metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.241D14, C4⋊Q8⋊20D7, (C4×D7)⋊5Q8, C4.40(Q8×D7), D14.5(C2×Q8), C28.54(C2×Q8), C4⋊C4.219D14, C28⋊2Q8⋊36C2, (Q8×Dic7)⋊22C2, (C2×Q8).147D14, C4.Dic14⋊42C2, Dic7.17(C2×Q8), (D7×C42).10C2, Dic7⋊3Q8⋊42C2, C28.136(C4○D4), C4.41(D4⋊2D7), C14.48(C22×Q8), (C4×C28).213C22, (C2×C28).105C23, (C2×C14).272C24, D14⋊C4.51C22, D14⋊2Q8.13C2, D14⋊3Q8.11C2, C4.22(Q8⋊2D7), Dic7⋊C4.61C22, C4⋊Dic7.251C22, (Q8×C14).139C22, C22.293(C23×D7), C7⋊6(C23.37C23), (C4×Dic7).161C22, (C2×Dic7).143C23, (C22×D7).233C23, (C2×Dic14).190C22, C2.31(C2×Q8×D7), (C7×C4⋊Q8)⋊14C2, C4⋊C4⋊7D7.14C2, C14.100(C2×C4○D4), C2.64(C2×D4⋊2D7), C2.29(C2×Q8⋊2D7), (C2×C4×D7).252C22, (C7×C4⋊C4).215C22, (C2×C4).600(C22×D7), SmallGroup(448,1181)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 844 in 222 conjugacy classes, 111 normal (33 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×12], C22, C22 [×4], C7, C2×C4 [×3], C2×C4 [×4], C2×C4 [×15], Q8 [×8], C23, D7 [×2], C14 [×3], C42, C42 [×7], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×3], C2×Q8 [×2], C2×Q8 [×2], Dic7 [×2], Dic7 [×6], C28 [×6], C28 [×4], D14 [×2], D14 [×2], C2×C14, C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8, C4⋊Q8, Dic14 [×4], C4×D7 [×4], C4×D7 [×4], C2×Dic7 [×3], C2×Dic7 [×4], C2×C28 [×3], C2×C28 [×4], C7×Q8 [×4], C22×D7, C23.37C23, C4×Dic7 [×3], C4×Dic7 [×4], Dic7⋊C4 [×4], C4⋊Dic7 [×8], D14⋊C4 [×4], C4×C28, C7×C4⋊C4 [×4], C2×Dic14 [×2], C2×C4×D7 [×3], Q8×C14 [×2], C28⋊2Q8, D7×C42, Dic7⋊3Q8 [×2], C4.Dic14 [×2], C4⋊C4⋊7D7 [×2], D14⋊2Q8 [×2], Q8×Dic7 [×2], D14⋊3Q8 [×2], C7×C4⋊Q8, C42.241D14
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D7, C2×Q8 [×6], C4○D4 [×4], C24, D14 [×7], C22×Q8, C2×C4○D4 [×2], C22×D7 [×7], C23.37C23, D4⋊2D7 [×2], Q8×D7 [×2], Q8⋊2D7 [×2], C23×D7, C2×D4⋊2D7, C2×Q8×D7, C2×Q8⋊2D7, C42.241D14
Generators and relations
G = < a,b,c,d | a4=b4=1, c14=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2c13 >
►On 224 points
Generators in S224
(1 59 190 135)(2 136 191 60)(3 61 192 137)(4 138 193 62)(5 63 194 139)(6 140 195 64)(7 65 196 113)(8 114 169 66)(9 67 170 115)(10 116 171 68)(11 69 172 117)(12 118 173 70)(13 71 174 119)(14 120 175 72)(15 73 176 121)(16 122 177 74)(17 75 178 123)(18 124 179 76)(19 77 180 125)(20 126 181 78)(21 79 182 127)(22 128 183 80)(23 81 184 129)(24 130 185 82)(25 83 186 131)(26 132 187 84)(27 57 188 133)(28 134 189 58)(29 215 145 102)(30 103 146 216)(31 217 147 104)(32 105 148 218)(33 219 149 106)(34 107 150 220)(35 221 151 108)(36 109 152 222)(37 223 153 110)(38 111 154 224)(39 197 155 112)(40 85 156 198)(41 199 157 86)(42 87 158 200)(43 201 159 88)(44 89 160 202)(45 203 161 90)(46 91 162 204)(47 205 163 92)(48 93 164 206)(49 207 165 94)(50 95 166 208)(51 209 167 96)(52 97 168 210)(53 211 141 98)(54 99 142 212)(55 213 143 100)(56 101 144 214)
(1 218 15 204)(2 205 16 219)(3 220 17 206)(4 207 18 221)(5 222 19 208)(6 209 20 223)(7 224 21 210)(8 211 22 197)(9 198 23 212)(10 213 24 199)(11 200 25 214)(12 215 26 201)(13 202 27 216)(14 217 28 203)(29 84 43 70)(30 71 44 57)(31 58 45 72)(32 73 46 59)(33 60 47 74)(34 75 48 61)(35 62 49 76)(36 77 50 63)(37 64 51 78)(38 79 52 65)(39 66 53 80)(40 81 54 67)(41 68 55 82)(42 83 56 69)(85 184 99 170)(86 171 100 185)(87 186 101 172)(88 173 102 187)(89 188 103 174)(90 175 104 189)(91 190 105 176)(92 177 106 191)(93 192 107 178)(94 179 108 193)(95 194 109 180)(96 181 110 195)(97 196 111 182)(98 183 112 169)(113 154 127 168)(114 141 128 155)(115 156 129 142)(116 143 130 157)(117 158 131 144)(118 145 132 159)(119 160 133 146)(120 147 134 161)(121 162 135 148)(122 149 136 163)(123 164 137 150)(124 151 138 165)(125 166 139 152)(126 153 140 167)
(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 28 176 175)(2 174 177 27)(3 26 178 173)(4 172 179 25)(5 24 180 171)(6 170 181 23)(7 22 182 169)(8 196 183 21)(9 20 184 195)(10 194 185 19)(11 18 186 193)(12 192 187 17)(13 16 188 191)(14 190 189 15)(29 164 159 34)(30 33 160 163)(31 162 161 32)(35 158 165 56)(36 55 166 157)(37 156 167 54)(38 53 168 155)(39 154 141 52)(40 51 142 153)(41 152 143 50)(42 49 144 151)(43 150 145 48)(44 47 146 149)(45 148 147 46)(57 60 119 122)(58 121 120 59)(61 84 123 118)(62 117 124 83)(63 82 125 116)(64 115 126 81)(65 80 127 114)(66 113 128 79)(67 78 129 140)(68 139 130 77)(69 76 131 138)(70 137 132 75)(71 74 133 136)(72 135 134 73)(85 96 212 223)(86 222 213 95)(87 94 214 221)(88 220 215 93)(89 92 216 219)(90 218 217 91)(97 112 224 211)(98 210 197 111)(99 110 198 209)(100 208 199 109)(101 108 200 207)(102 206 201 107)(103 106 202 205)(104 204 203 105)
G:=sub<Sym(224)| (1,59,190,135)(2,136,191,60)(3,61,192,137)(4,138,193,62)(5,63,194,139)(6,140,195,64)(7,65,196,113)(8,114,169,66)(9,67,170,115)(10,116,171,68)(11,69,172,117)(12,118,173,70)(13,71,174,119)(14,120,175,72)(15,73,176,121)(16,122,177,74)(17,75,178,123)(18,124,179,76)(19,77,180,125)(20,126,181,78)(21,79,182,127)(22,128,183,80)(23,81,184,129)(24,130,185,82)(25,83,186,131)(26,132,187,84)(27,57,188,133)(28,134,189,58)(29,215,145,102)(30,103,146,216)(31,217,147,104)(32,105,148,218)(33,219,149,106)(34,107,150,220)(35,221,151,108)(36,109,152,222)(37,223,153,110)(38,111,154,224)(39,197,155,112)(40,85,156,198)(41,199,157,86)(42,87,158,200)(43,201,159,88)(44,89,160,202)(45,203,161,90)(46,91,162,204)(47,205,163,92)(48,93,164,206)(49,207,165,94)(50,95,166,208)(51,209,167,96)(52,97,168,210)(53,211,141,98)(54,99,142,212)(55,213,143,100)(56,101,144,214), (1,218,15,204)(2,205,16,219)(3,220,17,206)(4,207,18,221)(5,222,19,208)(6,209,20,223)(7,224,21,210)(8,211,22,197)(9,198,23,212)(10,213,24,199)(11,200,25,214)(12,215,26,201)(13,202,27,216)(14,217,28,203)(29,84,43,70)(30,71,44,57)(31,58,45,72)(32,73,46,59)(33,60,47,74)(34,75,48,61)(35,62,49,76)(36,77,50,63)(37,64,51,78)(38,79,52,65)(39,66,53,80)(40,81,54,67)(41,68,55,82)(42,83,56,69)(85,184,99,170)(86,171,100,185)(87,186,101,172)(88,173,102,187)(89,188,103,174)(90,175,104,189)(91,190,105,176)(92,177,106,191)(93,192,107,178)(94,179,108,193)(95,194,109,180)(96,181,110,195)(97,196,111,182)(98,183,112,169)(113,154,127,168)(114,141,128,155)(115,156,129,142)(116,143,130,157)(117,158,131,144)(118,145,132,159)(119,160,133,146)(120,147,134,161)(121,162,135,148)(122,149,136,163)(123,164,137,150)(124,151,138,165)(125,166,139,152)(126,153,140,167), (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), (1,28,176,175)(2,174,177,27)(3,26,178,173)(4,172,179,25)(5,24,180,171)(6,170,181,23)(7,22,182,169)(8,196,183,21)(9,20,184,195)(10,194,185,19)(11,18,186,193)(12,192,187,17)(13,16,188,191)(14,190,189,15)(29,164,159,34)(30,33,160,163)(31,162,161,32)(35,158,165,56)(36,55,166,157)(37,156,167,54)(38,53,168,155)(39,154,141,52)(40,51,142,153)(41,152,143,50)(42,49,144,151)(43,150,145,48)(44,47,146,149)(45,148,147,46)(57,60,119,122)(58,121,120,59)(61,84,123,118)(62,117,124,83)(63,82,125,116)(64,115,126,81)(65,80,127,114)(66,113,128,79)(67,78,129,140)(68,139,130,77)(69,76,131,138)(70,137,132,75)(71,74,133,136)(72,135,134,73)(85,96,212,223)(86,222,213,95)(87,94,214,221)(88,220,215,93)(89,92,216,219)(90,218,217,91)(97,112,224,211)(98,210,197,111)(99,110,198,209)(100,208,199,109)(101,108,200,207)(102,206,201,107)(103,106,202,205)(104,204,203,105)>;
G:=Group( (1,59,190,135)(2,136,191,60)(3,61,192,137)(4,138,193,62)(5,63,194,139)(6,140,195,64)(7,65,196,113)(8,114,169,66)(9,67,170,115)(10,116,171,68)(11,69,172,117)(12,118,173,70)(13,71,174,119)(14,120,175,72)(15,73,176,121)(16,122,177,74)(17,75,178,123)(18,124,179,76)(19,77,180,125)(20,126,181,78)(21,79,182,127)(22,128,183,80)(23,81,184,129)(24,130,185,82)(25,83,186,131)(26,132,187,84)(27,57,188,133)(28,134,189,58)(29,215,145,102)(30,103,146,216)(31,217,147,104)(32,105,148,218)(33,219,149,106)(34,107,150,220)(35,221,151,108)(36,109,152,222)(37,223,153,110)(38,111,154,224)(39,197,155,112)(40,85,156,198)(41,199,157,86)(42,87,158,200)(43,201,159,88)(44,89,160,202)(45,203,161,90)(46,91,162,204)(47,205,163,92)(48,93,164,206)(49,207,165,94)(50,95,166,208)(51,209,167,96)(52,97,168,210)(53,211,141,98)(54,99,142,212)(55,213,143,100)(56,101,144,214), (1,218,15,204)(2,205,16,219)(3,220,17,206)(4,207,18,221)(5,222,19,208)(6,209,20,223)(7,224,21,210)(8,211,22,197)(9,198,23,212)(10,213,24,199)(11,200,25,214)(12,215,26,201)(13,202,27,216)(14,217,28,203)(29,84,43,70)(30,71,44,57)(31,58,45,72)(32,73,46,59)(33,60,47,74)(34,75,48,61)(35,62,49,76)(36,77,50,63)(37,64,51,78)(38,79,52,65)(39,66,53,80)(40,81,54,67)(41,68,55,82)(42,83,56,69)(85,184,99,170)(86,171,100,185)(87,186,101,172)(88,173,102,187)(89,188,103,174)(90,175,104,189)(91,190,105,176)(92,177,106,191)(93,192,107,178)(94,179,108,193)(95,194,109,180)(96,181,110,195)(97,196,111,182)(98,183,112,169)(113,154,127,168)(114,141,128,155)(115,156,129,142)(116,143,130,157)(117,158,131,144)(118,145,132,159)(119,160,133,146)(120,147,134,161)(121,162,135,148)(122,149,136,163)(123,164,137,150)(124,151,138,165)(125,166,139,152)(126,153,140,167), (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), (1,28,176,175)(2,174,177,27)(3,26,178,173)(4,172,179,25)(5,24,180,171)(6,170,181,23)(7,22,182,169)(8,196,183,21)(9,20,184,195)(10,194,185,19)(11,18,186,193)(12,192,187,17)(13,16,188,191)(14,190,189,15)(29,164,159,34)(30,33,160,163)(31,162,161,32)(35,158,165,56)(36,55,166,157)(37,156,167,54)(38,53,168,155)(39,154,141,52)(40,51,142,153)(41,152,143,50)(42,49,144,151)(43,150,145,48)(44,47,146,149)(45,148,147,46)(57,60,119,122)(58,121,120,59)(61,84,123,118)(62,117,124,83)(63,82,125,116)(64,115,126,81)(65,80,127,114)(66,113,128,79)(67,78,129,140)(68,139,130,77)(69,76,131,138)(70,137,132,75)(71,74,133,136)(72,135,134,73)(85,96,212,223)(86,222,213,95)(87,94,214,221)(88,220,215,93)(89,92,216,219)(90,218,217,91)(97,112,224,211)(98,210,197,111)(99,110,198,209)(100,208,199,109)(101,108,200,207)(102,206,201,107)(103,106,202,205)(104,204,203,105) );
G=PermutationGroup([(1,59,190,135),(2,136,191,60),(3,61,192,137),(4,138,193,62),(5,63,194,139),(6,140,195,64),(7,65,196,113),(8,114,169,66),(9,67,170,115),(10,116,171,68),(11,69,172,117),(12,118,173,70),(13,71,174,119),(14,120,175,72),(15,73,176,121),(16,122,177,74),(17,75,178,123),(18,124,179,76),(19,77,180,125),(20,126,181,78),(21,79,182,127),(22,128,183,80),(23,81,184,129),(24,130,185,82),(25,83,186,131),(26,132,187,84),(27,57,188,133),(28,134,189,58),(29,215,145,102),(30,103,146,216),(31,217,147,104),(32,105,148,218),(33,219,149,106),(34,107,150,220),(35,221,151,108),(36,109,152,222),(37,223,153,110),(38,111,154,224),(39,197,155,112),(40,85,156,198),(41,199,157,86),(42,87,158,200),(43,201,159,88),(44,89,160,202),(45,203,161,90),(46,91,162,204),(47,205,163,92),(48,93,164,206),(49,207,165,94),(50,95,166,208),(51,209,167,96),(52,97,168,210),(53,211,141,98),(54,99,142,212),(55,213,143,100),(56,101,144,214)], [(1,218,15,204),(2,205,16,219),(3,220,17,206),(4,207,18,221),(5,222,19,208),(6,209,20,223),(7,224,21,210),(8,211,22,197),(9,198,23,212),(10,213,24,199),(11,200,25,214),(12,215,26,201),(13,202,27,216),(14,217,28,203),(29,84,43,70),(30,71,44,57),(31,58,45,72),(32,73,46,59),(33,60,47,74),(34,75,48,61),(35,62,49,76),(36,77,50,63),(37,64,51,78),(38,79,52,65),(39,66,53,80),(40,81,54,67),(41,68,55,82),(42,83,56,69),(85,184,99,170),(86,171,100,185),(87,186,101,172),(88,173,102,187),(89,188,103,174),(90,175,104,189),(91,190,105,176),(92,177,106,191),(93,192,107,178),(94,179,108,193),(95,194,109,180),(96,181,110,195),(97,196,111,182),(98,183,112,169),(113,154,127,168),(114,141,128,155),(115,156,129,142),(116,143,130,157),(117,158,131,144),(118,145,132,159),(119,160,133,146),(120,147,134,161),(121,162,135,148),(122,149,136,163),(123,164,137,150),(124,151,138,165),(125,166,139,152),(126,153,140,167)], [(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)], [(1,28,176,175),(2,174,177,27),(3,26,178,173),(4,172,179,25),(5,24,180,171),(6,170,181,23),(7,22,182,169),(8,196,183,21),(9,20,184,195),(10,194,185,19),(11,18,186,193),(12,192,187,17),(13,16,188,191),(14,190,189,15),(29,164,159,34),(30,33,160,163),(31,162,161,32),(35,158,165,56),(36,55,166,157),(37,156,167,54),(38,53,168,155),(39,154,141,52),(40,51,142,153),(41,152,143,50),(42,49,144,151),(43,150,145,48),(44,47,146,149),(45,148,147,46),(57,60,119,122),(58,121,120,59),(61,84,123,118),(62,117,124,83),(63,82,125,116),(64,115,126,81),(65,80,127,114),(66,113,128,79),(67,78,129,140),(68,139,130,77),(69,76,131,138),(70,137,132,75),(71,74,133,136),(72,135,134,73),(85,96,212,223),(86,222,213,95),(87,94,214,221),(88,220,215,93),(89,92,216,219),(90,218,217,91),(97,112,224,211),(98,210,197,111),(99,110,198,209),(100,208,199,109),(101,108,200,207),(102,206,201,107),(103,106,202,205),(104,204,203,105)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 17 | 0 | 0 | 0 | 0 | 0 |
| 4 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 23 | 22 | 0 | 0 | 0 | 0 |
| 5 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 7 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 6 | 7 | 0 | 0 | 0 | 0 |
| 3 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 4 | 0 | 0 |
| 0 | 0 | 17 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(29))| [17,4,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,12],[23,5,0,0,0,0,22,6,0,0,0,0,0,0,0,7,0,0,0,0,4,22,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[6,3,0,0,0,0,7,23,0,0,0,0,0,0,7,17,0,0,0,0,4,22,0,0,0,0,0,0,0,1,0,0,0,0,28,0] >;
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 4U | 4V | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | C4○D4 | D14 | D14 | D14 | D4⋊2D7 | Q8×D7 | Q8⋊2D7 |
| kernel | C42.241D14 | C28⋊2Q8 | D7×C42 | Dic7⋊3Q8 | C4.Dic14 | C4⋊C4⋊7D7 | D14⋊2Q8 | Q8×Dic7 | D14⋊3Q8 | C7×C4⋊Q8 | C4×D7 | C4⋊Q8 | C28 | C42 | C4⋊C4 | C2×Q8 | C4 | C4 | C4 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 4 | 3 | 8 | 3 | 12 | 6 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4^2._{241}D_{14} % in TeX
G:=Group("C4^2.241D14"); // GroupNames label
G:=SmallGroup(448,1181);
// by ID
G=gap.SmallGroup(448,1181);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,100,1123,570,185,80,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^13>;
// generators/relations
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