metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.113D14, C14.192+ (1+4), (C4×D4)⋊20D7, (D4×C28)⋊22C2, (C4×D28)⋊32C2, C28⋊2D4⋊9C2, D14⋊D4⋊9C2, C4⋊C4.318D14, (C2×D4).219D14, C4.65(C4○D28), C28.6Q8⋊16C2, (C22×C4).47D14, D14.29(C4○D4), C28.110(C4○D4), (C4×C28).157C22, (C2×C28).700C23, (C2×C14).102C24, D14⋊C4.86C22, C22⋊C4.115D14, C23.D14⋊8C2, C2.20(D4⋊6D14), C23.99(C22×D7), (D4×C14).262C22, (C2×D28).213C22, C4⋊Dic7.200C22, (C2×Dic7).43C23, (C22×D7).36C23, C22.127(C23×D7), C23.D7.14C22, C23.23D14⋊17C2, Dic7⋊C4.100C22, (C22×C28).364C22, (C22×C14).172C23, C7⋊4(C22.47C24), (C4×Dic7).205C22, (D7×C4⋊C4)⋊16C2, (C4×C7⋊D4)⋊44C2, C2.25(D7×C4○D4), C4⋊C4⋊7D7⋊15C2, C2.51(C2×C4○D28), (C2×C4×D7).66C22, C14.142(C2×C4○D4), (C7×C4⋊C4).331C22, (C2×C4).285(C22×D7), (C2×C7⋊D4).17C22, (C7×C22⋊C4).126C22, SmallGroup(448,1011)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1076 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×13], C7, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×10], C23 [×2], C23 [×2], D7 [×3], C14 [×3], C14 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×5], Dic7 [×6], C28 [×2], C28 [×4], D14 [×2], D14 [×5], C2×C14, C2×C14 [×6], C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4 [×3], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×D7 [×6], D28 [×2], C2×Dic7 [×2], C2×Dic7 [×4], C7⋊D4 [×6], C2×C28 [×3], C2×C28 [×2], C2×C28 [×2], C7×D4 [×2], C22×D7 [×2], C22×C14 [×2], C22.47C24, C4×Dic7 [×2], Dic7⋊C4 [×6], C4⋊Dic7 [×3], D14⋊C4 [×2], D14⋊C4 [×2], C23.D7 [×4], C4×C28, C7×C22⋊C4 [×2], C7×C4⋊C4, C2×C4×D7 [×2], C2×C4×D7 [×2], C2×D28, C2×C7⋊D4 [×4], C22×C28 [×2], D4×C14, C28.6Q8, C4×D28, C23.D14 [×2], D14⋊D4 [×2], D7×C4⋊C4, C4⋊C4⋊7D7, C4×C7⋊D4 [×2], C23.23D14 [×2], C28⋊2D4 [×2], D4×C28, C42.113D14
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], C22.47C24, C4○D28 [×2], C23×D7, C2×C4○D28, D4⋊6D14, D7×C4○D4, C42.113D14
Generators and relations
G = < a,b,c,d | a4=b4=1, c14=d2=b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c13 >
►On 224 points
Generators in S224
(1 197 76 49)(2 50 77 198)(3 199 78 51)(4 52 79 200)(5 201 80 53)(6 54 81 202)(7 203 82 55)(8 56 83 204)(9 205 84 29)(10 30 57 206)(11 207 58 31)(12 32 59 208)(13 209 60 33)(14 34 61 210)(15 211 62 35)(16 36 63 212)(17 213 64 37)(18 38 65 214)(19 215 66 39)(20 40 67 216)(21 217 68 41)(22 42 69 218)(23 219 70 43)(24 44 71 220)(25 221 72 45)(26 46 73 222)(27 223 74 47)(28 48 75 224)(85 178 160 113)(86 114 161 179)(87 180 162 115)(88 116 163 181)(89 182 164 117)(90 118 165 183)(91 184 166 119)(92 120 167 185)(93 186 168 121)(94 122 141 187)(95 188 142 123)(96 124 143 189)(97 190 144 125)(98 126 145 191)(99 192 146 127)(100 128 147 193)(101 194 148 129)(102 130 149 195)(103 196 150 131)(104 132 151 169)(105 170 152 133)(106 134 153 171)(107 172 154 135)(108 136 155 173)(109 174 156 137)(110 138 157 175)(111 176 158 139)(112 140 159 177)
(1 99 15 85)(2 100 16 86)(3 101 17 87)(4 102 18 88)(5 103 19 89)(6 104 20 90)(7 105 21 91)(8 106 22 92)(9 107 23 93)(10 108 24 94)(11 109 25 95)(12 110 26 96)(13 111 27 97)(14 112 28 98)(29 135 43 121)(30 136 44 122)(31 137 45 123)(32 138 46 124)(33 139 47 125)(34 140 48 126)(35 113 49 127)(36 114 50 128)(37 115 51 129)(38 116 52 130)(39 117 53 131)(40 118 54 132)(41 119 55 133)(42 120 56 134)(57 155 71 141)(58 156 72 142)(59 157 73 143)(60 158 74 144)(61 159 75 145)(62 160 76 146)(63 161 77 147)(64 162 78 148)(65 163 79 149)(66 164 80 150)(67 165 81 151)(68 166 82 152)(69 167 83 153)(70 168 84 154)(169 216 183 202)(170 217 184 203)(171 218 185 204)(172 219 186 205)(173 220 187 206)(174 221 188 207)(175 222 189 208)(176 223 190 209)(177 224 191 210)(178 197 192 211)(179 198 193 212)(180 199 194 213)(181 200 195 214)(182 201 196 215)
(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 61 15 75)(2 74 16 60)(3 59 17 73)(4 72 18 58)(5 57 19 71)(6 70 20 84)(7 83 21 69)(8 68 22 82)(9 81 23 67)(10 66 24 80)(11 79 25 65)(12 64 26 78)(13 77 27 63)(14 62 28 76)(29 202 43 216)(30 215 44 201)(31 200 45 214)(32 213 46 199)(33 198 47 212)(34 211 48 197)(35 224 49 210)(36 209 50 223)(37 222 51 208)(38 207 52 221)(39 220 53 206)(40 205 54 219)(41 218 55 204)(42 203 56 217)(85 112 99 98)(86 97 100 111)(87 110 101 96)(88 95 102 109)(89 108 103 94)(90 93 104 107)(91 106 105 92)(113 140 127 126)(114 125 128 139)(115 138 129 124)(116 123 130 137)(117 136 131 122)(118 121 132 135)(119 134 133 120)(141 164 155 150)(142 149 156 163)(143 162 157 148)(144 147 158 161)(145 160 159 146)(151 154 165 168)(152 167 166 153)(169 172 183 186)(170 185 184 171)(173 196 187 182)(174 181 188 195)(175 194 189 180)(176 179 190 193)(177 192 191 178)
G:=sub<Sym(224)| (1,197,76,49)(2,50,77,198)(3,199,78,51)(4,52,79,200)(5,201,80,53)(6,54,81,202)(7,203,82,55)(8,56,83,204)(9,205,84,29)(10,30,57,206)(11,207,58,31)(12,32,59,208)(13,209,60,33)(14,34,61,210)(15,211,62,35)(16,36,63,212)(17,213,64,37)(18,38,65,214)(19,215,66,39)(20,40,67,216)(21,217,68,41)(22,42,69,218)(23,219,70,43)(24,44,71,220)(25,221,72,45)(26,46,73,222)(27,223,74,47)(28,48,75,224)(85,178,160,113)(86,114,161,179)(87,180,162,115)(88,116,163,181)(89,182,164,117)(90,118,165,183)(91,184,166,119)(92,120,167,185)(93,186,168,121)(94,122,141,187)(95,188,142,123)(96,124,143,189)(97,190,144,125)(98,126,145,191)(99,192,146,127)(100,128,147,193)(101,194,148,129)(102,130,149,195)(103,196,150,131)(104,132,151,169)(105,170,152,133)(106,134,153,171)(107,172,154,135)(108,136,155,173)(109,174,156,137)(110,138,157,175)(111,176,158,139)(112,140,159,177), (1,99,15,85)(2,100,16,86)(3,101,17,87)(4,102,18,88)(5,103,19,89)(6,104,20,90)(7,105,21,91)(8,106,22,92)(9,107,23,93)(10,108,24,94)(11,109,25,95)(12,110,26,96)(13,111,27,97)(14,112,28,98)(29,135,43,121)(30,136,44,122)(31,137,45,123)(32,138,46,124)(33,139,47,125)(34,140,48,126)(35,113,49,127)(36,114,50,128)(37,115,51,129)(38,116,52,130)(39,117,53,131)(40,118,54,132)(41,119,55,133)(42,120,56,134)(57,155,71,141)(58,156,72,142)(59,157,73,143)(60,158,74,144)(61,159,75,145)(62,160,76,146)(63,161,77,147)(64,162,78,148)(65,163,79,149)(66,164,80,150)(67,165,81,151)(68,166,82,152)(69,167,83,153)(70,168,84,154)(169,216,183,202)(170,217,184,203)(171,218,185,204)(172,219,186,205)(173,220,187,206)(174,221,188,207)(175,222,189,208)(176,223,190,209)(177,224,191,210)(178,197,192,211)(179,198,193,212)(180,199,194,213)(181,200,195,214)(182,201,196,215), (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), (1,61,15,75)(2,74,16,60)(3,59,17,73)(4,72,18,58)(5,57,19,71)(6,70,20,84)(7,83,21,69)(8,68,22,82)(9,81,23,67)(10,66,24,80)(11,79,25,65)(12,64,26,78)(13,77,27,63)(14,62,28,76)(29,202,43,216)(30,215,44,201)(31,200,45,214)(32,213,46,199)(33,198,47,212)(34,211,48,197)(35,224,49,210)(36,209,50,223)(37,222,51,208)(38,207,52,221)(39,220,53,206)(40,205,54,219)(41,218,55,204)(42,203,56,217)(85,112,99,98)(86,97,100,111)(87,110,101,96)(88,95,102,109)(89,108,103,94)(90,93,104,107)(91,106,105,92)(113,140,127,126)(114,125,128,139)(115,138,129,124)(116,123,130,137)(117,136,131,122)(118,121,132,135)(119,134,133,120)(141,164,155,150)(142,149,156,163)(143,162,157,148)(144,147,158,161)(145,160,159,146)(151,154,165,168)(152,167,166,153)(169,172,183,186)(170,185,184,171)(173,196,187,182)(174,181,188,195)(175,194,189,180)(176,179,190,193)(177,192,191,178)>;
G:=Group( (1,197,76,49)(2,50,77,198)(3,199,78,51)(4,52,79,200)(5,201,80,53)(6,54,81,202)(7,203,82,55)(8,56,83,204)(9,205,84,29)(10,30,57,206)(11,207,58,31)(12,32,59,208)(13,209,60,33)(14,34,61,210)(15,211,62,35)(16,36,63,212)(17,213,64,37)(18,38,65,214)(19,215,66,39)(20,40,67,216)(21,217,68,41)(22,42,69,218)(23,219,70,43)(24,44,71,220)(25,221,72,45)(26,46,73,222)(27,223,74,47)(28,48,75,224)(85,178,160,113)(86,114,161,179)(87,180,162,115)(88,116,163,181)(89,182,164,117)(90,118,165,183)(91,184,166,119)(92,120,167,185)(93,186,168,121)(94,122,141,187)(95,188,142,123)(96,124,143,189)(97,190,144,125)(98,126,145,191)(99,192,146,127)(100,128,147,193)(101,194,148,129)(102,130,149,195)(103,196,150,131)(104,132,151,169)(105,170,152,133)(106,134,153,171)(107,172,154,135)(108,136,155,173)(109,174,156,137)(110,138,157,175)(111,176,158,139)(112,140,159,177), (1,99,15,85)(2,100,16,86)(3,101,17,87)(4,102,18,88)(5,103,19,89)(6,104,20,90)(7,105,21,91)(8,106,22,92)(9,107,23,93)(10,108,24,94)(11,109,25,95)(12,110,26,96)(13,111,27,97)(14,112,28,98)(29,135,43,121)(30,136,44,122)(31,137,45,123)(32,138,46,124)(33,139,47,125)(34,140,48,126)(35,113,49,127)(36,114,50,128)(37,115,51,129)(38,116,52,130)(39,117,53,131)(40,118,54,132)(41,119,55,133)(42,120,56,134)(57,155,71,141)(58,156,72,142)(59,157,73,143)(60,158,74,144)(61,159,75,145)(62,160,76,146)(63,161,77,147)(64,162,78,148)(65,163,79,149)(66,164,80,150)(67,165,81,151)(68,166,82,152)(69,167,83,153)(70,168,84,154)(169,216,183,202)(170,217,184,203)(171,218,185,204)(172,219,186,205)(173,220,187,206)(174,221,188,207)(175,222,189,208)(176,223,190,209)(177,224,191,210)(178,197,192,211)(179,198,193,212)(180,199,194,213)(181,200,195,214)(182,201,196,215), (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), (1,61,15,75)(2,74,16,60)(3,59,17,73)(4,72,18,58)(5,57,19,71)(6,70,20,84)(7,83,21,69)(8,68,22,82)(9,81,23,67)(10,66,24,80)(11,79,25,65)(12,64,26,78)(13,77,27,63)(14,62,28,76)(29,202,43,216)(30,215,44,201)(31,200,45,214)(32,213,46,199)(33,198,47,212)(34,211,48,197)(35,224,49,210)(36,209,50,223)(37,222,51,208)(38,207,52,221)(39,220,53,206)(40,205,54,219)(41,218,55,204)(42,203,56,217)(85,112,99,98)(86,97,100,111)(87,110,101,96)(88,95,102,109)(89,108,103,94)(90,93,104,107)(91,106,105,92)(113,140,127,126)(114,125,128,139)(115,138,129,124)(116,123,130,137)(117,136,131,122)(118,121,132,135)(119,134,133,120)(141,164,155,150)(142,149,156,163)(143,162,157,148)(144,147,158,161)(145,160,159,146)(151,154,165,168)(152,167,166,153)(169,172,183,186)(170,185,184,171)(173,196,187,182)(174,181,188,195)(175,194,189,180)(176,179,190,193)(177,192,191,178) );
G=PermutationGroup([(1,197,76,49),(2,50,77,198),(3,199,78,51),(4,52,79,200),(5,201,80,53),(6,54,81,202),(7,203,82,55),(8,56,83,204),(9,205,84,29),(10,30,57,206),(11,207,58,31),(12,32,59,208),(13,209,60,33),(14,34,61,210),(15,211,62,35),(16,36,63,212),(17,213,64,37),(18,38,65,214),(19,215,66,39),(20,40,67,216),(21,217,68,41),(22,42,69,218),(23,219,70,43),(24,44,71,220),(25,221,72,45),(26,46,73,222),(27,223,74,47),(28,48,75,224),(85,178,160,113),(86,114,161,179),(87,180,162,115),(88,116,163,181),(89,182,164,117),(90,118,165,183),(91,184,166,119),(92,120,167,185),(93,186,168,121),(94,122,141,187),(95,188,142,123),(96,124,143,189),(97,190,144,125),(98,126,145,191),(99,192,146,127),(100,128,147,193),(101,194,148,129),(102,130,149,195),(103,196,150,131),(104,132,151,169),(105,170,152,133),(106,134,153,171),(107,172,154,135),(108,136,155,173),(109,174,156,137),(110,138,157,175),(111,176,158,139),(112,140,159,177)], [(1,99,15,85),(2,100,16,86),(3,101,17,87),(4,102,18,88),(5,103,19,89),(6,104,20,90),(7,105,21,91),(8,106,22,92),(9,107,23,93),(10,108,24,94),(11,109,25,95),(12,110,26,96),(13,111,27,97),(14,112,28,98),(29,135,43,121),(30,136,44,122),(31,137,45,123),(32,138,46,124),(33,139,47,125),(34,140,48,126),(35,113,49,127),(36,114,50,128),(37,115,51,129),(38,116,52,130),(39,117,53,131),(40,118,54,132),(41,119,55,133),(42,120,56,134),(57,155,71,141),(58,156,72,142),(59,157,73,143),(60,158,74,144),(61,159,75,145),(62,160,76,146),(63,161,77,147),(64,162,78,148),(65,163,79,149),(66,164,80,150),(67,165,81,151),(68,166,82,152),(69,167,83,153),(70,168,84,154),(169,216,183,202),(170,217,184,203),(171,218,185,204),(172,219,186,205),(173,220,187,206),(174,221,188,207),(175,222,189,208),(176,223,190,209),(177,224,191,210),(178,197,192,211),(179,198,193,212),(180,199,194,213),(181,200,195,214),(182,201,196,215)], [(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)], [(1,61,15,75),(2,74,16,60),(3,59,17,73),(4,72,18,58),(5,57,19,71),(6,70,20,84),(7,83,21,69),(8,68,22,82),(9,81,23,67),(10,66,24,80),(11,79,25,65),(12,64,26,78),(13,77,27,63),(14,62,28,76),(29,202,43,216),(30,215,44,201),(31,200,45,214),(32,213,46,199),(33,198,47,212),(34,211,48,197),(35,224,49,210),(36,209,50,223),(37,222,51,208),(38,207,52,221),(39,220,53,206),(40,205,54,219),(41,218,55,204),(42,203,56,217),(85,112,99,98),(86,97,100,111),(87,110,101,96),(88,95,102,109),(89,108,103,94),(90,93,104,107),(91,106,105,92),(113,140,127,126),(114,125,128,139),(115,138,129,124),(116,123,130,137),(117,136,131,122),(118,121,132,135),(119,134,133,120),(141,164,155,150),(142,149,156,163),(143,162,157,148),(144,147,158,161),(145,160,159,146),(151,154,165,168),(152,167,166,153),(169,172,183,186),(170,185,184,171),(173,196,187,182),(174,181,188,195),(175,194,189,180),(176,179,190,193),(177,192,191,178)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 13 |
| 0 | 0 | 7 | 13 |
| 22 | 24 | 0 | 0 |
| 10 | 7 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 |
| 2 | 19 | 0 | 0 |
| 20 | 1 | 0 | 0 |
| 0 | 0 | 11 | 18 |
| 0 | 0 | 19 | 18 |
| 16 | 16 | 0 | 0 |
| 22 | 13 | 0 | 0 |
| 0 | 0 | 18 | 11 |
| 0 | 0 | 10 | 11 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,16,7,0,0,13,13],[22,10,0,0,24,7,0,0,0,0,17,0,0,0,0,17],[2,20,0,0,19,1,0,0,0,0,11,19,0,0,18,18],[16,22,0,0,16,13,0,0,0,0,18,10,0,0,11,11] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | 4J | 4K | 4L | ··· | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | ··· | 2 | 4 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | D14 | C4○D28 | 2+ (1+4) | D4⋊6D14 | D7×C4○D4 |
| kernel | C42.113D14 | C28.6Q8 | C4×D28 | C23.D14 | D14⋊D4 | D7×C4⋊C4 | C4⋊C4⋊7D7 | C4×C7⋊D4 | C23.23D14 | C28⋊2D4 | D4×C28 | C4×D4 | C28 | D14 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 3 | 4 | 4 | 3 | 6 | 3 | 6 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4^2._{113}D_{14} % in TeX
G:=Group("C4^2.113D14"); // GroupNames label
G:=SmallGroup(448,1011);
// by ID
G=gap.SmallGroup(448,1011);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,100,1571,570,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^13>;
// generators/relations
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