metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.139D14, C14.882- (1+4), (Q8×Dic7)⋊18C2, C4.4D4.8D7, (C4×Dic14)⋊44C2, (C2×D4).169D14, (C2×C28).77C23, (C2×Q8).135D14, C22⋊C4.33D14, (D4×Dic7).14C2, Dic7⋊Q8⋊22C2, C28.124(C4○D4), C4.15(D4⋊2D7), (C2×C14).215C24, (C4×C28).184C22, C28.17D4.9C2, C23.37(C22×D7), Dic7.28(C4○D4), C22⋊Dic14⋊38C2, (D4×C14).151C22, C23.D14⋊37C2, Dic7⋊C4.48C22, C4⋊Dic7.233C22, (C22×C14).45C23, (Q8×C14).124C22, C22.236(C23×D7), C23.D7.52C22, C23.11D14⋊18C2, C7⋊6(C22.50C24), (C2×Dic7).252C23, (C4×Dic7).131C22, C2.49(D4.10D14), (C2×Dic14).296C22, (C22×Dic7).140C22, C2.74(D7×C4○D4), C14.93(C2×C4○D4), C2.55(C2×D4⋊2D7), (C7×C4.4D4).6C2, (C2×C4).299(C22×D7), (C7×C22⋊C4).62C22, SmallGroup(448,1124)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 780 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×13], C22, C22 [×6], C7, C2×C4 [×3], C2×C4 [×2], C2×C4 [×12], D4 [×2], Q8 [×6], C23 [×2], C14 [×3], C14 [×2], C42, C42 [×6], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×12], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic7 [×2], Dic7 [×7], C28 [×2], C28 [×4], C2×C14, C2×C14 [×6], C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4, C4.4D4, C42⋊2C2 [×4], C4⋊Q8, Dic14 [×4], C2×Dic7 [×4], C2×Dic7 [×4], C2×Dic7 [×4], C2×C28 [×3], C2×C28 [×2], C7×D4 [×2], C7×Q8 [×2], C22×C14 [×2], C22.50C24, C4×Dic7 [×2], C4×Dic7 [×4], Dic7⋊C4 [×2], Dic7⋊C4 [×6], C4⋊Dic7 [×2], C4⋊Dic7 [×2], C23.D7 [×6], C4×C28, C7×C22⋊C4 [×4], C2×Dic14 [×2], C22×Dic7 [×2], D4×C14, Q8×C14, C4×Dic14 [×2], C23.11D14 [×2], C22⋊Dic14 [×2], C23.D14 [×4], D4×Dic7, C28.17D4, Dic7⋊Q8, Q8×Dic7, C7×C4.4D4, C42.139D14
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, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, D7×C4○D4, D4.10D14, C42.139D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=1, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=c-1 >
►On 224 points
Generators in S224
(1 187 35 194)(2 195 29 188)(3 189 30 196)(4 183 31 190)(5 191 32 184)(6 185 33 192)(7 193 34 186)(8 94 15 87)(9 88 16 95)(10 96 17 89)(11 90 18 97)(12 98 19 91)(13 92 20 85)(14 86 21 93)(22 214 49 221)(23 222 43 215)(24 216 44 223)(25 224 45 217)(26 218 46 211)(27 212 47 219)(28 220 48 213)(36 117 51 124)(37 125 52 118)(38 119 53 126)(39 113 54 120)(40 121 55 114)(41 115 56 122)(42 123 50 116)(57 81 103 151)(58 152 104 82)(59 83 105 153)(60 154 106 84)(61 71 107 141)(62 142 108 72)(63 73 109 143)(64 144 110 74)(65 75 111 145)(66 146 112 76)(67 77 99 147)(68 148 100 78)(69 79 101 149)(70 150 102 80)(127 166 173 199)(128 200 174 167)(129 168 175 201)(130 202 176 155)(131 156 177 203)(132 204 178 157)(133 158 179 205)(134 206 180 159)(135 160 181 207)(136 208 182 161)(137 162 169 209)(138 210 170 163)(139 164 171 197)(140 198 172 165)
(1 133 50 172)(2 127 51 180)(3 135 52 174)(4 129 53 182)(5 137 54 176)(6 131 55 170)(7 139 56 178)(8 78 22 141)(9 72 23 149)(10 80 24 143)(11 74 25 151)(12 82 26 145)(13 76 27 153)(14 84 28 147)(15 148 49 71)(16 142 43 79)(17 150 44 73)(18 144 45 81)(19 152 46 75)(20 146 47 83)(21 154 48 77)(29 173 36 134)(30 181 37 128)(31 175 38 136)(32 169 39 130)(33 177 40 138)(34 171 41 132)(35 179 42 140)(57 90 64 224)(58 218 65 98)(59 92 66 212)(60 220 67 86)(61 94 68 214)(62 222 69 88)(63 96 70 216)(85 112 219 105)(87 100 221 107)(89 102 223 109)(91 104 211 111)(93 106 213 99)(95 108 215 101)(97 110 217 103)(113 202 184 209)(114 163 185 156)(115 204 186 197)(116 165 187 158)(117 206 188 199)(118 167 189 160)(119 208 190 201)(120 155 191 162)(121 210 192 203)(122 157 193 164)(123 198 194 205)(124 159 195 166)(125 200 196 207)(126 161 183 168)
(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 47 42 13)(2 46 36 12)(3 45 37 11)(4 44 38 10)(5 43 39 9)(6 49 40 8)(7 48 41 14)(15 33 22 55)(16 32 23 54)(17 31 24 53)(18 30 25 52)(19 29 26 51)(20 35 27 50)(21 34 28 56)(57 200 110 160)(58 199 111 159)(59 198 112 158)(60 197 99 157)(61 210 100 156)(62 209 101 155)(63 208 102 168)(64 207 103 167)(65 206 104 166)(66 205 105 165)(67 204 106 164)(68 203 107 163)(69 202 108 162)(70 201 109 161)(71 138 78 131)(72 137 79 130)(73 136 80 129)(74 135 81 128)(75 134 82 127)(76 133 83 140)(77 132 84 139)(85 187 212 123)(86 186 213 122)(87 185 214 121)(88 184 215 120)(89 183 216 119)(90 196 217 118)(91 195 218 117)(92 194 219 116)(93 193 220 115)(94 192 221 114)(95 191 222 113)(96 190 223 126)(97 189 224 125)(98 188 211 124)(141 170 148 177)(142 169 149 176)(143 182 150 175)(144 181 151 174)(145 180 152 173)(146 179 153 172)(147 178 154 171)
G:=sub<Sym(224)| (1,187,35,194)(2,195,29,188)(3,189,30,196)(4,183,31,190)(5,191,32,184)(6,185,33,192)(7,193,34,186)(8,94,15,87)(9,88,16,95)(10,96,17,89)(11,90,18,97)(12,98,19,91)(13,92,20,85)(14,86,21,93)(22,214,49,221)(23,222,43,215)(24,216,44,223)(25,224,45,217)(26,218,46,211)(27,212,47,219)(28,220,48,213)(36,117,51,124)(37,125,52,118)(38,119,53,126)(39,113,54,120)(40,121,55,114)(41,115,56,122)(42,123,50,116)(57,81,103,151)(58,152,104,82)(59,83,105,153)(60,154,106,84)(61,71,107,141)(62,142,108,72)(63,73,109,143)(64,144,110,74)(65,75,111,145)(66,146,112,76)(67,77,99,147)(68,148,100,78)(69,79,101,149)(70,150,102,80)(127,166,173,199)(128,200,174,167)(129,168,175,201)(130,202,176,155)(131,156,177,203)(132,204,178,157)(133,158,179,205)(134,206,180,159)(135,160,181,207)(136,208,182,161)(137,162,169,209)(138,210,170,163)(139,164,171,197)(140,198,172,165), (1,133,50,172)(2,127,51,180)(3,135,52,174)(4,129,53,182)(5,137,54,176)(6,131,55,170)(7,139,56,178)(8,78,22,141)(9,72,23,149)(10,80,24,143)(11,74,25,151)(12,82,26,145)(13,76,27,153)(14,84,28,147)(15,148,49,71)(16,142,43,79)(17,150,44,73)(18,144,45,81)(19,152,46,75)(20,146,47,83)(21,154,48,77)(29,173,36,134)(30,181,37,128)(31,175,38,136)(32,169,39,130)(33,177,40,138)(34,171,41,132)(35,179,42,140)(57,90,64,224)(58,218,65,98)(59,92,66,212)(60,220,67,86)(61,94,68,214)(62,222,69,88)(63,96,70,216)(85,112,219,105)(87,100,221,107)(89,102,223,109)(91,104,211,111)(93,106,213,99)(95,108,215,101)(97,110,217,103)(113,202,184,209)(114,163,185,156)(115,204,186,197)(116,165,187,158)(117,206,188,199)(118,167,189,160)(119,208,190,201)(120,155,191,162)(121,210,192,203)(122,157,193,164)(123,198,194,205)(124,159,195,166)(125,200,196,207)(126,161,183,168), (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), (1,47,42,13)(2,46,36,12)(3,45,37,11)(4,44,38,10)(5,43,39,9)(6,49,40,8)(7,48,41,14)(15,33,22,55)(16,32,23,54)(17,31,24,53)(18,30,25,52)(19,29,26,51)(20,35,27,50)(21,34,28,56)(57,200,110,160)(58,199,111,159)(59,198,112,158)(60,197,99,157)(61,210,100,156)(62,209,101,155)(63,208,102,168)(64,207,103,167)(65,206,104,166)(66,205,105,165)(67,204,106,164)(68,203,107,163)(69,202,108,162)(70,201,109,161)(71,138,78,131)(72,137,79,130)(73,136,80,129)(74,135,81,128)(75,134,82,127)(76,133,83,140)(77,132,84,139)(85,187,212,123)(86,186,213,122)(87,185,214,121)(88,184,215,120)(89,183,216,119)(90,196,217,118)(91,195,218,117)(92,194,219,116)(93,193,220,115)(94,192,221,114)(95,191,222,113)(96,190,223,126)(97,189,224,125)(98,188,211,124)(141,170,148,177)(142,169,149,176)(143,182,150,175)(144,181,151,174)(145,180,152,173)(146,179,153,172)(147,178,154,171)>;
G:=Group( (1,187,35,194)(2,195,29,188)(3,189,30,196)(4,183,31,190)(5,191,32,184)(6,185,33,192)(7,193,34,186)(8,94,15,87)(9,88,16,95)(10,96,17,89)(11,90,18,97)(12,98,19,91)(13,92,20,85)(14,86,21,93)(22,214,49,221)(23,222,43,215)(24,216,44,223)(25,224,45,217)(26,218,46,211)(27,212,47,219)(28,220,48,213)(36,117,51,124)(37,125,52,118)(38,119,53,126)(39,113,54,120)(40,121,55,114)(41,115,56,122)(42,123,50,116)(57,81,103,151)(58,152,104,82)(59,83,105,153)(60,154,106,84)(61,71,107,141)(62,142,108,72)(63,73,109,143)(64,144,110,74)(65,75,111,145)(66,146,112,76)(67,77,99,147)(68,148,100,78)(69,79,101,149)(70,150,102,80)(127,166,173,199)(128,200,174,167)(129,168,175,201)(130,202,176,155)(131,156,177,203)(132,204,178,157)(133,158,179,205)(134,206,180,159)(135,160,181,207)(136,208,182,161)(137,162,169,209)(138,210,170,163)(139,164,171,197)(140,198,172,165), (1,133,50,172)(2,127,51,180)(3,135,52,174)(4,129,53,182)(5,137,54,176)(6,131,55,170)(7,139,56,178)(8,78,22,141)(9,72,23,149)(10,80,24,143)(11,74,25,151)(12,82,26,145)(13,76,27,153)(14,84,28,147)(15,148,49,71)(16,142,43,79)(17,150,44,73)(18,144,45,81)(19,152,46,75)(20,146,47,83)(21,154,48,77)(29,173,36,134)(30,181,37,128)(31,175,38,136)(32,169,39,130)(33,177,40,138)(34,171,41,132)(35,179,42,140)(57,90,64,224)(58,218,65,98)(59,92,66,212)(60,220,67,86)(61,94,68,214)(62,222,69,88)(63,96,70,216)(85,112,219,105)(87,100,221,107)(89,102,223,109)(91,104,211,111)(93,106,213,99)(95,108,215,101)(97,110,217,103)(113,202,184,209)(114,163,185,156)(115,204,186,197)(116,165,187,158)(117,206,188,199)(118,167,189,160)(119,208,190,201)(120,155,191,162)(121,210,192,203)(122,157,193,164)(123,198,194,205)(124,159,195,166)(125,200,196,207)(126,161,183,168), (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), (1,47,42,13)(2,46,36,12)(3,45,37,11)(4,44,38,10)(5,43,39,9)(6,49,40,8)(7,48,41,14)(15,33,22,55)(16,32,23,54)(17,31,24,53)(18,30,25,52)(19,29,26,51)(20,35,27,50)(21,34,28,56)(57,200,110,160)(58,199,111,159)(59,198,112,158)(60,197,99,157)(61,210,100,156)(62,209,101,155)(63,208,102,168)(64,207,103,167)(65,206,104,166)(66,205,105,165)(67,204,106,164)(68,203,107,163)(69,202,108,162)(70,201,109,161)(71,138,78,131)(72,137,79,130)(73,136,80,129)(74,135,81,128)(75,134,82,127)(76,133,83,140)(77,132,84,139)(85,187,212,123)(86,186,213,122)(87,185,214,121)(88,184,215,120)(89,183,216,119)(90,196,217,118)(91,195,218,117)(92,194,219,116)(93,193,220,115)(94,192,221,114)(95,191,222,113)(96,190,223,126)(97,189,224,125)(98,188,211,124)(141,170,148,177)(142,169,149,176)(143,182,150,175)(144,181,151,174)(145,180,152,173)(146,179,153,172)(147,178,154,171) );
G=PermutationGroup([(1,187,35,194),(2,195,29,188),(3,189,30,196),(4,183,31,190),(5,191,32,184),(6,185,33,192),(7,193,34,186),(8,94,15,87),(9,88,16,95),(10,96,17,89),(11,90,18,97),(12,98,19,91),(13,92,20,85),(14,86,21,93),(22,214,49,221),(23,222,43,215),(24,216,44,223),(25,224,45,217),(26,218,46,211),(27,212,47,219),(28,220,48,213),(36,117,51,124),(37,125,52,118),(38,119,53,126),(39,113,54,120),(40,121,55,114),(41,115,56,122),(42,123,50,116),(57,81,103,151),(58,152,104,82),(59,83,105,153),(60,154,106,84),(61,71,107,141),(62,142,108,72),(63,73,109,143),(64,144,110,74),(65,75,111,145),(66,146,112,76),(67,77,99,147),(68,148,100,78),(69,79,101,149),(70,150,102,80),(127,166,173,199),(128,200,174,167),(129,168,175,201),(130,202,176,155),(131,156,177,203),(132,204,178,157),(133,158,179,205),(134,206,180,159),(135,160,181,207),(136,208,182,161),(137,162,169,209),(138,210,170,163),(139,164,171,197),(140,198,172,165)], [(1,133,50,172),(2,127,51,180),(3,135,52,174),(4,129,53,182),(5,137,54,176),(6,131,55,170),(7,139,56,178),(8,78,22,141),(9,72,23,149),(10,80,24,143),(11,74,25,151),(12,82,26,145),(13,76,27,153),(14,84,28,147),(15,148,49,71),(16,142,43,79),(17,150,44,73),(18,144,45,81),(19,152,46,75),(20,146,47,83),(21,154,48,77),(29,173,36,134),(30,181,37,128),(31,175,38,136),(32,169,39,130),(33,177,40,138),(34,171,41,132),(35,179,42,140),(57,90,64,224),(58,218,65,98),(59,92,66,212),(60,220,67,86),(61,94,68,214),(62,222,69,88),(63,96,70,216),(85,112,219,105),(87,100,221,107),(89,102,223,109),(91,104,211,111),(93,106,213,99),(95,108,215,101),(97,110,217,103),(113,202,184,209),(114,163,185,156),(115,204,186,197),(116,165,187,158),(117,206,188,199),(118,167,189,160),(119,208,190,201),(120,155,191,162),(121,210,192,203),(122,157,193,164),(123,198,194,205),(124,159,195,166),(125,200,196,207),(126,161,183,168)], [(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)], [(1,47,42,13),(2,46,36,12),(3,45,37,11),(4,44,38,10),(5,43,39,9),(6,49,40,8),(7,48,41,14),(15,33,22,55),(16,32,23,54),(17,31,24,53),(18,30,25,52),(19,29,26,51),(20,35,27,50),(21,34,28,56),(57,200,110,160),(58,199,111,159),(59,198,112,158),(60,197,99,157),(61,210,100,156),(62,209,101,155),(63,208,102,168),(64,207,103,167),(65,206,104,166),(66,205,105,165),(67,204,106,164),(68,203,107,163),(69,202,108,162),(70,201,109,161),(71,138,78,131),(72,137,79,130),(73,136,80,129),(74,135,81,128),(75,134,82,127),(76,133,83,140),(77,132,84,139),(85,187,212,123),(86,186,213,122),(87,185,214,121),(88,184,215,120),(89,183,216,119),(90,196,217,118),(91,195,218,117),(92,194,219,116),(93,193,220,115),(94,192,221,114),(95,191,222,113),(96,190,223,126),(97,189,224,125),(98,188,211,124),(141,170,148,177),(142,169,149,176),(143,182,150,175),(144,181,151,174),(145,180,152,173),(146,179,153,172),(147,178,154,171)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 10 | 25 | 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 | 28 |
| 19 | 11 | 0 | 0 | 0 | 0 |
| 20 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,28,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[7,10,0,0,0,0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[19,20,0,0,0,0,11,10,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,28] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 4P | 4Q | 4R | 4S | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| 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 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | 2- (1+4) | D4⋊2D7 | D7×C4○D4 | D4.10D14 |
| kernel | C42.139D14 | C4×Dic14 | C23.11D14 | C22⋊Dic14 | C23.D14 | D4×Dic7 | C28.17D4 | Dic7⋊Q8 | Q8×Dic7 | C7×C4.4D4 | C4.4D4 | Dic7 | C28 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 3 | 12 | 3 | 3 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4^2._{139}D_{14} % in TeX
G:=Group("C4^2.139D14"); // GroupNames label
G:=SmallGroup(448,1124);
// by ID
G=gap.SmallGroup(448,1124);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,387,100,794,297,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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