metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.168D14, C14.792+ (1+4), C4⋊1D4⋊8D7, C28⋊2D4⋊38C2, (D4×Dic7)⋊36C2, (C2×D4).179D14, C42⋊D7⋊24C2, C28.6Q8⋊23C2, Dic7⋊D4⋊38C2, C28.134(C4○D4), C4.40(D4⋊2D7), (C2×C28).637C23, (C4×C28).205C22, (C2×C14).263C24, C2.83(D4⋊6D14), C23.69(C22×D7), D14⋊C4.150C22, (D4×C14).215C22, Dic7⋊C4.87C22, C4⋊Dic7.249C22, (C22×C14).77C23, C22.284(C23×D7), C23.D7.74C22, C23.18D14⋊27C2, C7⋊7(C22.34C24), (C4×Dic7).156C22, (C2×Dic7).137C23, (C22×D7).117C23, (C22×Dic7).159C22, (C7×C4⋊1D4)⋊10C2, C14.98(C2×C4○D4), C2.62(C2×D4⋊2D7), (C2×C4×D7).140C22, (C2×C4).215(C22×D7), (C2×C7⋊D4).79C22, SmallGroup(448,1172)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1068 in 240 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C7, C2×C4, C2×C4 [×2], C2×C4 [×13], D4 [×12], C23 [×4], C23, D7, C14, C14 [×2], C14 [×4], C42, C42, C22⋊C4 [×10], C4⋊C4 [×8], C22×C4 [×5], C2×D4 [×6], C2×D4 [×4], Dic7 [×7], C28 [×2], C28 [×2], D14 [×3], C2×C14, C2×C14 [×12], C42⋊C2, C4×D4 [×2], C4⋊D4 [×6], C22.D4 [×4], C42.C2, C4⋊1D4, C4×D7 [×2], C2×Dic7, C2×Dic7 [×6], C2×Dic7 [×4], C7⋊D4 [×4], C2×C28, C2×C28 [×2], C7×D4 [×8], C22×D7, C22×C14 [×4], C22.34C24, C4×Dic7, Dic7⋊C4 [×6], C4⋊Dic7 [×2], D14⋊C4 [×2], C23.D7 [×8], C4×C28, C2×C4×D7, C22×Dic7 [×4], C2×C7⋊D4 [×4], D4×C14 [×6], C28.6Q8, C42⋊D7, D4×Dic7 [×2], C23.18D14 [×4], C28⋊2D4 [×2], Dic7⋊D4 [×4], C7×C4⋊1D4, C42.168D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D7 [×7], C22.34C24, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, D4⋊6D14 [×2], C42.168D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=b2c-1 >
►On 224 points
Generators in S224
(1 108 157 154)(2 141 158 109)(3 110 159 142)(4 143 160 111)(5 112 161 144)(6 145 162 99)(7 100 163 146)(8 147 164 101)(9 102 165 148)(10 149 166 103)(11 104 167 150)(12 151 168 105)(13 106 155 152)(14 153 156 107)(15 200 130 57)(16 58 131 201)(17 202 132 59)(18 60 133 203)(19 204 134 61)(20 62 135 205)(21 206 136 63)(22 64 137 207)(23 208 138 65)(24 66 139 209)(25 210 140 67)(26 68 127 197)(27 198 128 69)(28 70 129 199)(29 173 219 194)(30 195 220 174)(31 175 221 196)(32 183 222 176)(33 177 223 184)(34 185 224 178)(35 179 211 186)(36 187 212 180)(37 181 213 188)(38 189 214 182)(39 169 215 190)(40 191 216 170)(41 171 217 192)(42 193 218 172)(43 121 80 91)(44 92 81 122)(45 123 82 93)(46 94 83 124)(47 125 84 95)(48 96 71 126)(49 113 72 97)(50 98 73 114)(51 115 74 85)(52 86 75 116)(53 117 76 87)(54 88 77 118)(55 119 78 89)(56 90 79 120)
(1 101 51 122)(2 123 52 102)(3 103 53 124)(4 125 54 104)(5 105 55 126)(6 113 56 106)(7 107 43 114)(8 115 44 108)(9 109 45 116)(10 117 46 110)(11 111 47 118)(12 119 48 112)(13 99 49 120)(14 121 50 100)(15 207 171 224)(16 211 172 208)(17 209 173 212)(18 213 174 210)(19 197 175 214)(20 215 176 198)(21 199 177 216)(22 217 178 200)(23 201 179 218)(24 219 180 202)(25 203 181 220)(26 221 182 204)(27 205 169 222)(28 223 170 206)(29 187 59 139)(30 140 60 188)(31 189 61 127)(32 128 62 190)(33 191 63 129)(34 130 64 192)(35 193 65 131)(36 132 66 194)(37 195 67 133)(38 134 68 196)(39 183 69 135)(40 136 70 184)(41 185 57 137)(42 138 58 186)(71 144 168 89)(72 90 155 145)(73 146 156 91)(74 92 157 147)(75 148 158 93)(76 94 159 149)(77 150 160 95)(78 96 161 151)(79 152 162 97)(80 98 163 153)(81 154 164 85)(82 86 165 141)(83 142 166 87)(84 88 167 143)
(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)
(1 186 51 138)(2 137 52 185)(3 184 53 136)(4 135 54 183)(5 196 55 134)(6 133 56 195)(7 194 43 132)(8 131 44 193)(9 192 45 130)(10 129 46 191)(11 190 47 128)(12 127 48 189)(13 188 49 140)(14 139 50 187)(15 165 171 82)(16 81 172 164)(17 163 173 80)(18 79 174 162)(19 161 175 78)(20 77 176 160)(21 159 177 76)(22 75 178 158)(23 157 179 74)(24 73 180 156)(25 155 181 72)(26 71 182 168)(27 167 169 84)(28 83 170 166)(29 91 59 146)(30 145 60 90)(31 89 61 144)(32 143 62 88)(33 87 63 142)(34 141 64 86)(35 85 65 154)(36 153 66 98)(37 97 67 152)(38 151 68 96)(39 95 69 150)(40 149 70 94)(41 93 57 148)(42 147 58 92)(99 203 120 220)(100 219 121 202)(101 201 122 218)(102 217 123 200)(103 199 124 216)(104 215 125 198)(105 197 126 214)(106 213 113 210)(107 209 114 212)(108 211 115 208)(109 207 116 224)(110 223 117 206)(111 205 118 222)(112 221 119 204)
G:=sub<Sym(224)| (1,108,157,154)(2,141,158,109)(3,110,159,142)(4,143,160,111)(5,112,161,144)(6,145,162,99)(7,100,163,146)(8,147,164,101)(9,102,165,148)(10,149,166,103)(11,104,167,150)(12,151,168,105)(13,106,155,152)(14,153,156,107)(15,200,130,57)(16,58,131,201)(17,202,132,59)(18,60,133,203)(19,204,134,61)(20,62,135,205)(21,206,136,63)(22,64,137,207)(23,208,138,65)(24,66,139,209)(25,210,140,67)(26,68,127,197)(27,198,128,69)(28,70,129,199)(29,173,219,194)(30,195,220,174)(31,175,221,196)(32,183,222,176)(33,177,223,184)(34,185,224,178)(35,179,211,186)(36,187,212,180)(37,181,213,188)(38,189,214,182)(39,169,215,190)(40,191,216,170)(41,171,217,192)(42,193,218,172)(43,121,80,91)(44,92,81,122)(45,123,82,93)(46,94,83,124)(47,125,84,95)(48,96,71,126)(49,113,72,97)(50,98,73,114)(51,115,74,85)(52,86,75,116)(53,117,76,87)(54,88,77,118)(55,119,78,89)(56,90,79,120), (1,101,51,122)(2,123,52,102)(3,103,53,124)(4,125,54,104)(5,105,55,126)(6,113,56,106)(7,107,43,114)(8,115,44,108)(9,109,45,116)(10,117,46,110)(11,111,47,118)(12,119,48,112)(13,99,49,120)(14,121,50,100)(15,207,171,224)(16,211,172,208)(17,209,173,212)(18,213,174,210)(19,197,175,214)(20,215,176,198)(21,199,177,216)(22,217,178,200)(23,201,179,218)(24,219,180,202)(25,203,181,220)(26,221,182,204)(27,205,169,222)(28,223,170,206)(29,187,59,139)(30,140,60,188)(31,189,61,127)(32,128,62,190)(33,191,63,129)(34,130,64,192)(35,193,65,131)(36,132,66,194)(37,195,67,133)(38,134,68,196)(39,183,69,135)(40,136,70,184)(41,185,57,137)(42,138,58,186)(71,144,168,89)(72,90,155,145)(73,146,156,91)(74,92,157,147)(75,148,158,93)(76,94,159,149)(77,150,160,95)(78,96,161,151)(79,152,162,97)(80,98,163,153)(81,154,164,85)(82,86,165,141)(83,142,166,87)(84,88,167,143), (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), (1,186,51,138)(2,137,52,185)(3,184,53,136)(4,135,54,183)(5,196,55,134)(6,133,56,195)(7,194,43,132)(8,131,44,193)(9,192,45,130)(10,129,46,191)(11,190,47,128)(12,127,48,189)(13,188,49,140)(14,139,50,187)(15,165,171,82)(16,81,172,164)(17,163,173,80)(18,79,174,162)(19,161,175,78)(20,77,176,160)(21,159,177,76)(22,75,178,158)(23,157,179,74)(24,73,180,156)(25,155,181,72)(26,71,182,168)(27,167,169,84)(28,83,170,166)(29,91,59,146)(30,145,60,90)(31,89,61,144)(32,143,62,88)(33,87,63,142)(34,141,64,86)(35,85,65,154)(36,153,66,98)(37,97,67,152)(38,151,68,96)(39,95,69,150)(40,149,70,94)(41,93,57,148)(42,147,58,92)(99,203,120,220)(100,219,121,202)(101,201,122,218)(102,217,123,200)(103,199,124,216)(104,215,125,198)(105,197,126,214)(106,213,113,210)(107,209,114,212)(108,211,115,208)(109,207,116,224)(110,223,117,206)(111,205,118,222)(112,221,119,204)>;
G:=Group( (1,108,157,154)(2,141,158,109)(3,110,159,142)(4,143,160,111)(5,112,161,144)(6,145,162,99)(7,100,163,146)(8,147,164,101)(9,102,165,148)(10,149,166,103)(11,104,167,150)(12,151,168,105)(13,106,155,152)(14,153,156,107)(15,200,130,57)(16,58,131,201)(17,202,132,59)(18,60,133,203)(19,204,134,61)(20,62,135,205)(21,206,136,63)(22,64,137,207)(23,208,138,65)(24,66,139,209)(25,210,140,67)(26,68,127,197)(27,198,128,69)(28,70,129,199)(29,173,219,194)(30,195,220,174)(31,175,221,196)(32,183,222,176)(33,177,223,184)(34,185,224,178)(35,179,211,186)(36,187,212,180)(37,181,213,188)(38,189,214,182)(39,169,215,190)(40,191,216,170)(41,171,217,192)(42,193,218,172)(43,121,80,91)(44,92,81,122)(45,123,82,93)(46,94,83,124)(47,125,84,95)(48,96,71,126)(49,113,72,97)(50,98,73,114)(51,115,74,85)(52,86,75,116)(53,117,76,87)(54,88,77,118)(55,119,78,89)(56,90,79,120), (1,101,51,122)(2,123,52,102)(3,103,53,124)(4,125,54,104)(5,105,55,126)(6,113,56,106)(7,107,43,114)(8,115,44,108)(9,109,45,116)(10,117,46,110)(11,111,47,118)(12,119,48,112)(13,99,49,120)(14,121,50,100)(15,207,171,224)(16,211,172,208)(17,209,173,212)(18,213,174,210)(19,197,175,214)(20,215,176,198)(21,199,177,216)(22,217,178,200)(23,201,179,218)(24,219,180,202)(25,203,181,220)(26,221,182,204)(27,205,169,222)(28,223,170,206)(29,187,59,139)(30,140,60,188)(31,189,61,127)(32,128,62,190)(33,191,63,129)(34,130,64,192)(35,193,65,131)(36,132,66,194)(37,195,67,133)(38,134,68,196)(39,183,69,135)(40,136,70,184)(41,185,57,137)(42,138,58,186)(71,144,168,89)(72,90,155,145)(73,146,156,91)(74,92,157,147)(75,148,158,93)(76,94,159,149)(77,150,160,95)(78,96,161,151)(79,152,162,97)(80,98,163,153)(81,154,164,85)(82,86,165,141)(83,142,166,87)(84,88,167,143), (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), (1,186,51,138)(2,137,52,185)(3,184,53,136)(4,135,54,183)(5,196,55,134)(6,133,56,195)(7,194,43,132)(8,131,44,193)(9,192,45,130)(10,129,46,191)(11,190,47,128)(12,127,48,189)(13,188,49,140)(14,139,50,187)(15,165,171,82)(16,81,172,164)(17,163,173,80)(18,79,174,162)(19,161,175,78)(20,77,176,160)(21,159,177,76)(22,75,178,158)(23,157,179,74)(24,73,180,156)(25,155,181,72)(26,71,182,168)(27,167,169,84)(28,83,170,166)(29,91,59,146)(30,145,60,90)(31,89,61,144)(32,143,62,88)(33,87,63,142)(34,141,64,86)(35,85,65,154)(36,153,66,98)(37,97,67,152)(38,151,68,96)(39,95,69,150)(40,149,70,94)(41,93,57,148)(42,147,58,92)(99,203,120,220)(100,219,121,202)(101,201,122,218)(102,217,123,200)(103,199,124,216)(104,215,125,198)(105,197,126,214)(106,213,113,210)(107,209,114,212)(108,211,115,208)(109,207,116,224)(110,223,117,206)(111,205,118,222)(112,221,119,204) );
G=PermutationGroup([(1,108,157,154),(2,141,158,109),(3,110,159,142),(4,143,160,111),(5,112,161,144),(6,145,162,99),(7,100,163,146),(8,147,164,101),(9,102,165,148),(10,149,166,103),(11,104,167,150),(12,151,168,105),(13,106,155,152),(14,153,156,107),(15,200,130,57),(16,58,131,201),(17,202,132,59),(18,60,133,203),(19,204,134,61),(20,62,135,205),(21,206,136,63),(22,64,137,207),(23,208,138,65),(24,66,139,209),(25,210,140,67),(26,68,127,197),(27,198,128,69),(28,70,129,199),(29,173,219,194),(30,195,220,174),(31,175,221,196),(32,183,222,176),(33,177,223,184),(34,185,224,178),(35,179,211,186),(36,187,212,180),(37,181,213,188),(38,189,214,182),(39,169,215,190),(40,191,216,170),(41,171,217,192),(42,193,218,172),(43,121,80,91),(44,92,81,122),(45,123,82,93),(46,94,83,124),(47,125,84,95),(48,96,71,126),(49,113,72,97),(50,98,73,114),(51,115,74,85),(52,86,75,116),(53,117,76,87),(54,88,77,118),(55,119,78,89),(56,90,79,120)], [(1,101,51,122),(2,123,52,102),(3,103,53,124),(4,125,54,104),(5,105,55,126),(6,113,56,106),(7,107,43,114),(8,115,44,108),(9,109,45,116),(10,117,46,110),(11,111,47,118),(12,119,48,112),(13,99,49,120),(14,121,50,100),(15,207,171,224),(16,211,172,208),(17,209,173,212),(18,213,174,210),(19,197,175,214),(20,215,176,198),(21,199,177,216),(22,217,178,200),(23,201,179,218),(24,219,180,202),(25,203,181,220),(26,221,182,204),(27,205,169,222),(28,223,170,206),(29,187,59,139),(30,140,60,188),(31,189,61,127),(32,128,62,190),(33,191,63,129),(34,130,64,192),(35,193,65,131),(36,132,66,194),(37,195,67,133),(38,134,68,196),(39,183,69,135),(40,136,70,184),(41,185,57,137),(42,138,58,186),(71,144,168,89),(72,90,155,145),(73,146,156,91),(74,92,157,147),(75,148,158,93),(76,94,159,149),(77,150,160,95),(78,96,161,151),(79,152,162,97),(80,98,163,153),(81,154,164,85),(82,86,165,141),(83,142,166,87),(84,88,167,143)], [(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)], [(1,186,51,138),(2,137,52,185),(3,184,53,136),(4,135,54,183),(5,196,55,134),(6,133,56,195),(7,194,43,132),(8,131,44,193),(9,192,45,130),(10,129,46,191),(11,190,47,128),(12,127,48,189),(13,188,49,140),(14,139,50,187),(15,165,171,82),(16,81,172,164),(17,163,173,80),(18,79,174,162),(19,161,175,78),(20,77,176,160),(21,159,177,76),(22,75,178,158),(23,157,179,74),(24,73,180,156),(25,155,181,72),(26,71,182,168),(27,167,169,84),(28,83,170,166),(29,91,59,146),(30,145,60,90),(31,89,61,144),(32,143,62,88),(33,87,63,142),(34,141,64,86),(35,85,65,154),(36,153,66,98),(37,97,67,152),(38,151,68,96),(39,95,69,150),(40,149,70,94),(41,93,57,148),(42,147,58,92),(99,203,120,220),(100,219,121,202),(101,201,122,218),(102,217,123,200),(103,199,124,216),(104,215,125,198),(105,197,126,214),(106,213,113,210),(107,209,114,212),(108,211,115,208),(109,207,116,224),(110,223,117,206),(111,205,118,222),(112,221,119,204)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 27 |
| 0 | 0 | 0 | 0 | 2 | 18 |
| 0 | 0 | 18 | 2 | 0 | 0 |
| 0 | 0 | 27 | 11 | 0 | 0 |
| 8 | 28 | 0 | 0 | 0 | 0 |
| 7 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 2 | 0 | 0 |
| 0 | 0 | 27 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 2 |
| 0 | 0 | 0 | 0 | 27 | 11 |
| 8 | 28 | 0 | 0 | 0 | 0 |
| 5 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 25 |
| 0 | 0 | 0 | 0 | 4 | 25 |
| 0 | 0 | 11 | 25 | 0 | 0 |
| 0 | 0 | 4 | 25 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 18 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 16 | 0 | 0 |
| 0 | 0 | 26 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 16 |
| 0 | 0 | 0 | 0 | 26 | 7 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,18,27,0,0,0,0,2,11,0,0,11,2,0,0,0,0,27,18,0,0],[8,7,0,0,0,0,28,21,0,0,0,0,0,0,18,27,0,0,0,0,2,11,0,0,0,0,0,0,18,27,0,0,0,0,2,11],[8,5,0,0,0,0,28,21,0,0,0,0,0,0,0,0,11,4,0,0,0,0,25,25,0,0,11,4,0,0,0,0,25,25,0,0],[12,18,0,0,0,0,0,17,0,0,0,0,0,0,22,26,0,0,0,0,16,7,0,0,0,0,0,0,22,26,0,0,0,0,16,7] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4M | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | 2+ (1+4) | D4⋊2D7 | D4⋊6D14 |
| kernel | C42.168D14 | C28.6Q8 | C42⋊D7 | D4×Dic7 | C23.18D14 | C28⋊2D4 | Dic7⋊D4 | C7×C4⋊1D4 | C4⋊1D4 | C28 | C42 | C2×D4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 1 | 3 | 4 | 3 | 18 | 2 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2._{168}D_{14} % in TeX
G:=Group("C4^2.168D14"); // GroupNames label
G:=SmallGroup(448,1172);
// by ID
G=gap.SmallGroup(448,1172);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,100,675,570,185,18822]);
// 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=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations
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