metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.156D14, C14.1342+ (1+4), C4⋊D28⋊15C2, C28⋊1D4⋊34C2, C4⋊C4.113D14, C42.C2⋊12D7, D28⋊C4⋊38C2, C42⋊D7⋊22C2, D14.5D4⋊36C2, C28.132(C4○D4), (C2×C14).242C24, (C2×C28).189C23, (C4×C28).201C22, C4.21(Q8⋊2D7), C2.59(D4⋊8D14), D14⋊C4.113C22, (C2×D28).166C22, C22.263(C23×D7), Dic7⋊C4.125C22, C7⋊6(C22.34C24), (C4×Dic7).147C22, (C2×Dic7).262C23, (C22×D7).107C23, C14.119(C2×C4○D4), (C7×C42.C2)⋊15C2, C2.26(C2×Q8⋊2D7), (C2×C4×D7).132C22, (C7×C4⋊C4).197C22, (C2×C4).594(C22×D7), SmallGroup(448,1151)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1404 in 240 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C7, C2×C4, C2×C4 [×6], C2×C4 [×9], D4 [×12], C23 [×5], D7 [×5], C14, C14 [×2], C42, C42, C22⋊C4 [×10], C4⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×5], C2×D4 [×10], Dic7 [×3], C28 [×2], C28 [×6], D14 [×15], C2×C14, C42⋊C2, C4×D4 [×2], C4⋊D4 [×6], C22.D4 [×4], C42.C2, C4⋊1D4, C4×D7 [×6], D28 [×12], C2×Dic7, C2×Dic7 [×2], C2×C28, C2×C28 [×6], C22×D7, C22×D7 [×4], C22.34C24, C4×Dic7, Dic7⋊C4 [×2], D14⋊C4 [×10], C4×C28, C7×C4⋊C4 [×6], C2×C4×D7, C2×C4×D7 [×4], C2×D28 [×10], C42⋊D7, C4⋊D28, D28⋊C4 [×2], D14.5D4 [×4], C28⋊1D4 [×6], C7×C42.C2, C42.156D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D7 [×7], C22.34C24, Q8⋊2D7 [×2], C23×D7, C2×Q8⋊2D7, D4⋊8D14 [×2], C42.156D14
Generators and relations
G = < a,b,c,d | a4=b4=1, c14=d2=a2, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c13 >
►On 224 points
Generators in S224
(1 113 15 127)(2 60 16 74)(3 115 17 129)(4 62 18 76)(5 117 19 131)(6 64 20 78)(7 119 21 133)(8 66 22 80)(9 121 23 135)(10 68 24 82)(11 123 25 137)(12 70 26 84)(13 125 27 139)(14 72 28 58)(29 191 43 177)(30 144 44 158)(31 193 45 179)(32 146 46 160)(33 195 47 181)(34 148 48 162)(35 169 49 183)(36 150 50 164)(37 171 51 185)(38 152 52 166)(39 173 53 187)(40 154 54 168)(41 175 55 189)(42 156 56 142)(57 207 71 221)(59 209 73 223)(61 211 75 197)(63 213 77 199)(65 215 79 201)(67 217 81 203)(69 219 83 205)(85 149 99 163)(86 170 100 184)(87 151 101 165)(88 172 102 186)(89 153 103 167)(90 174 104 188)(91 155 105 141)(92 176 106 190)(93 157 107 143)(94 178 108 192)(95 159 109 145)(96 180 110 194)(97 161 111 147)(98 182 112 196)(114 210 128 224)(116 212 130 198)(118 214 132 200)(120 216 134 202)(122 218 136 204)(124 220 138 206)(126 222 140 208)
(1 148 223 196)(2 163 224 183)(3 150 197 170)(4 165 198 185)(5 152 199 172)(6 167 200 187)(7 154 201 174)(8 141 202 189)(9 156 203 176)(10 143 204 191)(11 158 205 178)(12 145 206 193)(13 160 207 180)(14 147 208 195)(15 162 209 182)(16 149 210 169)(17 164 211 184)(18 151 212 171)(19 166 213 186)(20 153 214 173)(21 168 215 188)(22 155 216 175)(23 142 217 190)(24 157 218 177)(25 144 219 192)(26 159 220 179)(27 146 221 194)(28 161 222 181)(29 82 107 136)(30 69 108 123)(31 84 109 138)(32 71 110 125)(33 58 111 140)(34 73 112 127)(35 60 85 114)(36 75 86 129)(37 62 87 116)(38 77 88 131)(39 64 89 118)(40 79 90 133)(41 66 91 120)(42 81 92 135)(43 68 93 122)(44 83 94 137)(45 70 95 124)(46 57 96 139)(47 72 97 126)(48 59 98 113)(49 74 99 128)(50 61 100 115)(51 76 101 130)(52 63 102 117)(53 78 103 132)(54 65 104 119)(55 80 105 134)(56 67 106 121)
(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 97 15 111)(2 110 16 96)(3 95 17 109)(4 108 18 94)(5 93 19 107)(6 106 20 92)(7 91 21 105)(8 104 22 90)(9 89 23 103)(10 102 24 88)(11 87 25 101)(12 100 26 86)(13 85 27 99)(14 98 28 112)(29 199 43 213)(30 212 44 198)(31 197 45 211)(32 210 46 224)(33 223 47 209)(34 208 48 222)(35 221 49 207)(36 206 50 220)(37 219 51 205)(38 204 52 218)(39 217 53 203)(40 202 54 216)(41 215 55 201)(42 200 56 214)(57 149 71 163)(58 162 72 148)(59 147 73 161)(60 160 74 146)(61 145 75 159)(62 158 76 144)(63 143 77 157)(64 156 78 142)(65 141 79 155)(66 154 80 168)(67 167 81 153)(68 152 82 166)(69 165 83 151)(70 150 84 164)(113 195 127 181)(114 180 128 194)(115 193 129 179)(116 178 130 192)(117 191 131 177)(118 176 132 190)(119 189 133 175)(120 174 134 188)(121 187 135 173)(122 172 136 186)(123 185 137 171)(124 170 138 184)(125 183 139 169)(126 196 140 182)
G:=sub<Sym(224)| (1,113,15,127)(2,60,16,74)(3,115,17,129)(4,62,18,76)(5,117,19,131)(6,64,20,78)(7,119,21,133)(8,66,22,80)(9,121,23,135)(10,68,24,82)(11,123,25,137)(12,70,26,84)(13,125,27,139)(14,72,28,58)(29,191,43,177)(30,144,44,158)(31,193,45,179)(32,146,46,160)(33,195,47,181)(34,148,48,162)(35,169,49,183)(36,150,50,164)(37,171,51,185)(38,152,52,166)(39,173,53,187)(40,154,54,168)(41,175,55,189)(42,156,56,142)(57,207,71,221)(59,209,73,223)(61,211,75,197)(63,213,77,199)(65,215,79,201)(67,217,81,203)(69,219,83,205)(85,149,99,163)(86,170,100,184)(87,151,101,165)(88,172,102,186)(89,153,103,167)(90,174,104,188)(91,155,105,141)(92,176,106,190)(93,157,107,143)(94,178,108,192)(95,159,109,145)(96,180,110,194)(97,161,111,147)(98,182,112,196)(114,210,128,224)(116,212,130,198)(118,214,132,200)(120,216,134,202)(122,218,136,204)(124,220,138,206)(126,222,140,208), (1,148,223,196)(2,163,224,183)(3,150,197,170)(4,165,198,185)(5,152,199,172)(6,167,200,187)(7,154,201,174)(8,141,202,189)(9,156,203,176)(10,143,204,191)(11,158,205,178)(12,145,206,193)(13,160,207,180)(14,147,208,195)(15,162,209,182)(16,149,210,169)(17,164,211,184)(18,151,212,171)(19,166,213,186)(20,153,214,173)(21,168,215,188)(22,155,216,175)(23,142,217,190)(24,157,218,177)(25,144,219,192)(26,159,220,179)(27,146,221,194)(28,161,222,181)(29,82,107,136)(30,69,108,123)(31,84,109,138)(32,71,110,125)(33,58,111,140)(34,73,112,127)(35,60,85,114)(36,75,86,129)(37,62,87,116)(38,77,88,131)(39,64,89,118)(40,79,90,133)(41,66,91,120)(42,81,92,135)(43,68,93,122)(44,83,94,137)(45,70,95,124)(46,57,96,139)(47,72,97,126)(48,59,98,113)(49,74,99,128)(50,61,100,115)(51,76,101,130)(52,63,102,117)(53,78,103,132)(54,65,104,119)(55,80,105,134)(56,67,106,121), (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), (1,97,15,111)(2,110,16,96)(3,95,17,109)(4,108,18,94)(5,93,19,107)(6,106,20,92)(7,91,21,105)(8,104,22,90)(9,89,23,103)(10,102,24,88)(11,87,25,101)(12,100,26,86)(13,85,27,99)(14,98,28,112)(29,199,43,213)(30,212,44,198)(31,197,45,211)(32,210,46,224)(33,223,47,209)(34,208,48,222)(35,221,49,207)(36,206,50,220)(37,219,51,205)(38,204,52,218)(39,217,53,203)(40,202,54,216)(41,215,55,201)(42,200,56,214)(57,149,71,163)(58,162,72,148)(59,147,73,161)(60,160,74,146)(61,145,75,159)(62,158,76,144)(63,143,77,157)(64,156,78,142)(65,141,79,155)(66,154,80,168)(67,167,81,153)(68,152,82,166)(69,165,83,151)(70,150,84,164)(113,195,127,181)(114,180,128,194)(115,193,129,179)(116,178,130,192)(117,191,131,177)(118,176,132,190)(119,189,133,175)(120,174,134,188)(121,187,135,173)(122,172,136,186)(123,185,137,171)(124,170,138,184)(125,183,139,169)(126,196,140,182)>;
G:=Group( (1,113,15,127)(2,60,16,74)(3,115,17,129)(4,62,18,76)(5,117,19,131)(6,64,20,78)(7,119,21,133)(8,66,22,80)(9,121,23,135)(10,68,24,82)(11,123,25,137)(12,70,26,84)(13,125,27,139)(14,72,28,58)(29,191,43,177)(30,144,44,158)(31,193,45,179)(32,146,46,160)(33,195,47,181)(34,148,48,162)(35,169,49,183)(36,150,50,164)(37,171,51,185)(38,152,52,166)(39,173,53,187)(40,154,54,168)(41,175,55,189)(42,156,56,142)(57,207,71,221)(59,209,73,223)(61,211,75,197)(63,213,77,199)(65,215,79,201)(67,217,81,203)(69,219,83,205)(85,149,99,163)(86,170,100,184)(87,151,101,165)(88,172,102,186)(89,153,103,167)(90,174,104,188)(91,155,105,141)(92,176,106,190)(93,157,107,143)(94,178,108,192)(95,159,109,145)(96,180,110,194)(97,161,111,147)(98,182,112,196)(114,210,128,224)(116,212,130,198)(118,214,132,200)(120,216,134,202)(122,218,136,204)(124,220,138,206)(126,222,140,208), (1,148,223,196)(2,163,224,183)(3,150,197,170)(4,165,198,185)(5,152,199,172)(6,167,200,187)(7,154,201,174)(8,141,202,189)(9,156,203,176)(10,143,204,191)(11,158,205,178)(12,145,206,193)(13,160,207,180)(14,147,208,195)(15,162,209,182)(16,149,210,169)(17,164,211,184)(18,151,212,171)(19,166,213,186)(20,153,214,173)(21,168,215,188)(22,155,216,175)(23,142,217,190)(24,157,218,177)(25,144,219,192)(26,159,220,179)(27,146,221,194)(28,161,222,181)(29,82,107,136)(30,69,108,123)(31,84,109,138)(32,71,110,125)(33,58,111,140)(34,73,112,127)(35,60,85,114)(36,75,86,129)(37,62,87,116)(38,77,88,131)(39,64,89,118)(40,79,90,133)(41,66,91,120)(42,81,92,135)(43,68,93,122)(44,83,94,137)(45,70,95,124)(46,57,96,139)(47,72,97,126)(48,59,98,113)(49,74,99,128)(50,61,100,115)(51,76,101,130)(52,63,102,117)(53,78,103,132)(54,65,104,119)(55,80,105,134)(56,67,106,121), (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), (1,97,15,111)(2,110,16,96)(3,95,17,109)(4,108,18,94)(5,93,19,107)(6,106,20,92)(7,91,21,105)(8,104,22,90)(9,89,23,103)(10,102,24,88)(11,87,25,101)(12,100,26,86)(13,85,27,99)(14,98,28,112)(29,199,43,213)(30,212,44,198)(31,197,45,211)(32,210,46,224)(33,223,47,209)(34,208,48,222)(35,221,49,207)(36,206,50,220)(37,219,51,205)(38,204,52,218)(39,217,53,203)(40,202,54,216)(41,215,55,201)(42,200,56,214)(57,149,71,163)(58,162,72,148)(59,147,73,161)(60,160,74,146)(61,145,75,159)(62,158,76,144)(63,143,77,157)(64,156,78,142)(65,141,79,155)(66,154,80,168)(67,167,81,153)(68,152,82,166)(69,165,83,151)(70,150,84,164)(113,195,127,181)(114,180,128,194)(115,193,129,179)(116,178,130,192)(117,191,131,177)(118,176,132,190)(119,189,133,175)(120,174,134,188)(121,187,135,173)(122,172,136,186)(123,185,137,171)(124,170,138,184)(125,183,139,169)(126,196,140,182) );
G=PermutationGroup([(1,113,15,127),(2,60,16,74),(3,115,17,129),(4,62,18,76),(5,117,19,131),(6,64,20,78),(7,119,21,133),(8,66,22,80),(9,121,23,135),(10,68,24,82),(11,123,25,137),(12,70,26,84),(13,125,27,139),(14,72,28,58),(29,191,43,177),(30,144,44,158),(31,193,45,179),(32,146,46,160),(33,195,47,181),(34,148,48,162),(35,169,49,183),(36,150,50,164),(37,171,51,185),(38,152,52,166),(39,173,53,187),(40,154,54,168),(41,175,55,189),(42,156,56,142),(57,207,71,221),(59,209,73,223),(61,211,75,197),(63,213,77,199),(65,215,79,201),(67,217,81,203),(69,219,83,205),(85,149,99,163),(86,170,100,184),(87,151,101,165),(88,172,102,186),(89,153,103,167),(90,174,104,188),(91,155,105,141),(92,176,106,190),(93,157,107,143),(94,178,108,192),(95,159,109,145),(96,180,110,194),(97,161,111,147),(98,182,112,196),(114,210,128,224),(116,212,130,198),(118,214,132,200),(120,216,134,202),(122,218,136,204),(124,220,138,206),(126,222,140,208)], [(1,148,223,196),(2,163,224,183),(3,150,197,170),(4,165,198,185),(5,152,199,172),(6,167,200,187),(7,154,201,174),(8,141,202,189),(9,156,203,176),(10,143,204,191),(11,158,205,178),(12,145,206,193),(13,160,207,180),(14,147,208,195),(15,162,209,182),(16,149,210,169),(17,164,211,184),(18,151,212,171),(19,166,213,186),(20,153,214,173),(21,168,215,188),(22,155,216,175),(23,142,217,190),(24,157,218,177),(25,144,219,192),(26,159,220,179),(27,146,221,194),(28,161,222,181),(29,82,107,136),(30,69,108,123),(31,84,109,138),(32,71,110,125),(33,58,111,140),(34,73,112,127),(35,60,85,114),(36,75,86,129),(37,62,87,116),(38,77,88,131),(39,64,89,118),(40,79,90,133),(41,66,91,120),(42,81,92,135),(43,68,93,122),(44,83,94,137),(45,70,95,124),(46,57,96,139),(47,72,97,126),(48,59,98,113),(49,74,99,128),(50,61,100,115),(51,76,101,130),(52,63,102,117),(53,78,103,132),(54,65,104,119),(55,80,105,134),(56,67,106,121)], [(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)], [(1,97,15,111),(2,110,16,96),(3,95,17,109),(4,108,18,94),(5,93,19,107),(6,106,20,92),(7,91,21,105),(8,104,22,90),(9,89,23,103),(10,102,24,88),(11,87,25,101),(12,100,26,86),(13,85,27,99),(14,98,28,112),(29,199,43,213),(30,212,44,198),(31,197,45,211),(32,210,46,224),(33,223,47,209),(34,208,48,222),(35,221,49,207),(36,206,50,220),(37,219,51,205),(38,204,52,218),(39,217,53,203),(40,202,54,216),(41,215,55,201),(42,200,56,214),(57,149,71,163),(58,162,72,148),(59,147,73,161),(60,160,74,146),(61,145,75,159),(62,158,76,144),(63,143,77,157),(64,156,78,142),(65,141,79,155),(66,154,80,168),(67,167,81,153),(68,152,82,166),(69,165,83,151),(70,150,84,164),(113,195,127,181),(114,180,128,194),(115,193,129,179),(116,178,130,192),(117,191,131,177),(118,176,132,190),(119,189,133,175),(120,174,134,188),(121,187,135,173),(122,172,136,186),(123,185,137,171),(124,170,138,184),(125,183,139,169),(126,196,140,182)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 1 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 2 | 0 | 0 | 0 | 0 |
| 28 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 14 | 0 | 4 |
| 0 | 0 | 0 | 0 | 20 | 2 | 25 | 11 |
| 0 | 0 | 0 | 0 | 19 | 15 | 22 | 5 |
| 0 | 0 | 0 | 0 | 1 | 14 | 14 | 11 |
| 28 | 0 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 27 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 14 | 8 | 6 |
| 0 | 0 | 0 | 0 | 14 | 0 | 23 | 21 |
| 10 | 9 | 25 | 1 | 0 | 0 | 0 | 0 |
| 11 | 11 | 27 | 9 | 0 | 0 | 0 | 0 |
| 17 | 6 | 19 | 20 | 0 | 0 | 0 | 0 |
| 17 | 8 | 18 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 | 16 | 20 |
| 0 | 0 | 0 | 0 | 16 | 21 | 28 | 13 |
| 0 | 0 | 0 | 0 | 21 | 4 | 6 | 10 |
| 0 | 0 | 0 | 0 | 18 | 12 | 27 | 2 |
| 4 | 22 | 28 | 6 | 0 | 0 | 0 | 0 |
| 9 | 25 | 28 | 1 | 0 | 0 | 0 | 0 |
| 15 | 26 | 5 | 16 | 0 | 0 | 0 | 0 |
| 15 | 14 | 10 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 10 | 8 | 28 |
| 0 | 0 | 0 | 0 | 9 | 6 | 21 | 18 |
| 0 | 0 | 0 | 0 | 10 | 24 | 28 | 24 |
| 0 | 0 | 0 | 0 | 17 | 15 | 24 | 10 |
G:=sub<GL(8,GF(29))| [1,0,28,0,0,0,0,0,0,1,0,28,0,0,0,0,2,0,28,0,0,0,0,0,0,2,0,28,0,0,0,0,0,0,0,0,23,20,19,1,0,0,0,0,14,2,15,14,0,0,0,0,0,25,22,14,0,0,0,0,4,11,5,11],[28,0,1,0,0,0,0,0,0,28,0,1,0,0,0,0,27,0,1,0,0,0,0,0,0,27,0,1,0,0,0,0,0,0,0,0,19,17,16,14,0,0,0,0,6,10,14,0,0,0,0,0,0,0,8,23,0,0,0,0,0,0,6,21],[10,11,17,17,0,0,0,0,9,11,6,8,0,0,0,0,25,27,19,18,0,0,0,0,1,9,20,18,0,0,0,0,0,0,0,0,0,16,21,18,0,0,0,0,27,21,4,12,0,0,0,0,16,28,6,27,0,0,0,0,20,13,10,2],[4,9,15,15,0,0,0,0,22,25,26,14,0,0,0,0,28,28,5,10,0,0,0,0,6,1,16,24,0,0,0,0,0,0,0,0,14,9,10,17,0,0,0,0,10,6,24,15,0,0,0,0,8,21,28,24,0,0,0,0,28,18,24,10] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2H | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 28 | ··· | 28 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | 2+ (1+4) | Q8⋊2D7 | D4⋊8D14 |
| kernel | C42.156D14 | C42⋊D7 | C4⋊D28 | D28⋊C4 | D14.5D4 | C28⋊1D4 | C7×C42.C2 | C42.C2 | C28 | C42 | C4⋊C4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 4 | 6 | 1 | 3 | 4 | 3 | 18 | 2 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2._{156}D_{14} % in TeX
G:=Group("C4^2.156D14"); // GroupNames label
G:=SmallGroup(448,1151);
// by ID
G=gap.SmallGroup(448,1151);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,184,675,570,80,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^13>;
// generators/relations
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