metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.160D14, C14.992- (1+4), C28⋊Q8⋊40C2, C4⋊C4.117D14, C42⋊D7⋊5C2, C42⋊2C2⋊1D7, D14⋊Q8⋊39C2, (C4×Dic14)⋊14C2, (C4×C28).32C22, C22⋊C4.75D14, Dic7⋊3Q8⋊40C2, (C2×C28).192C23, (C2×C14).246C24, Dic7⋊4D4.4C2, C23.52(C22×D7), D14⋊C4.139C22, Dic7.31(C4○D4), C22⋊Dic14⋊44C2, C23.D14⋊43C2, C4⋊Dic7.317C22, (C22×C14).60C23, Dic7.D4.4C2, C22.267(C23×D7), C23.D7.62C22, C23.11D14⋊20C2, Dic7⋊C4.145C22, C7⋊7(C22.50C24), (C4×Dic7).149C22, (C2×Dic7).263C23, (C22×D7).110C23, C2.63(D4.10D14), (C2×Dic14).254C22, (C22×Dic7).149C22, C2.93(D7×C4○D4), C4⋊C4⋊D7⋊39C2, (C7×C42⋊2C2)⋊1C2, C14.204(C2×C4○D4), (C2×C4×D7).218C22, (C2×C4).83(C22×D7), (C7×C4⋊C4).201C22, (C2×C7⋊D4).67C22, (C7×C22⋊C4).71C22, SmallGroup(448,1155)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 876 in 212 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×2], C4 [×15], C22, C22 [×6], C7, C2×C4 [×6], C2×C4 [×11], D4 [×2], Q8 [×6], C23, C23, D7, C14 [×3], C14, C42, C42 [×6], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×9], C22×C4 [×2], C2×D4, C2×Q8 [×3], Dic7 [×4], Dic7 [×5], C28 [×6], D14 [×3], C2×C14, C2×C14 [×3], C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2, C42⋊2C2 [×3], C4⋊Q8, Dic14 [×6], C4×D7 [×2], C2×Dic7 [×7], C2×Dic7 [×2], C7⋊D4 [×2], C2×C28 [×6], C22×D7, C22×C14, C22.50C24, C4×Dic7 [×6], Dic7⋊C4 [×7], C4⋊Dic7 [×2], 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, C4×Dic14, C42⋊D7, C23.11D14, C22⋊Dic14, C23.D14, Dic7⋊4D4, Dic7.D4 [×2], Dic7⋊3Q8 [×2], C28⋊Q8, D14⋊Q8, C4⋊C4⋊D7 [×2], C7×C42⋊2C2, C42.160D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×4], C24, D14 [×7], C2×C4○D4 [×2], 2- (1+4), C22×D7 [×7], C22.50C24, C23×D7, D7×C4○D4 [×2], D4.10D14, C42.160D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=1, d2=b2, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd-1=a2b-1, dcd-1=c-1 >
►On 224 points
Generators in S224
(1 111 199 117)(2 89 200 135)(3 99 201 119)(4 91 202 137)(5 101 203 121)(6 93 204 139)(7 103 205 123)(8 95 206 127)(9 105 207 125)(10 97 208 129)(11 107 209 113)(12 85 210 131)(13 109 197 115)(14 87 198 133)(15 74 56 154)(16 195 43 159)(17 76 44 142)(18 183 45 161)(19 78 46 144)(20 185 47 163)(21 80 48 146)(22 187 49 165)(23 82 50 148)(24 189 51 167)(25 84 52 150)(26 191 53 155)(27 72 54 152)(28 193 55 157)(29 122 59 102)(30 140 60 94)(31 124 61 104)(32 128 62 96)(33 126 63 106)(34 130 64 98)(35 114 65 108)(36 132 66 86)(37 116 67 110)(38 134 68 88)(39 118 69 112)(40 136 70 90)(41 120 57 100)(42 138 58 92)(71 180 151 218)(73 182 153 220)(75 170 141 222)(77 172 143 224)(79 174 145 212)(81 176 147 214)(83 178 149 216)(156 219 192 181)(158 221 194 169)(160 223 196 171)(162 211 184 173)(164 213 186 175)(166 215 188 177)(168 217 190 179)
(1 223 68 17)(2 45 69 172)(3 211 70 19)(4 47 57 174)(5 213 58 21)(6 49 59 176)(7 215 60 23)(8 51 61 178)(9 217 62 25)(10 53 63 180)(11 219 64 27)(12 55 65 182)(13 221 66 15)(14 43 67 170)(16 37 222 198)(18 39 224 200)(20 41 212 202)(22 29 214 204)(24 31 216 206)(26 33 218 208)(28 35 220 210)(30 50 205 177)(32 52 207 179)(34 54 209 181)(36 56 197 169)(38 44 199 171)(40 46 201 173)(42 48 203 175)(71 129 191 126)(72 107 192 98)(73 131 193 114)(74 109 194 86)(75 133 195 116)(76 111 196 88)(77 135 183 118)(78 99 184 90)(79 137 185 120)(80 101 186 92)(81 139 187 122)(82 103 188 94)(83 127 189 124)(84 105 190 96)(85 157 108 153)(87 159 110 141)(89 161 112 143)(91 163 100 145)(93 165 102 147)(95 167 104 149)(97 155 106 151)(113 156 130 152)(115 158 132 154)(117 160 134 142)(119 162 136 144)(121 164 138 146)(123 166 140 148)(125 168 128 150)
(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 16 68 222)(2 15 69 221)(3 28 70 220)(4 27 57 219)(5 26 58 218)(6 25 59 217)(7 24 60 216)(8 23 61 215)(9 22 62 214)(10 21 63 213)(11 20 64 212)(12 19 65 211)(13 18 66 224)(14 17 67 223)(29 179 204 52)(30 178 205 51)(31 177 206 50)(32 176 207 49)(33 175 208 48)(34 174 209 47)(35 173 210 46)(36 172 197 45)(37 171 198 44)(38 170 199 43)(39 169 200 56)(40 182 201 55)(41 181 202 54)(42 180 203 53)(71 101 191 92)(72 100 192 91)(73 99 193 90)(74 112 194 89)(75 111 195 88)(76 110 196 87)(77 109 183 86)(78 108 184 85)(79 107 185 98)(80 106 186 97)(81 105 187 96)(82 104 188 95)(83 103 189 94)(84 102 190 93)(113 163 130 145)(114 162 131 144)(115 161 132 143)(116 160 133 142)(117 159 134 141)(118 158 135 154)(119 157 136 153)(120 156 137 152)(121 155 138 151)(122 168 139 150)(123 167 140 149)(124 166 127 148)(125 165 128 147)(126 164 129 146)
G:=sub<Sym(224)| (1,111,199,117)(2,89,200,135)(3,99,201,119)(4,91,202,137)(5,101,203,121)(6,93,204,139)(7,103,205,123)(8,95,206,127)(9,105,207,125)(10,97,208,129)(11,107,209,113)(12,85,210,131)(13,109,197,115)(14,87,198,133)(15,74,56,154)(16,195,43,159)(17,76,44,142)(18,183,45,161)(19,78,46,144)(20,185,47,163)(21,80,48,146)(22,187,49,165)(23,82,50,148)(24,189,51,167)(25,84,52,150)(26,191,53,155)(27,72,54,152)(28,193,55,157)(29,122,59,102)(30,140,60,94)(31,124,61,104)(32,128,62,96)(33,126,63,106)(34,130,64,98)(35,114,65,108)(36,132,66,86)(37,116,67,110)(38,134,68,88)(39,118,69,112)(40,136,70,90)(41,120,57,100)(42,138,58,92)(71,180,151,218)(73,182,153,220)(75,170,141,222)(77,172,143,224)(79,174,145,212)(81,176,147,214)(83,178,149,216)(156,219,192,181)(158,221,194,169)(160,223,196,171)(162,211,184,173)(164,213,186,175)(166,215,188,177)(168,217,190,179), (1,223,68,17)(2,45,69,172)(3,211,70,19)(4,47,57,174)(5,213,58,21)(6,49,59,176)(7,215,60,23)(8,51,61,178)(9,217,62,25)(10,53,63,180)(11,219,64,27)(12,55,65,182)(13,221,66,15)(14,43,67,170)(16,37,222,198)(18,39,224,200)(20,41,212,202)(22,29,214,204)(24,31,216,206)(26,33,218,208)(28,35,220,210)(30,50,205,177)(32,52,207,179)(34,54,209,181)(36,56,197,169)(38,44,199,171)(40,46,201,173)(42,48,203,175)(71,129,191,126)(72,107,192,98)(73,131,193,114)(74,109,194,86)(75,133,195,116)(76,111,196,88)(77,135,183,118)(78,99,184,90)(79,137,185,120)(80,101,186,92)(81,139,187,122)(82,103,188,94)(83,127,189,124)(84,105,190,96)(85,157,108,153)(87,159,110,141)(89,161,112,143)(91,163,100,145)(93,165,102,147)(95,167,104,149)(97,155,106,151)(113,156,130,152)(115,158,132,154)(117,160,134,142)(119,162,136,144)(121,164,138,146)(123,166,140,148)(125,168,128,150), (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), (1,16,68,222)(2,15,69,221)(3,28,70,220)(4,27,57,219)(5,26,58,218)(6,25,59,217)(7,24,60,216)(8,23,61,215)(9,22,62,214)(10,21,63,213)(11,20,64,212)(12,19,65,211)(13,18,66,224)(14,17,67,223)(29,179,204,52)(30,178,205,51)(31,177,206,50)(32,176,207,49)(33,175,208,48)(34,174,209,47)(35,173,210,46)(36,172,197,45)(37,171,198,44)(38,170,199,43)(39,169,200,56)(40,182,201,55)(41,181,202,54)(42,180,203,53)(71,101,191,92)(72,100,192,91)(73,99,193,90)(74,112,194,89)(75,111,195,88)(76,110,196,87)(77,109,183,86)(78,108,184,85)(79,107,185,98)(80,106,186,97)(81,105,187,96)(82,104,188,95)(83,103,189,94)(84,102,190,93)(113,163,130,145)(114,162,131,144)(115,161,132,143)(116,160,133,142)(117,159,134,141)(118,158,135,154)(119,157,136,153)(120,156,137,152)(121,155,138,151)(122,168,139,150)(123,167,140,149)(124,166,127,148)(125,165,128,147)(126,164,129,146)>;
G:=Group( (1,111,199,117)(2,89,200,135)(3,99,201,119)(4,91,202,137)(5,101,203,121)(6,93,204,139)(7,103,205,123)(8,95,206,127)(9,105,207,125)(10,97,208,129)(11,107,209,113)(12,85,210,131)(13,109,197,115)(14,87,198,133)(15,74,56,154)(16,195,43,159)(17,76,44,142)(18,183,45,161)(19,78,46,144)(20,185,47,163)(21,80,48,146)(22,187,49,165)(23,82,50,148)(24,189,51,167)(25,84,52,150)(26,191,53,155)(27,72,54,152)(28,193,55,157)(29,122,59,102)(30,140,60,94)(31,124,61,104)(32,128,62,96)(33,126,63,106)(34,130,64,98)(35,114,65,108)(36,132,66,86)(37,116,67,110)(38,134,68,88)(39,118,69,112)(40,136,70,90)(41,120,57,100)(42,138,58,92)(71,180,151,218)(73,182,153,220)(75,170,141,222)(77,172,143,224)(79,174,145,212)(81,176,147,214)(83,178,149,216)(156,219,192,181)(158,221,194,169)(160,223,196,171)(162,211,184,173)(164,213,186,175)(166,215,188,177)(168,217,190,179), (1,223,68,17)(2,45,69,172)(3,211,70,19)(4,47,57,174)(5,213,58,21)(6,49,59,176)(7,215,60,23)(8,51,61,178)(9,217,62,25)(10,53,63,180)(11,219,64,27)(12,55,65,182)(13,221,66,15)(14,43,67,170)(16,37,222,198)(18,39,224,200)(20,41,212,202)(22,29,214,204)(24,31,216,206)(26,33,218,208)(28,35,220,210)(30,50,205,177)(32,52,207,179)(34,54,209,181)(36,56,197,169)(38,44,199,171)(40,46,201,173)(42,48,203,175)(71,129,191,126)(72,107,192,98)(73,131,193,114)(74,109,194,86)(75,133,195,116)(76,111,196,88)(77,135,183,118)(78,99,184,90)(79,137,185,120)(80,101,186,92)(81,139,187,122)(82,103,188,94)(83,127,189,124)(84,105,190,96)(85,157,108,153)(87,159,110,141)(89,161,112,143)(91,163,100,145)(93,165,102,147)(95,167,104,149)(97,155,106,151)(113,156,130,152)(115,158,132,154)(117,160,134,142)(119,162,136,144)(121,164,138,146)(123,166,140,148)(125,168,128,150), (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), (1,16,68,222)(2,15,69,221)(3,28,70,220)(4,27,57,219)(5,26,58,218)(6,25,59,217)(7,24,60,216)(8,23,61,215)(9,22,62,214)(10,21,63,213)(11,20,64,212)(12,19,65,211)(13,18,66,224)(14,17,67,223)(29,179,204,52)(30,178,205,51)(31,177,206,50)(32,176,207,49)(33,175,208,48)(34,174,209,47)(35,173,210,46)(36,172,197,45)(37,171,198,44)(38,170,199,43)(39,169,200,56)(40,182,201,55)(41,181,202,54)(42,180,203,53)(71,101,191,92)(72,100,192,91)(73,99,193,90)(74,112,194,89)(75,111,195,88)(76,110,196,87)(77,109,183,86)(78,108,184,85)(79,107,185,98)(80,106,186,97)(81,105,187,96)(82,104,188,95)(83,103,189,94)(84,102,190,93)(113,163,130,145)(114,162,131,144)(115,161,132,143)(116,160,133,142)(117,159,134,141)(118,158,135,154)(119,157,136,153)(120,156,137,152)(121,155,138,151)(122,168,139,150)(123,167,140,149)(124,166,127,148)(125,165,128,147)(126,164,129,146) );
G=PermutationGroup([(1,111,199,117),(2,89,200,135),(3,99,201,119),(4,91,202,137),(5,101,203,121),(6,93,204,139),(7,103,205,123),(8,95,206,127),(9,105,207,125),(10,97,208,129),(11,107,209,113),(12,85,210,131),(13,109,197,115),(14,87,198,133),(15,74,56,154),(16,195,43,159),(17,76,44,142),(18,183,45,161),(19,78,46,144),(20,185,47,163),(21,80,48,146),(22,187,49,165),(23,82,50,148),(24,189,51,167),(25,84,52,150),(26,191,53,155),(27,72,54,152),(28,193,55,157),(29,122,59,102),(30,140,60,94),(31,124,61,104),(32,128,62,96),(33,126,63,106),(34,130,64,98),(35,114,65,108),(36,132,66,86),(37,116,67,110),(38,134,68,88),(39,118,69,112),(40,136,70,90),(41,120,57,100),(42,138,58,92),(71,180,151,218),(73,182,153,220),(75,170,141,222),(77,172,143,224),(79,174,145,212),(81,176,147,214),(83,178,149,216),(156,219,192,181),(158,221,194,169),(160,223,196,171),(162,211,184,173),(164,213,186,175),(166,215,188,177),(168,217,190,179)], [(1,223,68,17),(2,45,69,172),(3,211,70,19),(4,47,57,174),(5,213,58,21),(6,49,59,176),(7,215,60,23),(8,51,61,178),(9,217,62,25),(10,53,63,180),(11,219,64,27),(12,55,65,182),(13,221,66,15),(14,43,67,170),(16,37,222,198),(18,39,224,200),(20,41,212,202),(22,29,214,204),(24,31,216,206),(26,33,218,208),(28,35,220,210),(30,50,205,177),(32,52,207,179),(34,54,209,181),(36,56,197,169),(38,44,199,171),(40,46,201,173),(42,48,203,175),(71,129,191,126),(72,107,192,98),(73,131,193,114),(74,109,194,86),(75,133,195,116),(76,111,196,88),(77,135,183,118),(78,99,184,90),(79,137,185,120),(80,101,186,92),(81,139,187,122),(82,103,188,94),(83,127,189,124),(84,105,190,96),(85,157,108,153),(87,159,110,141),(89,161,112,143),(91,163,100,145),(93,165,102,147),(95,167,104,149),(97,155,106,151),(113,156,130,152),(115,158,132,154),(117,160,134,142),(119,162,136,144),(121,164,138,146),(123,166,140,148),(125,168,128,150)], [(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)], [(1,16,68,222),(2,15,69,221),(3,28,70,220),(4,27,57,219),(5,26,58,218),(6,25,59,217),(7,24,60,216),(8,23,61,215),(9,22,62,214),(10,21,63,213),(11,20,64,212),(12,19,65,211),(13,18,66,224),(14,17,67,223),(29,179,204,52),(30,178,205,51),(31,177,206,50),(32,176,207,49),(33,175,208,48),(34,174,209,47),(35,173,210,46),(36,172,197,45),(37,171,198,44),(38,170,199,43),(39,169,200,56),(40,182,201,55),(41,181,202,54),(42,180,203,53),(71,101,191,92),(72,100,192,91),(73,99,193,90),(74,112,194,89),(75,111,195,88),(76,110,196,87),(77,109,183,86),(78,108,184,85),(79,107,185,98),(80,106,186,97),(81,105,187,96),(82,104,188,95),(83,103,189,94),(84,102,190,93),(113,163,130,145),(114,162,131,144),(115,161,132,143),(116,160,133,142),(117,159,134,141),(118,158,135,154),(119,157,136,153),(120,156,137,152),(121,155,138,151),(122,168,139,150),(123,167,140,149),(124,166,127,148),(125,165,128,147),(126,164,129,146)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 15 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 10 | 0 | 0 |
| 0 | 0 | 19 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 5 |
| 0 | 0 | 0 | 0 | 22 | 23 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 5 |
| 0 | 0 | 0 | 0 | 22 | 23 |
G:=sub<GL(6,GF(29))| [0,1,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,15,0,0,0,0,0,28],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,22,19,0,0,0,0,10,10,0,0,0,0,0,0,6,22,0,0,0,0,5,23],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,6,22,0,0,0,0,5,23] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 7A | 7B | 7C | 14A | ··· | 14I | 14J | 14K | 14L | 28A | ··· | 28R | 28S | ··· | 28AA |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | 2- (1+4) | D7×C4○D4 | D4.10D14 |
| kernel | C42.160D14 | C4×Dic14 | C42⋊D7 | C23.11D14 | C22⋊Dic14 | C23.D14 | Dic7⋊4D4 | Dic7.D4 | Dic7⋊3Q8 | C28⋊Q8 | D14⋊Q8 | C4⋊C4⋊D7 | C7×C42⋊2C2 | C42⋊2C2 | Dic7 | C42 | C22⋊C4 | C4⋊C4 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 3 | 8 | 3 | 9 | 9 | 1 | 12 | 6 |
In GAP, Magma, Sage, TeX
C_4^2._{160}D_{14} % in TeX
G:=Group("C4^2.160D14"); // GroupNames label
G:=SmallGroup(448,1155);
// by ID
G=gap.SmallGroup(448,1155);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,387,100,794,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations
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