metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.161D14, C14.1002- (1+4), C14.1392+ (1+4), (C4×D28)⋊15C2, C4⋊C4.118D14, C42⋊2C2⋊4D7, C28.6Q8⋊9C2, D14⋊2Q8⋊41C2, D14⋊Q8⋊43C2, D14⋊D4.4C2, (C4×C28).33C22, C22⋊C4.79D14, C4.Dic14⋊39C2, D14.14(C4○D4), Dic7⋊4D4⋊36C2, D14.D4⋊51C2, D14.5D4⋊41C2, (C2×C14).251C24, (C2×C28).194C23, C4⋊Dic7.54C22, C2.64(D4⋊8D14), C23.57(C22×D7), D14⋊C4.114C22, C22⋊Dic14⋊45C2, (C2×D28).227C22, C22.D28⋊29C2, Dic7⋊C4.56C22, (C22×C14).65C23, C22.272(C23×D7), C23.D7.67C22, C7⋊9(C22.33C24), (C4×Dic7).151C22, (C2×Dic7).265C23, (C22×D7).225C23, C2.64(D4.10D14), (C2×Dic14).184C22, (C22×Dic7).151C22, (D7×C4⋊C4)⋊41C2, C2.98(D7×C4○D4), C4⋊C4⋊D7⋊42C2, (C7×C42⋊2C2)⋊6C2, C14.209(C2×C4○D4), (C2×C4×D7).219C22, (C7×C4⋊C4).203C22, (C2×C4).209(C22×D7), (C2×C7⋊D4).71C22, (C7×C22⋊C4).76C22, SmallGroup(448,1160)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1036 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×10], C7, C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×2], D7 [×3], C14 [×3], C14, C42, C42, C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×11], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic7 [×6], C28 [×6], D14 [×2], D14 [×5], C2×C14, C2×C14 [×3], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8 [×3], C22.D4 [×4], C42.C2 [×2], C42⋊2C2, C42⋊2C2, Dic14, C4×D7 [×5], D28 [×2], C2×Dic7 [×6], C2×Dic7, C7⋊D4 [×3], C2×C28 [×6], C22×D7 [×2], C22×C14, C22.33C24, C4×Dic7, Dic7⋊C4 [×6], C4⋊Dic7 [×5], D14⋊C4 [×6], C23.D7, C4×C28, C7×C22⋊C4 [×3], C7×C4⋊C4 [×3], C2×Dic14, C2×C4×D7 [×4], C2×D28, C22×Dic7, C2×C7⋊D4 [×2], C28.6Q8, C4×D28, C22⋊Dic14, Dic7⋊4D4, D14.D4 [×2], D14⋊D4, C22.D28, C4.Dic14, D7×C4⋊C4, D14.5D4, D14⋊Q8, D14⋊2Q8, C4⋊C4⋊D7, C7×C42⋊2C2, C42.161D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D7 [×7], C22.33C24, C23×D7, D7×C4○D4, D4⋊8D14, D4.10D14, C42.161D14
Generators and relations
G = < a,b,c,d | a4=b4=1, c14=d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c13 >
►On 224 points
Generators in S224
(1 139 223 69)(2 126 224 84)(3 113 197 71)(4 128 198 58)(5 115 199 73)(6 130 200 60)(7 117 201 75)(8 132 202 62)(9 119 203 77)(10 134 204 64)(11 121 205 79)(12 136 206 66)(13 123 207 81)(14 138 208 68)(15 125 209 83)(16 140 210 70)(17 127 211 57)(18 114 212 72)(19 129 213 59)(20 116 214 74)(21 131 215 61)(22 118 216 76)(23 133 217 63)(24 120 218 78)(25 135 219 65)(26 122 220 80)(27 137 221 67)(28 124 222 82)(29 196 150 86)(30 183 151 101)(31 170 152 88)(32 185 153 103)(33 172 154 90)(34 187 155 105)(35 174 156 92)(36 189 157 107)(37 176 158 94)(38 191 159 109)(39 178 160 96)(40 193 161 111)(41 180 162 98)(42 195 163 85)(43 182 164 100)(44 169 165 87)(45 184 166 102)(46 171 167 89)(47 186 168 104)(48 173 141 91)(49 188 142 106)(50 175 143 93)(51 190 144 108)(52 177 145 95)(53 192 146 110)(54 179 147 97)(55 194 148 112)(56 181 149 99)
(1 158 15 144)(2 52 16 38)(3 160 17 146)(4 54 18 40)(5 162 19 148)(6 56 20 42)(7 164 21 150)(8 30 22 44)(9 166 23 152)(10 32 24 46)(11 168 25 154)(12 34 26 48)(13 142 27 156)(14 36 28 50)(29 201 43 215)(31 203 45 217)(33 205 47 219)(35 207 49 221)(37 209 51 223)(39 211 53 197)(41 213 55 199)(57 192 71 178)(58 97 72 111)(59 194 73 180)(60 99 74 85)(61 196 75 182)(62 101 76 87)(63 170 77 184)(64 103 78 89)(65 172 79 186)(66 105 80 91)(67 174 81 188)(68 107 82 93)(69 176 83 190)(70 109 84 95)(86 117 100 131)(88 119 102 133)(90 121 104 135)(92 123 106 137)(94 125 108 139)(96 127 110 113)(98 129 112 115)(114 193 128 179)(116 195 130 181)(118 169 132 183)(120 171 134 185)(122 173 136 187)(124 175 138 189)(126 177 140 191)(141 206 155 220)(143 208 157 222)(145 210 159 224)(147 212 161 198)(149 214 163 200)(151 216 165 202)(153 218 167 204)
(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 14 15 28)(2 27 16 13)(3 12 17 26)(4 25 18 11)(5 10 19 24)(6 23 20 9)(7 8 21 22)(29 44 43 30)(31 42 45 56)(32 55 46 41)(33 40 47 54)(34 53 48 39)(35 38 49 52)(36 51 50 37)(57 66 71 80)(58 79 72 65)(59 64 73 78)(60 77 74 63)(61 62 75 76)(67 84 81 70)(68 69 82 83)(85 88 99 102)(86 101 100 87)(89 112 103 98)(90 97 104 111)(91 110 105 96)(92 95 106 109)(93 108 107 94)(113 122 127 136)(114 135 128 121)(115 120 129 134)(116 133 130 119)(117 118 131 132)(123 140 137 126)(124 125 138 139)(141 160 155 146)(142 145 156 159)(143 158 157 144)(147 154 161 168)(148 167 162 153)(149 152 163 166)(150 165 164 151)(169 196 183 182)(170 181 184 195)(171 194 185 180)(172 179 186 193)(173 192 187 178)(174 177 188 191)(175 190 189 176)(197 206 211 220)(198 219 212 205)(199 204 213 218)(200 217 214 203)(201 202 215 216)(207 224 221 210)(208 209 222 223)
G:=sub<Sym(224)| (1,139,223,69)(2,126,224,84)(3,113,197,71)(4,128,198,58)(5,115,199,73)(6,130,200,60)(7,117,201,75)(8,132,202,62)(9,119,203,77)(10,134,204,64)(11,121,205,79)(12,136,206,66)(13,123,207,81)(14,138,208,68)(15,125,209,83)(16,140,210,70)(17,127,211,57)(18,114,212,72)(19,129,213,59)(20,116,214,74)(21,131,215,61)(22,118,216,76)(23,133,217,63)(24,120,218,78)(25,135,219,65)(26,122,220,80)(27,137,221,67)(28,124,222,82)(29,196,150,86)(30,183,151,101)(31,170,152,88)(32,185,153,103)(33,172,154,90)(34,187,155,105)(35,174,156,92)(36,189,157,107)(37,176,158,94)(38,191,159,109)(39,178,160,96)(40,193,161,111)(41,180,162,98)(42,195,163,85)(43,182,164,100)(44,169,165,87)(45,184,166,102)(46,171,167,89)(47,186,168,104)(48,173,141,91)(49,188,142,106)(50,175,143,93)(51,190,144,108)(52,177,145,95)(53,192,146,110)(54,179,147,97)(55,194,148,112)(56,181,149,99), (1,158,15,144)(2,52,16,38)(3,160,17,146)(4,54,18,40)(5,162,19,148)(6,56,20,42)(7,164,21,150)(8,30,22,44)(9,166,23,152)(10,32,24,46)(11,168,25,154)(12,34,26,48)(13,142,27,156)(14,36,28,50)(29,201,43,215)(31,203,45,217)(33,205,47,219)(35,207,49,221)(37,209,51,223)(39,211,53,197)(41,213,55,199)(57,192,71,178)(58,97,72,111)(59,194,73,180)(60,99,74,85)(61,196,75,182)(62,101,76,87)(63,170,77,184)(64,103,78,89)(65,172,79,186)(66,105,80,91)(67,174,81,188)(68,107,82,93)(69,176,83,190)(70,109,84,95)(86,117,100,131)(88,119,102,133)(90,121,104,135)(92,123,106,137)(94,125,108,139)(96,127,110,113)(98,129,112,115)(114,193,128,179)(116,195,130,181)(118,169,132,183)(120,171,134,185)(122,173,136,187)(124,175,138,189)(126,177,140,191)(141,206,155,220)(143,208,157,222)(145,210,159,224)(147,212,161,198)(149,214,163,200)(151,216,165,202)(153,218,167,204), (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), (1,14,15,28)(2,27,16,13)(3,12,17,26)(4,25,18,11)(5,10,19,24)(6,23,20,9)(7,8,21,22)(29,44,43,30)(31,42,45,56)(32,55,46,41)(33,40,47,54)(34,53,48,39)(35,38,49,52)(36,51,50,37)(57,66,71,80)(58,79,72,65)(59,64,73,78)(60,77,74,63)(61,62,75,76)(67,84,81,70)(68,69,82,83)(85,88,99,102)(86,101,100,87)(89,112,103,98)(90,97,104,111)(91,110,105,96)(92,95,106,109)(93,108,107,94)(113,122,127,136)(114,135,128,121)(115,120,129,134)(116,133,130,119)(117,118,131,132)(123,140,137,126)(124,125,138,139)(141,160,155,146)(142,145,156,159)(143,158,157,144)(147,154,161,168)(148,167,162,153)(149,152,163,166)(150,165,164,151)(169,196,183,182)(170,181,184,195)(171,194,185,180)(172,179,186,193)(173,192,187,178)(174,177,188,191)(175,190,189,176)(197,206,211,220)(198,219,212,205)(199,204,213,218)(200,217,214,203)(201,202,215,216)(207,224,221,210)(208,209,222,223)>;
G:=Group( (1,139,223,69)(2,126,224,84)(3,113,197,71)(4,128,198,58)(5,115,199,73)(6,130,200,60)(7,117,201,75)(8,132,202,62)(9,119,203,77)(10,134,204,64)(11,121,205,79)(12,136,206,66)(13,123,207,81)(14,138,208,68)(15,125,209,83)(16,140,210,70)(17,127,211,57)(18,114,212,72)(19,129,213,59)(20,116,214,74)(21,131,215,61)(22,118,216,76)(23,133,217,63)(24,120,218,78)(25,135,219,65)(26,122,220,80)(27,137,221,67)(28,124,222,82)(29,196,150,86)(30,183,151,101)(31,170,152,88)(32,185,153,103)(33,172,154,90)(34,187,155,105)(35,174,156,92)(36,189,157,107)(37,176,158,94)(38,191,159,109)(39,178,160,96)(40,193,161,111)(41,180,162,98)(42,195,163,85)(43,182,164,100)(44,169,165,87)(45,184,166,102)(46,171,167,89)(47,186,168,104)(48,173,141,91)(49,188,142,106)(50,175,143,93)(51,190,144,108)(52,177,145,95)(53,192,146,110)(54,179,147,97)(55,194,148,112)(56,181,149,99), (1,158,15,144)(2,52,16,38)(3,160,17,146)(4,54,18,40)(5,162,19,148)(6,56,20,42)(7,164,21,150)(8,30,22,44)(9,166,23,152)(10,32,24,46)(11,168,25,154)(12,34,26,48)(13,142,27,156)(14,36,28,50)(29,201,43,215)(31,203,45,217)(33,205,47,219)(35,207,49,221)(37,209,51,223)(39,211,53,197)(41,213,55,199)(57,192,71,178)(58,97,72,111)(59,194,73,180)(60,99,74,85)(61,196,75,182)(62,101,76,87)(63,170,77,184)(64,103,78,89)(65,172,79,186)(66,105,80,91)(67,174,81,188)(68,107,82,93)(69,176,83,190)(70,109,84,95)(86,117,100,131)(88,119,102,133)(90,121,104,135)(92,123,106,137)(94,125,108,139)(96,127,110,113)(98,129,112,115)(114,193,128,179)(116,195,130,181)(118,169,132,183)(120,171,134,185)(122,173,136,187)(124,175,138,189)(126,177,140,191)(141,206,155,220)(143,208,157,222)(145,210,159,224)(147,212,161,198)(149,214,163,200)(151,216,165,202)(153,218,167,204), (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), (1,14,15,28)(2,27,16,13)(3,12,17,26)(4,25,18,11)(5,10,19,24)(6,23,20,9)(7,8,21,22)(29,44,43,30)(31,42,45,56)(32,55,46,41)(33,40,47,54)(34,53,48,39)(35,38,49,52)(36,51,50,37)(57,66,71,80)(58,79,72,65)(59,64,73,78)(60,77,74,63)(61,62,75,76)(67,84,81,70)(68,69,82,83)(85,88,99,102)(86,101,100,87)(89,112,103,98)(90,97,104,111)(91,110,105,96)(92,95,106,109)(93,108,107,94)(113,122,127,136)(114,135,128,121)(115,120,129,134)(116,133,130,119)(117,118,131,132)(123,140,137,126)(124,125,138,139)(141,160,155,146)(142,145,156,159)(143,158,157,144)(147,154,161,168)(148,167,162,153)(149,152,163,166)(150,165,164,151)(169,196,183,182)(170,181,184,195)(171,194,185,180)(172,179,186,193)(173,192,187,178)(174,177,188,191)(175,190,189,176)(197,206,211,220)(198,219,212,205)(199,204,213,218)(200,217,214,203)(201,202,215,216)(207,224,221,210)(208,209,222,223) );
G=PermutationGroup([(1,139,223,69),(2,126,224,84),(3,113,197,71),(4,128,198,58),(5,115,199,73),(6,130,200,60),(7,117,201,75),(8,132,202,62),(9,119,203,77),(10,134,204,64),(11,121,205,79),(12,136,206,66),(13,123,207,81),(14,138,208,68),(15,125,209,83),(16,140,210,70),(17,127,211,57),(18,114,212,72),(19,129,213,59),(20,116,214,74),(21,131,215,61),(22,118,216,76),(23,133,217,63),(24,120,218,78),(25,135,219,65),(26,122,220,80),(27,137,221,67),(28,124,222,82),(29,196,150,86),(30,183,151,101),(31,170,152,88),(32,185,153,103),(33,172,154,90),(34,187,155,105),(35,174,156,92),(36,189,157,107),(37,176,158,94),(38,191,159,109),(39,178,160,96),(40,193,161,111),(41,180,162,98),(42,195,163,85),(43,182,164,100),(44,169,165,87),(45,184,166,102),(46,171,167,89),(47,186,168,104),(48,173,141,91),(49,188,142,106),(50,175,143,93),(51,190,144,108),(52,177,145,95),(53,192,146,110),(54,179,147,97),(55,194,148,112),(56,181,149,99)], [(1,158,15,144),(2,52,16,38),(3,160,17,146),(4,54,18,40),(5,162,19,148),(6,56,20,42),(7,164,21,150),(8,30,22,44),(9,166,23,152),(10,32,24,46),(11,168,25,154),(12,34,26,48),(13,142,27,156),(14,36,28,50),(29,201,43,215),(31,203,45,217),(33,205,47,219),(35,207,49,221),(37,209,51,223),(39,211,53,197),(41,213,55,199),(57,192,71,178),(58,97,72,111),(59,194,73,180),(60,99,74,85),(61,196,75,182),(62,101,76,87),(63,170,77,184),(64,103,78,89),(65,172,79,186),(66,105,80,91),(67,174,81,188),(68,107,82,93),(69,176,83,190),(70,109,84,95),(86,117,100,131),(88,119,102,133),(90,121,104,135),(92,123,106,137),(94,125,108,139),(96,127,110,113),(98,129,112,115),(114,193,128,179),(116,195,130,181),(118,169,132,183),(120,171,134,185),(122,173,136,187),(124,175,138,189),(126,177,140,191),(141,206,155,220),(143,208,157,222),(145,210,159,224),(147,212,161,198),(149,214,163,200),(151,216,165,202),(153,218,167,204)], [(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)], [(1,14,15,28),(2,27,16,13),(3,12,17,26),(4,25,18,11),(5,10,19,24),(6,23,20,9),(7,8,21,22),(29,44,43,30),(31,42,45,56),(32,55,46,41),(33,40,47,54),(34,53,48,39),(35,38,49,52),(36,51,50,37),(57,66,71,80),(58,79,72,65),(59,64,73,78),(60,77,74,63),(61,62,75,76),(67,84,81,70),(68,69,82,83),(85,88,99,102),(86,101,100,87),(89,112,103,98),(90,97,104,111),(91,110,105,96),(92,95,106,109),(93,108,107,94),(113,122,127,136),(114,135,128,121),(115,120,129,134),(116,133,130,119),(117,118,131,132),(123,140,137,126),(124,125,138,139),(141,160,155,146),(142,145,156,159),(143,158,157,144),(147,154,161,168),(148,167,162,153),(149,152,163,166),(150,165,164,151),(169,196,183,182),(170,181,184,195),(171,194,185,180),(172,179,186,193),(173,192,187,178),(174,177,188,191),(175,190,189,176),(197,206,211,220),(198,219,212,205),(199,204,213,218),(200,217,214,203),(201,202,215,216),(207,224,221,210),(208,209,222,223)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 18 | 26 | 18 |
| 0 | 0 | 11 | 27 | 0 | 11 |
| 0 | 0 | 9 | 9 | 16 | 11 |
| 0 | 0 | 0 | 16 | 3 | 13 |
| 1 | 24 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 6 | 27 |
| 0 | 0 | 0 | 1 | 4 | 27 |
| 0 | 0 | 28 | 1 | 28 | 0 |
| 0 | 0 | 27 | 3 | 0 | 28 |
| 24 | 12 | 0 | 0 | 0 | 0 |
| 27 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 12 | 13 | 0 |
| 0 | 0 | 17 | 3 | 16 | 13 |
| 0 | 0 | 21 | 12 | 16 | 17 |
| 0 | 0 | 9 | 11 | 2 | 14 |
| 5 | 17 | 0 | 0 | 0 | 0 |
| 2 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 12 | 13 | 0 |
| 0 | 0 | 0 | 4 | 23 | 16 |
| 0 | 0 | 21 | 0 | 14 | 12 |
| 0 | 0 | 26 | 8 | 14 | 15 |
G:=sub<GL(6,GF(29))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,2,11,9,0,0,0,18,27,9,16,0,0,26,0,16,3,0,0,18,11,11,13],[1,0,0,0,0,0,24,28,0,0,0,0,0,0,1,0,28,27,0,0,0,1,1,3,0,0,6,4,28,0,0,0,27,27,0,28],[24,27,0,0,0,0,12,5,0,0,0,0,0,0,25,17,21,9,0,0,12,3,12,11,0,0,13,16,16,2,0,0,0,13,17,14],[5,2,0,0,0,0,17,24,0,0,0,0,0,0,25,0,21,26,0,0,12,4,0,8,0,0,13,23,14,14,0,0,0,16,12,15] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14I | 14J | 14K | 14L | 28A | ··· | 28R | 28S | ··· | 28AA |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 14 | 14 | 28 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D7×C4○D4 | D4⋊8D14 | D4.10D14 |
| kernel | C42.161D14 | C28.6Q8 | C4×D28 | C22⋊Dic14 | Dic7⋊4D4 | D14.D4 | D14⋊D4 | C22.D28 | C4.Dic14 | D7×C4⋊C4 | D14.5D4 | D14⋊Q8 | D14⋊2Q8 | C4⋊C4⋊D7 | C7×C42⋊2C2 | C42⋊2C2 | D14 | C42 | C22⋊C4 | C4⋊C4 | C14 | C14 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 3 | 9 | 9 | 1 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4^2._{161}D_{14} % in TeX
G:=Group("C4^2.161D14"); // GroupNames label
G:=SmallGroup(448,1160);
// by ID
G=gap.SmallGroup(448,1160);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,268,675,570,80,18822]);
// 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*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^13>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀