1=cac-1=dad=a-1, cbc-1=b-1, dbd=b5, dcd=c-1 >
Subgroups: 660 in 120 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C7⋊C8, C56, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C8⋊D4, C8⋊D7, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, D14⋊C4, D4.D7, C23.D7, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C2×C7⋊D4, D4×C14, C28.Q8, C28.44D4, C7×D4⋊C4, D14⋊2Q8, C2×C8⋊D7, C2×D4.D7, C28⋊2D4, C7⋊C8⋊1D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C8⋊C22, C8.C22, C22×D7, C8⋊D4, C4○D28, D4×D7, D14⋊D4, D8⋊D7, SD16⋊D7, C7⋊C8⋊1D4
►On 224 points
Generators in S224
(1 59 129 207 182 44 187)(2 188 45 183 208 130 60)(3 61 131 201 184 46 189)(4 190 47 177 202 132 62)(5 63 133 203 178 48 191)(6 192 41 179 204 134 64)(7 57 135 205 180 42 185)(8 186 43 181 206 136 58)(9 172 18 215 79 123 92)(10 93 124 80 216 19 173)(11 174 20 209 73 125 94)(12 95 126 74 210 21 175)(13 176 22 211 75 127 96)(14 89 128 76 212 23 169)(15 170 24 213 77 121 90)(16 91 122 78 214 17 171)(25 36 70 88 154 116 165)(26 166 117 155 81 71 37)(27 38 72 82 156 118 167)(28 168 119 157 83 65 39)(29 40 66 84 158 120 161)(30 162 113 159 85 67 33)(31 34 68 86 160 114 163)(32 164 115 153 87 69 35)(49 104 145 138 110 217 200)(50 193 218 111 139 146 97)(51 98 147 140 112 219 194)(52 195 220 105 141 148 99)(53 100 149 142 106 221 196)(54 197 222 107 143 150 101)(55 102 151 144 108 223 198)(56 199 224 109 137 152 103)
(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)
(1 164 217 80)(2 163 218 79)(3 162 219 78)(4 161 220 77)(5 168 221 76)(6 167 222 75)(7 166 223 74)(8 165 224 73)(9 208 68 97)(10 207 69 104)(11 206 70 103)(12 205 71 102)(13 204 72 101)(14 203 65 100)(15 202 66 99)(16 201 67 98)(17 46 159 140)(18 45 160 139)(19 44 153 138)(20 43 154 137)(21 42 155 144)(22 41 156 143)(23 48 157 142)(24 47 158 141)(25 199 125 58)(26 198 126 57)(27 197 127 64)(28 196 128 63)(29 195 121 62)(30 194 122 61)(31 193 123 60)(32 200 124 59)(33 51 91 131)(34 50 92 130)(35 49 93 129)(36 56 94 136)(37 55 95 135)(38 54 96 134)(39 53 89 133)(40 52 90 132)(81 151 175 180)(82 150 176 179)(83 149 169 178)(84 148 170 177)(85 147 171 184)(86 146 172 183)(87 145 173 182)(88 152 174 181)(105 213 190 120)(106 212 191 119)(107 211 192 118)(108 210 185 117)(109 209 186 116)(110 216 187 115)(111 215 188 114)(112 214 189 113)
(2 6)(4 8)(9 82)(10 87)(11 84)(12 81)(13 86)(14 83)(15 88)(16 85)(17 33)(18 38)(19 35)(20 40)(21 37)(22 34)(23 39)(24 36)(25 213)(26 210)(27 215)(28 212)(29 209)(30 214)(31 211)(32 216)(41 130)(42 135)(43 132)(44 129)(45 134)(46 131)(47 136)(48 133)(49 138)(50 143)(51 140)(52 137)(53 142)(54 139)(55 144)(56 141)(57 185)(58 190)(59 187)(60 192)(61 189)(62 186)(63 191)(64 188)(65 169)(66 174)(67 171)(68 176)(69 173)(70 170)(71 175)(72 172)(73 161)(74 166)(75 163)(76 168)(77 165)(78 162)(79 167)(80 164)(89 157)(90 154)(91 159)(92 156)(93 153)(94 158)(95 155)(96 160)(97 150)(98 147)(99 152)(100 149)(101 146)(102 151)(103 148)(104 145)(105 199)(106 196)(107 193)(108 198)(109 195)(110 200)(111 197)(112 194)(113 122)(114 127)(115 124)(116 121)(117 126)(118 123)(119 128)(120 125)(177 206)(178 203)(179 208)(180 205)(181 202)(182 207)(183 204)(184 201)(218 222)(220 224)
G:=sub<Sym(224)| (1,59,129,207,182,44,187)(2,188,45,183,208,130,60)(3,61,131,201,184,46,189)(4,190,47,177,202,132,62)(5,63,133,203,178,48,191)(6,192,41,179,204,134,64)(7,57,135,205,180,42,185)(8,186,43,181,206,136,58)(9,172,18,215,79,123,92)(10,93,124,80,216,19,173)(11,174,20,209,73,125,94)(12,95,126,74,210,21,175)(13,176,22,211,75,127,96)(14,89,128,76,212,23,169)(15,170,24,213,77,121,90)(16,91,122,78,214,17,171)(25,36,70,88,154,116,165)(26,166,117,155,81,71,37)(27,38,72,82,156,118,167)(28,168,119,157,83,65,39)(29,40,66,84,158,120,161)(30,162,113,159,85,67,33)(31,34,68,86,160,114,163)(32,164,115,153,87,69,35)(49,104,145,138,110,217,200)(50,193,218,111,139,146,97)(51,98,147,140,112,219,194)(52,195,220,105,141,148,99)(53,100,149,142,106,221,196)(54,197,222,107,143,150,101)(55,102,151,144,108,223,198)(56,199,224,109,137,152,103), (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), (1,164,217,80)(2,163,218,79)(3,162,219,78)(4,161,220,77)(5,168,221,76)(6,167,222,75)(7,166,223,74)(8,165,224,73)(9,208,68,97)(10,207,69,104)(11,206,70,103)(12,205,71,102)(13,204,72,101)(14,203,65,100)(15,202,66,99)(16,201,67,98)(17,46,159,140)(18,45,160,139)(19,44,153,138)(20,43,154,137)(21,42,155,144)(22,41,156,143)(23,48,157,142)(24,47,158,141)(25,199,125,58)(26,198,126,57)(27,197,127,64)(28,196,128,63)(29,195,121,62)(30,194,122,61)(31,193,123,60)(32,200,124,59)(33,51,91,131)(34,50,92,130)(35,49,93,129)(36,56,94,136)(37,55,95,135)(38,54,96,134)(39,53,89,133)(40,52,90,132)(81,151,175,180)(82,150,176,179)(83,149,169,178)(84,148,170,177)(85,147,171,184)(86,146,172,183)(87,145,173,182)(88,152,174,181)(105,213,190,120)(106,212,191,119)(107,211,192,118)(108,210,185,117)(109,209,186,116)(110,216,187,115)(111,215,188,114)(112,214,189,113), (2,6)(4,8)(9,82)(10,87)(11,84)(12,81)(13,86)(14,83)(15,88)(16,85)(17,33)(18,38)(19,35)(20,40)(21,37)(22,34)(23,39)(24,36)(25,213)(26,210)(27,215)(28,212)(29,209)(30,214)(31,211)(32,216)(41,130)(42,135)(43,132)(44,129)(45,134)(46,131)(47,136)(48,133)(49,138)(50,143)(51,140)(52,137)(53,142)(54,139)(55,144)(56,141)(57,185)(58,190)(59,187)(60,192)(61,189)(62,186)(63,191)(64,188)(65,169)(66,174)(67,171)(68,176)(69,173)(70,170)(71,175)(72,172)(73,161)(74,166)(75,163)(76,168)(77,165)(78,162)(79,167)(80,164)(89,157)(90,154)(91,159)(92,156)(93,153)(94,158)(95,155)(96,160)(97,150)(98,147)(99,152)(100,149)(101,146)(102,151)(103,148)(104,145)(105,199)(106,196)(107,193)(108,198)(109,195)(110,200)(111,197)(112,194)(113,122)(114,127)(115,124)(116,121)(117,126)(118,123)(119,128)(120,125)(177,206)(178,203)(179,208)(180,205)(181,202)(182,207)(183,204)(184,201)(218,222)(220,224)>;
G:=Group( (1,59,129,207,182,44,187)(2,188,45,183,208,130,60)(3,61,131,201,184,46,189)(4,190,47,177,202,132,62)(5,63,133,203,178,48,191)(6,192,41,179,204,134,64)(7,57,135,205,180,42,185)(8,186,43,181,206,136,58)(9,172,18,215,79,123,92)(10,93,124,80,216,19,173)(11,174,20,209,73,125,94)(12,95,126,74,210,21,175)(13,176,22,211,75,127,96)(14,89,128,76,212,23,169)(15,170,24,213,77,121,90)(16,91,122,78,214,17,171)(25,36,70,88,154,116,165)(26,166,117,155,81,71,37)(27,38,72,82,156,118,167)(28,168,119,157,83,65,39)(29,40,66,84,158,120,161)(30,162,113,159,85,67,33)(31,34,68,86,160,114,163)(32,164,115,153,87,69,35)(49,104,145,138,110,217,200)(50,193,218,111,139,146,97)(51,98,147,140,112,219,194)(52,195,220,105,141,148,99)(53,100,149,142,106,221,196)(54,197,222,107,143,150,101)(55,102,151,144,108,223,198)(56,199,224,109,137,152,103), (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), (1,164,217,80)(2,163,218,79)(3,162,219,78)(4,161,220,77)(5,168,221,76)(6,167,222,75)(7,166,223,74)(8,165,224,73)(9,208,68,97)(10,207,69,104)(11,206,70,103)(12,205,71,102)(13,204,72,101)(14,203,65,100)(15,202,66,99)(16,201,67,98)(17,46,159,140)(18,45,160,139)(19,44,153,138)(20,43,154,137)(21,42,155,144)(22,41,156,143)(23,48,157,142)(24,47,158,141)(25,199,125,58)(26,198,126,57)(27,197,127,64)(28,196,128,63)(29,195,121,62)(30,194,122,61)(31,193,123,60)(32,200,124,59)(33,51,91,131)(34,50,92,130)(35,49,93,129)(36,56,94,136)(37,55,95,135)(38,54,96,134)(39,53,89,133)(40,52,90,132)(81,151,175,180)(82,150,176,179)(83,149,169,178)(84,148,170,177)(85,147,171,184)(86,146,172,183)(87,145,173,182)(88,152,174,181)(105,213,190,120)(106,212,191,119)(107,211,192,118)(108,210,185,117)(109,209,186,116)(110,216,187,115)(111,215,188,114)(112,214,189,113), (2,6)(4,8)(9,82)(10,87)(11,84)(12,81)(13,86)(14,83)(15,88)(16,85)(17,33)(18,38)(19,35)(20,40)(21,37)(22,34)(23,39)(24,36)(25,213)(26,210)(27,215)(28,212)(29,209)(30,214)(31,211)(32,216)(41,130)(42,135)(43,132)(44,129)(45,134)(46,131)(47,136)(48,133)(49,138)(50,143)(51,140)(52,137)(53,142)(54,139)(55,144)(56,141)(57,185)(58,190)(59,187)(60,192)(61,189)(62,186)(63,191)(64,188)(65,169)(66,174)(67,171)(68,176)(69,173)(70,170)(71,175)(72,172)(73,161)(74,166)(75,163)(76,168)(77,165)(78,162)(79,167)(80,164)(89,157)(90,154)(91,159)(92,156)(93,153)(94,158)(95,155)(96,160)(97,150)(98,147)(99,152)(100,149)(101,146)(102,151)(103,148)(104,145)(105,199)(106,196)(107,193)(108,198)(109,195)(110,200)(111,197)(112,194)(113,122)(114,127)(115,124)(116,121)(117,126)(118,123)(119,128)(120,125)(177,206)(178,203)(179,208)(180,205)(181,202)(182,207)(183,204)(184,201)(218,222)(220,224) );
G=PermutationGroup([[(1,59,129,207,182,44,187),(2,188,45,183,208,130,60),(3,61,131,201,184,46,189),(4,190,47,177,202,132,62),(5,63,133,203,178,48,191),(6,192,41,179,204,134,64),(7,57,135,205,180,42,185),(8,186,43,181,206,136,58),(9,172,18,215,79,123,92),(10,93,124,80,216,19,173),(11,174,20,209,73,125,94),(12,95,126,74,210,21,175),(13,176,22,211,75,127,96),(14,89,128,76,212,23,169),(15,170,24,213,77,121,90),(16,91,122,78,214,17,171),(25,36,70,88,154,116,165),(26,166,117,155,81,71,37),(27,38,72,82,156,118,167),(28,168,119,157,83,65,39),(29,40,66,84,158,120,161),(30,162,113,159,85,67,33),(31,34,68,86,160,114,163),(32,164,115,153,87,69,35),(49,104,145,138,110,217,200),(50,193,218,111,139,146,97),(51,98,147,140,112,219,194),(52,195,220,105,141,148,99),(53,100,149,142,106,221,196),(54,197,222,107,143,150,101),(55,102,151,144,108,223,198),(56,199,224,109,137,152,103)], [(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)], [(1,164,217,80),(2,163,218,79),(3,162,219,78),(4,161,220,77),(5,168,221,76),(6,167,222,75),(7,166,223,74),(8,165,224,73),(9,208,68,97),(10,207,69,104),(11,206,70,103),(12,205,71,102),(13,204,72,101),(14,203,65,100),(15,202,66,99),(16,201,67,98),(17,46,159,140),(18,45,160,139),(19,44,153,138),(20,43,154,137),(21,42,155,144),(22,41,156,143),(23,48,157,142),(24,47,158,141),(25,199,125,58),(26,198,126,57),(27,197,127,64),(28,196,128,63),(29,195,121,62),(30,194,122,61),(31,193,123,60),(32,200,124,59),(33,51,91,131),(34,50,92,130),(35,49,93,129),(36,56,94,136),(37,55,95,135),(38,54,96,134),(39,53,89,133),(40,52,90,132),(81,151,175,180),(82,150,176,179),(83,149,169,178),(84,148,170,177),(85,147,171,184),(86,146,172,183),(87,145,173,182),(88,152,174,181),(105,213,190,120),(106,212,191,119),(107,211,192,118),(108,210,185,117),(109,209,186,116),(110,216,187,115),(111,215,188,114),(112,214,189,113)], [(2,6),(4,8),(9,82),(10,87),(11,84),(12,81),(13,86),(14,83),(15,88),(16,85),(17,33),(18,38),(19,35),(20,40),(21,37),(22,34),(23,39),(24,36),(25,213),(26,210),(27,215),(28,212),(29,209),(30,214),(31,211),(32,216),(41,130),(42,135),(43,132),(44,129),(45,134),(46,131),(47,136),(48,133),(49,138),(50,143),(51,140),(52,137),(53,142),(54,139),(55,144),(56,141),(57,185),(58,190),(59,187),(60,192),(61,189),(62,186),(63,191),(64,188),(65,169),(66,174),(67,171),(68,176),(69,173),(70,170),(71,175),(72,172),(73,161),(74,166),(75,163),(76,168),(77,165),(78,162),(79,167),(80,164),(89,157),(90,154),(91,159),(92,156),(93,153),(94,158),(95,155),(96,160),(97,150),(98,147),(99,152),(100,149),(101,146),(102,151),(103,148),(104,145),(105,199),(106,196),(107,193),(108,198),(109,195),(110,200),(111,197),(112,194),(113,122),(114,127),(115,124),(116,121),(117,126),(118,123),(119,128),(120,125),(177,206),(178,203),(179,208),(180,205),(181,202),(182,207),(183,204),(184,201),(218,222),(220,224)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 28 | 2 | 2 | 8 | 28 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4○D28 | C8⋊C22 | C8.C22 | D4×D7 | D4×D7 | D8⋊D7 | SD16⋊D7 |
| kernel | C7⋊C8⋊1D4 | C28.Q8 | C28.44D4 | C7×D4⋊C4 | D14⋊2Q8 | C2×C8⋊D7 | C2×D4.D7 | C28⋊2D4 | C7⋊C8 | C2×Dic7 | C22×D7 | D4⋊C4 | C28 | C4⋊C4 | C2×C8 | C2×D4 | C4 | C14 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 3 | 3 | 3 | 12 | 1 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of C7⋊C8⋊1D4 ►in GL8(𝔽113)
| 79 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 104 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 88 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 112 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 62 | 3 | 27 | 66 | 0 | 0 | 0 | 0 |
| 42 | 51 | 2 | 18 | 0 | 0 | 0 | 0 |
| 51 | 39 | 55 | 6 | 0 | 0 | 0 | 0 |
| 32 | 20 | 101 | 58 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 42 | 76 | 71 |
| 0 | 0 | 0 | 0 | 36 | 76 | 77 | 37 |
| 0 | 0 | 0 | 0 | 37 | 42 | 37 | 42 |
| 0 | 0 | 0 | 0 | 36 | 76 | 36 | 76 |
| 73 | 110 | 107 | 31 | 0 | 0 | 0 | 0 |
| 1 | 40 | 3 | 22 | 0 | 0 | 0 | 0 |
| 100 | 44 | 34 | 21 | 0 | 0 | 0 | 0 |
| 48 | 96 | 21 | 79 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 95 | 0 |
| 0 | 0 | 0 | 0 | 36 | 109 | 64 | 18 |
| 0 | 0 | 0 | 0 | 95 | 0 | 109 | 0 |
| 0 | 0 | 0 | 0 | 64 | 18 | 77 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 79 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 81 | 4 | 35 | 88 | 0 | 0 | 0 | 0 |
| 66 | 19 | 58 | 78 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 112 |
G:=sub<GL(8,GF(113))| [79,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,104,2,0,0,0,0,0,0,112,88,0,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,112,0],[62,42,51,32,0,0,0,0,3,51,39,20,0,0,0,0,27,2,55,101,0,0,0,0,66,18,6,58,0,0,0,0,0,0,0,0,37,36,37,36,0,0,0,0,42,76,42,76,0,0,0,0,76,77,37,36,0,0,0,0,71,37,42,76],[73,1,100,48,0,0,0,0,110,40,44,96,0,0,0,0,107,3,34,21,0,0,0,0,31,22,21,79,0,0,0,0,0,0,0,0,4,36,95,64,0,0,0,0,0,109,0,18,0,0,0,0,95,64,109,77,0,0,0,0,0,18,0,4],[1,79,81,66,0,0,0,0,0,112,4,19,0,0,0,0,0,0,35,58,0,0,0,0,0,0,88,78,0,0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,0,112] >;
C7⋊C8⋊1D4 in GAP, Magma, Sage, TeX
C_7\rtimes C_8\rtimes_1D_4
% in TeX
G:=Group("C7:C8:1D4"); // GroupNames label
G:=SmallGroup(448,314);
// by ID
G=gap.SmallGroup(448,314);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,1094,135,100,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=b^-1,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations