metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.233D14, (C4×D7)⋊6D4, C4.30(D4×D7), D14.6(C2×D4), C28.59(C2×D4), (D7×C42)⋊8C2, C28⋊D4⋊23C2, C4.4D4⋊18D7, D14⋊D4⋊37C2, Dic7⋊3(C4○D4), (C2×D4).168D14, C4.D28⋊22C2, (C2×C28).76C23, (C2×Q8).134D14, C22⋊C4.70D14, Dic7.65(C2×D4), C14.86(C22×D4), Dic7⋊4D4⋊27C2, Dic7⋊Q8⋊19C2, (C2×C14).212C24, (C4×C28).182C22, D14⋊C4.58C22, C23.34(C22×D7), (C2×D28).160C22, (D4×C14).150C22, Dic7⋊C4.47C22, (C22×C14).42C23, (Q8×C14).121C22, C7⋊4(C22.26C24), C22.233(C23×D7), (C2×Dic7).249C23, (C4×Dic7).296C22, (C22×D7).212C23, (C2×Dic14).173C22, (C22×Dic7).137C22, C2.59(C2×D4×D7), C2.71(D7×C4○D4), (C2×Q8⋊2D7)⋊9C2, (C7×C4.4D4)⋊6C2, (C2×D4⋊2D7)⋊18C2, C14.183(C2×C4○D4), (C2×C4×D7).118C22, (C2×C4).298(C22×D7), (C2×C7⋊D4).55C22, (C7×C22⋊C4).59C22, SmallGroup(448,1121)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.233D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=dbd=a2b, dcd=a2c-1 >
Subgroups: 1516 in 310 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, Dic14, C4×D7, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×D7, C22×C14, C22.26C24, C4×Dic7, C4×Dic7, Dic7⋊C4, D14⋊C4, C4×C28, C7×C22⋊C4, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, D4⋊2D7, Q8⋊2D7, C22×Dic7, C2×C7⋊D4, D4×C14, Q8×C14, D7×C42, C4.D28, Dic7⋊4D4, D14⋊D4, C28⋊D4, Dic7⋊Q8, C7×C4.4D4, C2×D4⋊2D7, C2×Q8⋊2D7, C42.233D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, C22×D7, C22.26C24, D4×D7, C23×D7, C2×D4×D7, D7×C4○D4, C42.233D14
►On 224 points
Generators in S224
(1 159 32 143)(2 213 33 132)(3 161 34 145)(4 215 35 134)(5 163 36 147)(6 217 37 136)(7 165 38 149)(8 219 39 138)(9 167 40 151)(10 221 41 140)(11 155 42 153)(12 223 29 128)(13 157 30 141)(14 211 31 130)(15 64 51 200)(16 194 52 116)(17 66 53 202)(18 196 54 118)(19 68 55 204)(20 184 56 120)(21 70 43 206)(22 186 44 122)(23 58 45 208)(24 188 46 124)(25 60 47 210)(26 190 48 126)(27 62 49 198)(28 192 50 114)(57 106 207 72)(59 108 209 74)(61 110 197 76)(63 112 199 78)(65 100 201 80)(67 102 203 82)(69 104 205 84)(71 185 105 121)(73 187 107 123)(75 189 109 125)(77 191 111 113)(79 193 99 115)(81 195 101 117)(83 183 103 119)(85 137 171 218)(86 150 172 166)(87 139 173 220)(88 152 174 168)(89 127 175 222)(90 154 176 156)(91 129 177 224)(92 142 178 158)(93 131 179 212)(94 144 180 160)(95 133 181 214)(96 146 182 162)(97 135 169 216)(98 148 170 164)
(1 59 179 188)(2 210 180 125)(3 61 181 190)(4 198 182 113)(5 63 169 192)(6 200 170 115)(7 65 171 194)(8 202 172 117)(9 67 173 196)(10 204 174 119)(11 69 175 184)(12 206 176 121)(13 57 177 186)(14 208 178 123)(15 164 79 217)(16 149 80 137)(17 166 81 219)(18 151 82 139)(19 168 83 221)(20 153 84 127)(21 156 71 223)(22 141 72 129)(23 158 73 211)(24 143 74 131)(25 160 75 213)(26 145 76 133)(27 162 77 215)(28 147 78 135)(29 70 90 185)(30 207 91 122)(31 58 92 187)(32 209 93 124)(33 60 94 189)(34 197 95 126)(35 62 96 191)(36 199 97 114)(37 64 98 193)(38 201 85 116)(39 66 86 195)(40 203 87 118)(41 68 88 183)(42 205 89 120)(43 154 105 128)(44 157 106 224)(45 142 107 130)(46 159 108 212)(47 144 109 132)(48 161 110 214)(49 146 111 134)(50 163 112 216)(51 148 99 136)(52 165 100 218)(53 150 101 138)(54 167 102 220)(55 152 103 140)(56 155 104 222)
(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)
(1 158)(2 141)(3 156)(4 153)(5 168)(6 151)(7 166)(8 149)(9 164)(10 147)(11 162)(12 145)(13 160)(14 143)(15 118)(16 195)(17 116)(18 193)(19 114)(20 191)(21 126)(22 189)(23 124)(24 187)(25 122)(26 185)(27 120)(28 183)(29 161)(30 144)(31 159)(32 142)(33 157)(34 154)(35 155)(36 152)(37 167)(38 150)(39 165)(40 148)(41 163)(42 146)(43 190)(44 125)(45 188)(46 123)(47 186)(48 121)(49 184)(50 119)(51 196)(52 117)(53 194)(54 115)(55 192)(56 113)(57 109)(58 74)(59 107)(60 72)(61 105)(62 84)(63 103)(64 82)(65 101)(66 80)(67 99)(68 78)(69 111)(70 76)(71 197)(73 209)(75 207)(77 205)(79 203)(81 201)(83 199)(85 138)(86 218)(87 136)(88 216)(89 134)(90 214)(91 132)(92 212)(93 130)(94 224)(95 128)(96 222)(97 140)(98 220)(100 202)(102 200)(104 198)(106 210)(108 208)(110 206)(112 204)(127 182)(129 180)(131 178)(133 176)(135 174)(137 172)(139 170)(169 221)(171 219)(173 217)(175 215)(177 213)(179 211)(181 223)
G:=sub<Sym(224)| (1,159,32,143)(2,213,33,132)(3,161,34,145)(4,215,35,134)(5,163,36,147)(6,217,37,136)(7,165,38,149)(8,219,39,138)(9,167,40,151)(10,221,41,140)(11,155,42,153)(12,223,29,128)(13,157,30,141)(14,211,31,130)(15,64,51,200)(16,194,52,116)(17,66,53,202)(18,196,54,118)(19,68,55,204)(20,184,56,120)(21,70,43,206)(22,186,44,122)(23,58,45,208)(24,188,46,124)(25,60,47,210)(26,190,48,126)(27,62,49,198)(28,192,50,114)(57,106,207,72)(59,108,209,74)(61,110,197,76)(63,112,199,78)(65,100,201,80)(67,102,203,82)(69,104,205,84)(71,185,105,121)(73,187,107,123)(75,189,109,125)(77,191,111,113)(79,193,99,115)(81,195,101,117)(83,183,103,119)(85,137,171,218)(86,150,172,166)(87,139,173,220)(88,152,174,168)(89,127,175,222)(90,154,176,156)(91,129,177,224)(92,142,178,158)(93,131,179,212)(94,144,180,160)(95,133,181,214)(96,146,182,162)(97,135,169,216)(98,148,170,164), (1,59,179,188)(2,210,180,125)(3,61,181,190)(4,198,182,113)(5,63,169,192)(6,200,170,115)(7,65,171,194)(8,202,172,117)(9,67,173,196)(10,204,174,119)(11,69,175,184)(12,206,176,121)(13,57,177,186)(14,208,178,123)(15,164,79,217)(16,149,80,137)(17,166,81,219)(18,151,82,139)(19,168,83,221)(20,153,84,127)(21,156,71,223)(22,141,72,129)(23,158,73,211)(24,143,74,131)(25,160,75,213)(26,145,76,133)(27,162,77,215)(28,147,78,135)(29,70,90,185)(30,207,91,122)(31,58,92,187)(32,209,93,124)(33,60,94,189)(34,197,95,126)(35,62,96,191)(36,199,97,114)(37,64,98,193)(38,201,85,116)(39,66,86,195)(40,203,87,118)(41,68,88,183)(42,205,89,120)(43,154,105,128)(44,157,106,224)(45,142,107,130)(46,159,108,212)(47,144,109,132)(48,161,110,214)(49,146,111,134)(50,163,112,216)(51,148,99,136)(52,165,100,218)(53,150,101,138)(54,167,102,220)(55,152,103,140)(56,155,104,222), (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), (1,158)(2,141)(3,156)(4,153)(5,168)(6,151)(7,166)(8,149)(9,164)(10,147)(11,162)(12,145)(13,160)(14,143)(15,118)(16,195)(17,116)(18,193)(19,114)(20,191)(21,126)(22,189)(23,124)(24,187)(25,122)(26,185)(27,120)(28,183)(29,161)(30,144)(31,159)(32,142)(33,157)(34,154)(35,155)(36,152)(37,167)(38,150)(39,165)(40,148)(41,163)(42,146)(43,190)(44,125)(45,188)(46,123)(47,186)(48,121)(49,184)(50,119)(51,196)(52,117)(53,194)(54,115)(55,192)(56,113)(57,109)(58,74)(59,107)(60,72)(61,105)(62,84)(63,103)(64,82)(65,101)(66,80)(67,99)(68,78)(69,111)(70,76)(71,197)(73,209)(75,207)(77,205)(79,203)(81,201)(83,199)(85,138)(86,218)(87,136)(88,216)(89,134)(90,214)(91,132)(92,212)(93,130)(94,224)(95,128)(96,222)(97,140)(98,220)(100,202)(102,200)(104,198)(106,210)(108,208)(110,206)(112,204)(127,182)(129,180)(131,178)(133,176)(135,174)(137,172)(139,170)(169,221)(171,219)(173,217)(175,215)(177,213)(179,211)(181,223)>;
G:=Group( (1,159,32,143)(2,213,33,132)(3,161,34,145)(4,215,35,134)(5,163,36,147)(6,217,37,136)(7,165,38,149)(8,219,39,138)(9,167,40,151)(10,221,41,140)(11,155,42,153)(12,223,29,128)(13,157,30,141)(14,211,31,130)(15,64,51,200)(16,194,52,116)(17,66,53,202)(18,196,54,118)(19,68,55,204)(20,184,56,120)(21,70,43,206)(22,186,44,122)(23,58,45,208)(24,188,46,124)(25,60,47,210)(26,190,48,126)(27,62,49,198)(28,192,50,114)(57,106,207,72)(59,108,209,74)(61,110,197,76)(63,112,199,78)(65,100,201,80)(67,102,203,82)(69,104,205,84)(71,185,105,121)(73,187,107,123)(75,189,109,125)(77,191,111,113)(79,193,99,115)(81,195,101,117)(83,183,103,119)(85,137,171,218)(86,150,172,166)(87,139,173,220)(88,152,174,168)(89,127,175,222)(90,154,176,156)(91,129,177,224)(92,142,178,158)(93,131,179,212)(94,144,180,160)(95,133,181,214)(96,146,182,162)(97,135,169,216)(98,148,170,164), (1,59,179,188)(2,210,180,125)(3,61,181,190)(4,198,182,113)(5,63,169,192)(6,200,170,115)(7,65,171,194)(8,202,172,117)(9,67,173,196)(10,204,174,119)(11,69,175,184)(12,206,176,121)(13,57,177,186)(14,208,178,123)(15,164,79,217)(16,149,80,137)(17,166,81,219)(18,151,82,139)(19,168,83,221)(20,153,84,127)(21,156,71,223)(22,141,72,129)(23,158,73,211)(24,143,74,131)(25,160,75,213)(26,145,76,133)(27,162,77,215)(28,147,78,135)(29,70,90,185)(30,207,91,122)(31,58,92,187)(32,209,93,124)(33,60,94,189)(34,197,95,126)(35,62,96,191)(36,199,97,114)(37,64,98,193)(38,201,85,116)(39,66,86,195)(40,203,87,118)(41,68,88,183)(42,205,89,120)(43,154,105,128)(44,157,106,224)(45,142,107,130)(46,159,108,212)(47,144,109,132)(48,161,110,214)(49,146,111,134)(50,163,112,216)(51,148,99,136)(52,165,100,218)(53,150,101,138)(54,167,102,220)(55,152,103,140)(56,155,104,222), (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), (1,158)(2,141)(3,156)(4,153)(5,168)(6,151)(7,166)(8,149)(9,164)(10,147)(11,162)(12,145)(13,160)(14,143)(15,118)(16,195)(17,116)(18,193)(19,114)(20,191)(21,126)(22,189)(23,124)(24,187)(25,122)(26,185)(27,120)(28,183)(29,161)(30,144)(31,159)(32,142)(33,157)(34,154)(35,155)(36,152)(37,167)(38,150)(39,165)(40,148)(41,163)(42,146)(43,190)(44,125)(45,188)(46,123)(47,186)(48,121)(49,184)(50,119)(51,196)(52,117)(53,194)(54,115)(55,192)(56,113)(57,109)(58,74)(59,107)(60,72)(61,105)(62,84)(63,103)(64,82)(65,101)(66,80)(67,99)(68,78)(69,111)(70,76)(71,197)(73,209)(75,207)(77,205)(79,203)(81,201)(83,199)(85,138)(86,218)(87,136)(88,216)(89,134)(90,214)(91,132)(92,212)(93,130)(94,224)(95,128)(96,222)(97,140)(98,220)(100,202)(102,200)(104,198)(106,210)(108,208)(110,206)(112,204)(127,182)(129,180)(131,178)(133,176)(135,174)(137,172)(139,170)(169,221)(171,219)(173,217)(175,215)(177,213)(179,211)(181,223) );
G=PermutationGroup([[(1,159,32,143),(2,213,33,132),(3,161,34,145),(4,215,35,134),(5,163,36,147),(6,217,37,136),(7,165,38,149),(8,219,39,138),(9,167,40,151),(10,221,41,140),(11,155,42,153),(12,223,29,128),(13,157,30,141),(14,211,31,130),(15,64,51,200),(16,194,52,116),(17,66,53,202),(18,196,54,118),(19,68,55,204),(20,184,56,120),(21,70,43,206),(22,186,44,122),(23,58,45,208),(24,188,46,124),(25,60,47,210),(26,190,48,126),(27,62,49,198),(28,192,50,114),(57,106,207,72),(59,108,209,74),(61,110,197,76),(63,112,199,78),(65,100,201,80),(67,102,203,82),(69,104,205,84),(71,185,105,121),(73,187,107,123),(75,189,109,125),(77,191,111,113),(79,193,99,115),(81,195,101,117),(83,183,103,119),(85,137,171,218),(86,150,172,166),(87,139,173,220),(88,152,174,168),(89,127,175,222),(90,154,176,156),(91,129,177,224),(92,142,178,158),(93,131,179,212),(94,144,180,160),(95,133,181,214),(96,146,182,162),(97,135,169,216),(98,148,170,164)], [(1,59,179,188),(2,210,180,125),(3,61,181,190),(4,198,182,113),(5,63,169,192),(6,200,170,115),(7,65,171,194),(8,202,172,117),(9,67,173,196),(10,204,174,119),(11,69,175,184),(12,206,176,121),(13,57,177,186),(14,208,178,123),(15,164,79,217),(16,149,80,137),(17,166,81,219),(18,151,82,139),(19,168,83,221),(20,153,84,127),(21,156,71,223),(22,141,72,129),(23,158,73,211),(24,143,74,131),(25,160,75,213),(26,145,76,133),(27,162,77,215),(28,147,78,135),(29,70,90,185),(30,207,91,122),(31,58,92,187),(32,209,93,124),(33,60,94,189),(34,197,95,126),(35,62,96,191),(36,199,97,114),(37,64,98,193),(38,201,85,116),(39,66,86,195),(40,203,87,118),(41,68,88,183),(42,205,89,120),(43,154,105,128),(44,157,106,224),(45,142,107,130),(46,159,108,212),(47,144,109,132),(48,161,110,214),(49,146,111,134),(50,163,112,216),(51,148,99,136),(52,165,100,218),(53,150,101,138),(54,167,102,220),(55,152,103,140),(56,155,104,222)], [(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)], [(1,158),(2,141),(3,156),(4,153),(5,168),(6,151),(7,166),(8,149),(9,164),(10,147),(11,162),(12,145),(13,160),(14,143),(15,118),(16,195),(17,116),(18,193),(19,114),(20,191),(21,126),(22,189),(23,124),(24,187),(25,122),(26,185),(27,120),(28,183),(29,161),(30,144),(31,159),(32,142),(33,157),(34,154),(35,155),(36,152),(37,167),(38,150),(39,165),(40,148),(41,163),(42,146),(43,190),(44,125),(45,188),(46,123),(47,186),(48,121),(49,184),(50,119),(51,196),(52,117),(53,194),(54,115),(55,192),(56,113),(57,109),(58,74),(59,107),(60,72),(61,105),(62,84),(63,103),(64,82),(65,101),(66,80),(67,99),(68,78),(69,111),(70,76),(71,197),(73,209),(75,207),(77,205),(79,203),(81,201),(83,199),(85,138),(86,218),(87,136),(88,216),(89,134),(90,214),(91,132),(92,212),(93,130),(94,224),(95,128),(96,222),(97,140),(98,220),(100,202),(102,200),(104,198),(106,210),(108,208),(110,206),(112,204),(127,182),(129,180),(131,178),(133,176),(135,174),(137,172),(139,170),(169,221),(171,219),(173,217),(175,215),(177,213),(179,211),(181,223)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | D4×D7 | D7×C4○D4 |
| kernel | C42.233D14 | D7×C42 | C4.D28 | Dic7⋊4D4 | D14⋊D4 | C28⋊D4 | Dic7⋊Q8 | C7×C4.4D4 | C2×D4⋊2D7 | C2×Q8⋊2D7 | C4×D7 | C4.4D4 | Dic7 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 8 | 3 | 12 | 3 | 3 | 6 | 12 |
Matrix representation of C42.233D14 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 27 | 1 |
| 25 | 19 | 0 | 0 | 0 | 0 |
| 15 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 17 |
| 0 | 0 | 0 | 0 | 24 | 17 |
| 11 | 22 | 0 | 0 | 0 | 0 |
| 13 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 28 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,28,27,0,0,0,0,0,1],[25,15,0,0,0,0,19,1,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,12,24,0,0,0,0,17,17],[11,13,0,0,0,0,22,18,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,28,28] >;
C42.233D14 in GAP, Magma, Sage, TeX
C_4^2._{233}D_{14} % in TeX
G:=Group("C4^2.233D14"); // GroupNames label
G:=SmallGroup(448,1121);
// by ID
G=gap.SmallGroup(448,1121);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,232,100,1123,346,297,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^2,c*b*c^-1=d*b*d=a^2*b,d*c*d=a^2*c^-1>;
// generators/relations