metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.165D14, C14.1432+ (1+4), C14.1032- (1+4), C28⋊Q8⋊42C2, C28⋊2Q8⋊9C2, C4⋊C4.120D14, C42⋊2C2⋊9D7, C42⋊2D7⋊2C2, (C4×C28).9C22, D14⋊2Q8⋊43C2, D14⋊Q8⋊46C2, Dic7.Q8⋊40C2, (C2×C28).97C23, C22⋊C4.42D14, C4.Dic14⋊41C2, (C2×C14).256C24, D14⋊C4.48C22, D14.D4.5C2, C4⋊Dic7.55C22, C2.68(D4⋊8D14), C23.62(C22×D7), C22⋊Dic14⋊48C2, C23.D14⋊47C2, Dic7⋊C4.11C22, (C22×C14).70C23, Dic7.D4.5C2, C22.D28.3C2, C22.277(C23×D7), C23.D7.70C22, C7⋊5(C22.57C24), (C2×Dic14).43C22, (C4×Dic7).153C22, (C2×Dic7).132C23, (C22×D7).115C23, C2.67(D4.10D14), (C22×Dic7).155C22, C4⋊C4⋊D7⋊45C2, (C2×C4×D7).137C22, (C7×C42⋊2C2)⋊11C2, (C7×C4⋊C4).207C22, (C2×C4).212(C22×D7), (C2×C7⋊D4).76C22, (C7×C22⋊C4).81C22, SmallGroup(448,1165)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 876 in 196 conjugacy classes, 91 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×13], C22, C22 [×6], C7, C2×C4 [×6], C2×C4 [×9], D4, Q8 [×3], C23, C23, D7, C14 [×3], C14, C42, C42 [×2], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4 [×2], C2×D4, C2×Q8 [×3], Dic7 [×7], C28 [×6], D14 [×3], C2×C14, C2×C14 [×3], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C42⋊2C2, C42⋊2C2 [×3], C4⋊Q8 [×2], Dic14 [×3], C4×D7, C2×Dic7 [×7], C2×Dic7, C7⋊D4, C2×C28 [×6], C22×D7, C22×C14, C22.57C24, C4×Dic7 [×2], Dic7⋊C4 [×7], C4⋊Dic7 [×6], D14⋊C4 [×5], C23.D7 [×2], C4×C28, C7×C22⋊C4 [×3], C7×C4⋊C4 [×3], C2×Dic14 [×3], C2×C4×D7, C22×Dic7, C2×C7⋊D4, C28⋊2Q8, C42⋊2D7, C22⋊Dic14 [×2], C23.D14, D14.D4, Dic7.D4, C22.D28, C28⋊Q8, Dic7.Q8, C4.Dic14, D14⋊Q8, D14⋊2Q8, C4⋊C4⋊D7, C7×C42⋊2C2, C42.165D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ (1+4), 2- (1+4) [×2], C22×D7 [×7], C22.57C24, C23×D7, D4⋊8D14, D4.10D14 [×2], C42.165D14
Generators and relations
G = < a,b,c,d | a4=b4=1, c14=d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c13 >
►On 224 points
Generators in S224
(1 161 154 8)(2 23 155 148)(3 163 156 10)(4 25 157 150)(5 165 158 12)(6 27 159 152)(7 167 160 14)(9 141 162 16)(11 143 164 18)(13 145 166 20)(15 147 168 22)(17 149 142 24)(19 151 144 26)(21 153 146 28)(29 91 98 50)(30 37 99 106)(31 93 100 52)(32 39 101 108)(33 95 102 54)(34 41 103 110)(35 97 104 56)(36 43 105 112)(38 45 107 86)(40 47 109 88)(42 49 111 90)(44 51 85 92)(46 53 87 94)(48 55 89 96)(57 205 212 78)(58 65 213 220)(59 207 214 80)(60 67 215 222)(61 209 216 82)(62 69 217 224)(63 211 218 84)(64 71 219 198)(66 73 221 200)(68 75 223 202)(70 77 197 204)(72 79 199 206)(74 81 201 208)(76 83 203 210)(113 174 195 120)(114 135 196 189)(115 176 169 122)(116 137 170 191)(117 178 171 124)(118 139 172 193)(119 180 173 126)(121 182 175 128)(123 184 177 130)(125 186 179 132)(127 188 181 134)(129 190 183 136)(131 192 185 138)(133 194 187 140)
(1 204 168 63)(2 219 141 78)(3 206 142 65)(4 221 143 80)(5 208 144 67)(6 223 145 82)(7 210 146 69)(8 197 147 84)(9 212 148 71)(10 199 149 58)(11 214 150 73)(12 201 151 60)(13 216 152 75)(14 203 153 62)(15 218 154 77)(16 205 155 64)(17 220 156 79)(18 207 157 66)(19 222 158 81)(20 209 159 68)(21 224 160 83)(22 211 161 70)(23 198 162 57)(24 213 163 72)(25 200 164 59)(26 215 165 74)(27 202 166 61)(28 217 167 76)(29 185 112 117)(30 172 85 132)(31 187 86 119)(32 174 87 134)(33 189 88 121)(34 176 89 136)(35 191 90 123)(36 178 91 138)(37 193 92 125)(38 180 93 140)(39 195 94 127)(40 182 95 114)(41 169 96 129)(42 184 97 116)(43 171 98 131)(44 186 99 118)(45 173 100 133)(46 188 101 120)(47 175 102 135)(48 190 103 122)(49 177 104 137)(50 192 105 124)(51 179 106 139)(52 194 107 126)(53 181 108 113)(54 196 109 128)(55 183 110 115)(56 170 111 130)
(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 93 15 107)(2 106 16 92)(3 91 17 105)(4 104 18 90)(5 89 19 103)(6 102 20 88)(7 87 21 101)(8 100 22 86)(9 85 23 99)(10 98 24 112)(11 111 25 97)(12 96 26 110)(13 109 27 95)(14 94 28 108)(29 149 43 163)(30 162 44 148)(31 147 45 161)(32 160 46 146)(33 145 47 159)(34 158 48 144)(35 143 49 157)(36 156 50 142)(37 141 51 155)(38 154 52 168)(39 167 53 153)(40 152 54 166)(41 165 55 151)(42 150 56 164)(57 118 71 132)(58 131 72 117)(59 116 73 130)(60 129 74 115)(61 114 75 128)(62 127 76 113)(63 140 77 126)(64 125 78 139)(65 138 79 124)(66 123 80 137)(67 136 81 122)(68 121 82 135)(69 134 83 120)(70 119 84 133)(169 215 183 201)(170 200 184 214)(171 213 185 199)(172 198 186 212)(173 211 187 197)(174 224 188 210)(175 209 189 223)(176 222 190 208)(177 207 191 221)(178 220 192 206)(179 205 193 219)(180 218 194 204)(181 203 195 217)(182 216 196 202)
G:=sub<Sym(224)| (1,161,154,8)(2,23,155,148)(3,163,156,10)(4,25,157,150)(5,165,158,12)(6,27,159,152)(7,167,160,14)(9,141,162,16)(11,143,164,18)(13,145,166,20)(15,147,168,22)(17,149,142,24)(19,151,144,26)(21,153,146,28)(29,91,98,50)(30,37,99,106)(31,93,100,52)(32,39,101,108)(33,95,102,54)(34,41,103,110)(35,97,104,56)(36,43,105,112)(38,45,107,86)(40,47,109,88)(42,49,111,90)(44,51,85,92)(46,53,87,94)(48,55,89,96)(57,205,212,78)(58,65,213,220)(59,207,214,80)(60,67,215,222)(61,209,216,82)(62,69,217,224)(63,211,218,84)(64,71,219,198)(66,73,221,200)(68,75,223,202)(70,77,197,204)(72,79,199,206)(74,81,201,208)(76,83,203,210)(113,174,195,120)(114,135,196,189)(115,176,169,122)(116,137,170,191)(117,178,171,124)(118,139,172,193)(119,180,173,126)(121,182,175,128)(123,184,177,130)(125,186,179,132)(127,188,181,134)(129,190,183,136)(131,192,185,138)(133,194,187,140), (1,204,168,63)(2,219,141,78)(3,206,142,65)(4,221,143,80)(5,208,144,67)(6,223,145,82)(7,210,146,69)(8,197,147,84)(9,212,148,71)(10,199,149,58)(11,214,150,73)(12,201,151,60)(13,216,152,75)(14,203,153,62)(15,218,154,77)(16,205,155,64)(17,220,156,79)(18,207,157,66)(19,222,158,81)(20,209,159,68)(21,224,160,83)(22,211,161,70)(23,198,162,57)(24,213,163,72)(25,200,164,59)(26,215,165,74)(27,202,166,61)(28,217,167,76)(29,185,112,117)(30,172,85,132)(31,187,86,119)(32,174,87,134)(33,189,88,121)(34,176,89,136)(35,191,90,123)(36,178,91,138)(37,193,92,125)(38,180,93,140)(39,195,94,127)(40,182,95,114)(41,169,96,129)(42,184,97,116)(43,171,98,131)(44,186,99,118)(45,173,100,133)(46,188,101,120)(47,175,102,135)(48,190,103,122)(49,177,104,137)(50,192,105,124)(51,179,106,139)(52,194,107,126)(53,181,108,113)(54,196,109,128)(55,183,110,115)(56,170,111,130), (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), (1,93,15,107)(2,106,16,92)(3,91,17,105)(4,104,18,90)(5,89,19,103)(6,102,20,88)(7,87,21,101)(8,100,22,86)(9,85,23,99)(10,98,24,112)(11,111,25,97)(12,96,26,110)(13,109,27,95)(14,94,28,108)(29,149,43,163)(30,162,44,148)(31,147,45,161)(32,160,46,146)(33,145,47,159)(34,158,48,144)(35,143,49,157)(36,156,50,142)(37,141,51,155)(38,154,52,168)(39,167,53,153)(40,152,54,166)(41,165,55,151)(42,150,56,164)(57,118,71,132)(58,131,72,117)(59,116,73,130)(60,129,74,115)(61,114,75,128)(62,127,76,113)(63,140,77,126)(64,125,78,139)(65,138,79,124)(66,123,80,137)(67,136,81,122)(68,121,82,135)(69,134,83,120)(70,119,84,133)(169,215,183,201)(170,200,184,214)(171,213,185,199)(172,198,186,212)(173,211,187,197)(174,224,188,210)(175,209,189,223)(176,222,190,208)(177,207,191,221)(178,220,192,206)(179,205,193,219)(180,218,194,204)(181,203,195,217)(182,216,196,202)>;
G:=Group( (1,161,154,8)(2,23,155,148)(3,163,156,10)(4,25,157,150)(5,165,158,12)(6,27,159,152)(7,167,160,14)(9,141,162,16)(11,143,164,18)(13,145,166,20)(15,147,168,22)(17,149,142,24)(19,151,144,26)(21,153,146,28)(29,91,98,50)(30,37,99,106)(31,93,100,52)(32,39,101,108)(33,95,102,54)(34,41,103,110)(35,97,104,56)(36,43,105,112)(38,45,107,86)(40,47,109,88)(42,49,111,90)(44,51,85,92)(46,53,87,94)(48,55,89,96)(57,205,212,78)(58,65,213,220)(59,207,214,80)(60,67,215,222)(61,209,216,82)(62,69,217,224)(63,211,218,84)(64,71,219,198)(66,73,221,200)(68,75,223,202)(70,77,197,204)(72,79,199,206)(74,81,201,208)(76,83,203,210)(113,174,195,120)(114,135,196,189)(115,176,169,122)(116,137,170,191)(117,178,171,124)(118,139,172,193)(119,180,173,126)(121,182,175,128)(123,184,177,130)(125,186,179,132)(127,188,181,134)(129,190,183,136)(131,192,185,138)(133,194,187,140), (1,204,168,63)(2,219,141,78)(3,206,142,65)(4,221,143,80)(5,208,144,67)(6,223,145,82)(7,210,146,69)(8,197,147,84)(9,212,148,71)(10,199,149,58)(11,214,150,73)(12,201,151,60)(13,216,152,75)(14,203,153,62)(15,218,154,77)(16,205,155,64)(17,220,156,79)(18,207,157,66)(19,222,158,81)(20,209,159,68)(21,224,160,83)(22,211,161,70)(23,198,162,57)(24,213,163,72)(25,200,164,59)(26,215,165,74)(27,202,166,61)(28,217,167,76)(29,185,112,117)(30,172,85,132)(31,187,86,119)(32,174,87,134)(33,189,88,121)(34,176,89,136)(35,191,90,123)(36,178,91,138)(37,193,92,125)(38,180,93,140)(39,195,94,127)(40,182,95,114)(41,169,96,129)(42,184,97,116)(43,171,98,131)(44,186,99,118)(45,173,100,133)(46,188,101,120)(47,175,102,135)(48,190,103,122)(49,177,104,137)(50,192,105,124)(51,179,106,139)(52,194,107,126)(53,181,108,113)(54,196,109,128)(55,183,110,115)(56,170,111,130), (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), (1,93,15,107)(2,106,16,92)(3,91,17,105)(4,104,18,90)(5,89,19,103)(6,102,20,88)(7,87,21,101)(8,100,22,86)(9,85,23,99)(10,98,24,112)(11,111,25,97)(12,96,26,110)(13,109,27,95)(14,94,28,108)(29,149,43,163)(30,162,44,148)(31,147,45,161)(32,160,46,146)(33,145,47,159)(34,158,48,144)(35,143,49,157)(36,156,50,142)(37,141,51,155)(38,154,52,168)(39,167,53,153)(40,152,54,166)(41,165,55,151)(42,150,56,164)(57,118,71,132)(58,131,72,117)(59,116,73,130)(60,129,74,115)(61,114,75,128)(62,127,76,113)(63,140,77,126)(64,125,78,139)(65,138,79,124)(66,123,80,137)(67,136,81,122)(68,121,82,135)(69,134,83,120)(70,119,84,133)(169,215,183,201)(170,200,184,214)(171,213,185,199)(172,198,186,212)(173,211,187,197)(174,224,188,210)(175,209,189,223)(176,222,190,208)(177,207,191,221)(178,220,192,206)(179,205,193,219)(180,218,194,204)(181,203,195,217)(182,216,196,202) );
G=PermutationGroup([(1,161,154,8),(2,23,155,148),(3,163,156,10),(4,25,157,150),(5,165,158,12),(6,27,159,152),(7,167,160,14),(9,141,162,16),(11,143,164,18),(13,145,166,20),(15,147,168,22),(17,149,142,24),(19,151,144,26),(21,153,146,28),(29,91,98,50),(30,37,99,106),(31,93,100,52),(32,39,101,108),(33,95,102,54),(34,41,103,110),(35,97,104,56),(36,43,105,112),(38,45,107,86),(40,47,109,88),(42,49,111,90),(44,51,85,92),(46,53,87,94),(48,55,89,96),(57,205,212,78),(58,65,213,220),(59,207,214,80),(60,67,215,222),(61,209,216,82),(62,69,217,224),(63,211,218,84),(64,71,219,198),(66,73,221,200),(68,75,223,202),(70,77,197,204),(72,79,199,206),(74,81,201,208),(76,83,203,210),(113,174,195,120),(114,135,196,189),(115,176,169,122),(116,137,170,191),(117,178,171,124),(118,139,172,193),(119,180,173,126),(121,182,175,128),(123,184,177,130),(125,186,179,132),(127,188,181,134),(129,190,183,136),(131,192,185,138),(133,194,187,140)], [(1,204,168,63),(2,219,141,78),(3,206,142,65),(4,221,143,80),(5,208,144,67),(6,223,145,82),(7,210,146,69),(8,197,147,84),(9,212,148,71),(10,199,149,58),(11,214,150,73),(12,201,151,60),(13,216,152,75),(14,203,153,62),(15,218,154,77),(16,205,155,64),(17,220,156,79),(18,207,157,66),(19,222,158,81),(20,209,159,68),(21,224,160,83),(22,211,161,70),(23,198,162,57),(24,213,163,72),(25,200,164,59),(26,215,165,74),(27,202,166,61),(28,217,167,76),(29,185,112,117),(30,172,85,132),(31,187,86,119),(32,174,87,134),(33,189,88,121),(34,176,89,136),(35,191,90,123),(36,178,91,138),(37,193,92,125),(38,180,93,140),(39,195,94,127),(40,182,95,114),(41,169,96,129),(42,184,97,116),(43,171,98,131),(44,186,99,118),(45,173,100,133),(46,188,101,120),(47,175,102,135),(48,190,103,122),(49,177,104,137),(50,192,105,124),(51,179,106,139),(52,194,107,126),(53,181,108,113),(54,196,109,128),(55,183,110,115),(56,170,111,130)], [(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)], [(1,93,15,107),(2,106,16,92),(3,91,17,105),(4,104,18,90),(5,89,19,103),(6,102,20,88),(7,87,21,101),(8,100,22,86),(9,85,23,99),(10,98,24,112),(11,111,25,97),(12,96,26,110),(13,109,27,95),(14,94,28,108),(29,149,43,163),(30,162,44,148),(31,147,45,161),(32,160,46,146),(33,145,47,159),(34,158,48,144),(35,143,49,157),(36,156,50,142),(37,141,51,155),(38,154,52,168),(39,167,53,153),(40,152,54,166),(41,165,55,151),(42,150,56,164),(57,118,71,132),(58,131,72,117),(59,116,73,130),(60,129,74,115),(61,114,75,128),(62,127,76,113),(63,140,77,126),(64,125,78,139),(65,138,79,124),(66,123,80,137),(67,136,81,122),(68,121,82,135),(69,134,83,120),(70,119,84,133),(169,215,183,201),(170,200,184,214),(171,213,185,199),(172,198,186,212),(173,211,187,197),(174,224,188,210),(175,209,189,223),(176,222,190,208),(177,207,191,221),(178,220,192,206),(179,205,193,219),(180,218,194,204),(181,203,195,217),(182,216,196,202)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 3 | 11 | 23 | 7 | 0 | 0 | 0 | 0 |
| 7 | 26 | 15 | 6 | 0 | 0 | 0 | 0 |
| 3 | 11 | 26 | 18 | 0 | 0 | 0 | 0 |
| 7 | 26 | 22 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 5 | 25 | 0 |
| 0 | 0 | 0 | 0 | 24 | 16 | 0 | 25 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 24 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 13 |
| 1 | 0 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 27 | 0 | 0 | 0 | 0 |
| 1 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 24 | 16 |
| 4 | 12 | 21 | 5 | 0 | 0 | 0 | 0 |
| 5 | 8 | 19 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 3 | 16 | 16 |
| 0 | 0 | 0 | 0 | 26 | 22 | 13 | 14 |
| 0 | 0 | 0 | 0 | 2 | 2 | 3 | 26 |
| 0 | 0 | 0 | 0 | 27 | 9 | 3 | 7 |
| 22 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 22 | 13 | 7 | 16 | 0 | 0 | 0 | 0 |
| 14 | 7 | 15 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 25 | 17 | 12 |
| 0 | 0 | 0 | 0 | 11 | 4 | 28 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 25 | 25 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 4 |
G:=sub<GL(8,GF(29))| [3,7,3,7,0,0,0,0,11,26,11,26,0,0,0,0,23,15,26,22,0,0,0,0,7,6,18,3,0,0,0,0,0,0,0,0,13,24,0,0,0,0,0,0,5,16,0,0,0,0,0,0,25,0,16,5,0,0,0,0,0,25,24,13],[1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,27,0,28,0,0,0,0,0,0,27,0,28,0,0,0,0,0,0,0,0,13,24,0,0,0,0,0,0,5,16,0,0,0,0,0,0,0,0,13,24,0,0,0,0,0,0,5,16],[4,5,0,0,0,0,0,0,12,8,0,0,0,0,0,0,21,19,25,24,0,0,0,0,5,13,17,21,0,0,0,0,0,0,0,0,26,26,2,27,0,0,0,0,3,22,2,9,0,0,0,0,16,13,3,3,0,0,0,0,16,14,26,7],[22,14,22,14,0,0,0,0,13,7,13,7,0,0,0,0,0,0,7,15,0,0,0,0,0,0,16,22,0,0,0,0,0,0,0,0,25,11,0,0,0,0,0,0,25,4,0,0,0,0,0,0,17,28,25,11,0,0,0,0,12,12,25,4] >;
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | ··· | 4M | 7A | 7B | 7C | 14A | ··· | 14I | 14J | 14K | 14L | 28A | ··· | 28R | 28S | ··· | 28AA |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 28 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D4⋊8D14 | D4.10D14 |
| kernel | C42.165D14 | C28⋊2Q8 | C42⋊2D7 | C22⋊Dic14 | C23.D14 | D14.D4 | Dic7.D4 | C22.D28 | C28⋊Q8 | Dic7.Q8 | C4.Dic14 | D14⋊Q8 | D14⋊2Q8 | C4⋊C4⋊D7 | C7×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 9 | 9 | 1 | 2 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2._{165}D_{14} % in TeX
G:=Group("C4^2.165D14"); // GroupNames label
G:=SmallGroup(448,1165);
// by ID
G=gap.SmallGroup(448,1165);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,1571,570,297,80,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^13>;
// generators/relations
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ℤ
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