metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.162D14, C14.1012- (1+4), C4⋊C4.213D14, C42⋊2C2⋊5D7, C42⋊D7⋊7C2, D14⋊Q8⋊44C2, (C4×Dic14)⋊15C2, Dic7.Q8⋊38C2, (C4×C28).34C22, C22⋊C4.80D14, C4.Dic14⋊40C2, D14.27(C4○D4), (C2×C14).252C24, (C2×C28).604C23, Dic7⋊4D4.5C2, D14.D4.4C2, C23.58(C22×D7), D14⋊C4.140C22, Dic7.32(C4○D4), C22⋊Dic14⋊46C2, C23.D14⋊46C2, C4⋊Dic7.247C22, (C22×C14).66C23, C22.273(C23×D7), C23.D7.68C22, C23.11D14⋊22C2, Dic7⋊C4.146C22, (C2×Dic7).130C23, (C4×Dic7).218C22, (C22×D7).226C23, C2.65(D4.10D14), C7⋊11(C22.46C24), (C2×Dic14).255C22, (C22×Dic7).152C22, (D7×C4⋊C4)⋊42C2, C2.99(D7×C4○D4), C4⋊C4⋊7D7⋊41C2, (C7×C42⋊2C2)⋊7C2, C14.210(C2×C4○D4), (C2×C4×D7).220C22, (C2×C4).88(C22×D7), (C7×C4⋊C4).204C22, (C2×C7⋊D4).72C22, (C7×C22⋊C4).77C22, SmallGroup(448,1161)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 876 in 214 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×14], C22, C22 [×7], C7, C2×C4 [×6], C2×C4 [×15], D4 [×2], Q8 [×2], C23, C23, D7 [×2], C14 [×3], C14, C42, C42 [×4], C22⋊C4 [×3], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4 [×4], C2×D4, C2×Q8, Dic7 [×2], Dic7 [×6], C28 [×6], D14 [×2], D14 [×2], C2×C14, C2×C14 [×3], C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C42⋊2C2, C42⋊2C2, Dic14 [×2], C4×D7 [×6], C2×Dic7 [×7], C2×Dic7 [×2], C7⋊D4 [×2], C2×C28 [×6], C22×D7, C22×C14, C22.46C24, C4×Dic7 [×4], Dic7⋊C4 [×9], C4⋊Dic7 [×4], D14⋊C4 [×3], C23.D7 [×2], C4×C28, C7×C22⋊C4 [×3], C7×C4⋊C4 [×3], C2×Dic14, C2×C4×D7 [×3], C22×Dic7, C2×C7⋊D4, C4×Dic14, C42⋊D7, C23.11D14, C22⋊Dic14, C23.D14, Dic7⋊4D4, D14.D4 [×2], Dic7.Q8 [×2], C4.Dic14, D7×C4⋊C4, C4⋊C4⋊7D7, D14⋊Q8, C7×C42⋊2C2, C42.162D14
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], C22.46C24, C23×D7, D7×C4○D4 [×2], D4.10D14, C42.162D14
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=b-1, dcd-1=c13 >
►On 224 points
Generators in S224
(1 177 143 204)(2 192 144 219)(3 179 145 206)(4 194 146 221)(5 181 147 208)(6 196 148 223)(7 183 149 210)(8 170 150 197)(9 185 151 212)(10 172 152 199)(11 187 153 214)(12 174 154 201)(13 189 155 216)(14 176 156 203)(15 191 157 218)(16 178 158 205)(17 193 159 220)(18 180 160 207)(19 195 161 222)(20 182 162 209)(21 169 163 224)(22 184 164 211)(23 171 165 198)(24 186 166 213)(25 173 167 200)(26 188 168 215)(27 175 141 202)(28 190 142 217)(29 124 77 105)(30 139 78 92)(31 126 79 107)(32 113 80 94)(33 128 81 109)(34 115 82 96)(35 130 83 111)(36 117 84 98)(37 132 57 85)(38 119 58 100)(39 134 59 87)(40 121 60 102)(41 136 61 89)(42 123 62 104)(43 138 63 91)(44 125 64 106)(45 140 65 93)(46 127 66 108)(47 114 67 95)(48 129 68 110)(49 116 69 97)(50 131 70 112)(51 118 71 99)(52 133 72 86)(53 120 73 101)(54 135 74 88)(55 122 75 103)(56 137 76 90)
(1 127 15 113)(2 95 16 109)(3 129 17 115)(4 97 18 111)(5 131 19 117)(6 99 20 85)(7 133 21 119)(8 101 22 87)(9 135 23 121)(10 103 24 89)(11 137 25 123)(12 105 26 91)(13 139 27 125)(14 107 28 93)(29 188 43 174)(30 202 44 216)(31 190 45 176)(32 204 46 218)(33 192 47 178)(34 206 48 220)(35 194 49 180)(36 208 50 222)(37 196 51 182)(38 210 52 224)(39 170 53 184)(40 212 54 198)(41 172 55 186)(42 214 56 200)(57 223 71 209)(58 183 72 169)(59 197 73 211)(60 185 74 171)(61 199 75 213)(62 187 76 173)(63 201 77 215)(64 189 78 175)(65 203 79 217)(66 191 80 177)(67 205 81 219)(68 193 82 179)(69 207 83 221)(70 195 84 181)(86 163 100 149)(88 165 102 151)(90 167 104 153)(92 141 106 155)(94 143 108 157)(96 145 110 159)(98 147 112 161)(114 158 128 144)(116 160 130 146)(118 162 132 148)(120 164 134 150)(122 166 136 152)(124 168 138 154)(126 142 140 156)
(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 82 43 68)(30 67 44 81)(31 80 45 66)(32 65 46 79)(33 78 47 64)(34 63 48 77)(35 76 49 62)(36 61 50 75)(37 74 51 60)(38 59 52 73)(39 72 53 58)(40 57 54 71)(41 70 55 84)(42 83 56 69)(85 121 99 135)(86 134 100 120)(87 119 101 133)(88 132 102 118)(89 117 103 131)(90 130 104 116)(91 115 105 129)(92 128 106 114)(93 113 107 127)(94 126 108 140)(95 139 109 125)(96 124 110 138)(97 137 111 123)(98 122 112 136)(141 158 155 144)(142 143 156 157)(145 154 159 168)(146 167 160 153)(147 152 161 166)(148 165 162 151)(149 150 163 164)(169 170 183 184)(171 196 185 182)(172 181 186 195)(173 194 187 180)(174 179 188 193)(175 192 189 178)(176 177 190 191)(197 210 211 224)(198 223 212 209)(199 208 213 222)(200 221 214 207)(201 206 215 220)(202 219 216 205)(203 204 217 218)
G:=sub<Sym(224)| (1,177,143,204)(2,192,144,219)(3,179,145,206)(4,194,146,221)(5,181,147,208)(6,196,148,223)(7,183,149,210)(8,170,150,197)(9,185,151,212)(10,172,152,199)(11,187,153,214)(12,174,154,201)(13,189,155,216)(14,176,156,203)(15,191,157,218)(16,178,158,205)(17,193,159,220)(18,180,160,207)(19,195,161,222)(20,182,162,209)(21,169,163,224)(22,184,164,211)(23,171,165,198)(24,186,166,213)(25,173,167,200)(26,188,168,215)(27,175,141,202)(28,190,142,217)(29,124,77,105)(30,139,78,92)(31,126,79,107)(32,113,80,94)(33,128,81,109)(34,115,82,96)(35,130,83,111)(36,117,84,98)(37,132,57,85)(38,119,58,100)(39,134,59,87)(40,121,60,102)(41,136,61,89)(42,123,62,104)(43,138,63,91)(44,125,64,106)(45,140,65,93)(46,127,66,108)(47,114,67,95)(48,129,68,110)(49,116,69,97)(50,131,70,112)(51,118,71,99)(52,133,72,86)(53,120,73,101)(54,135,74,88)(55,122,75,103)(56,137,76,90), (1,127,15,113)(2,95,16,109)(3,129,17,115)(4,97,18,111)(5,131,19,117)(6,99,20,85)(7,133,21,119)(8,101,22,87)(9,135,23,121)(10,103,24,89)(11,137,25,123)(12,105,26,91)(13,139,27,125)(14,107,28,93)(29,188,43,174)(30,202,44,216)(31,190,45,176)(32,204,46,218)(33,192,47,178)(34,206,48,220)(35,194,49,180)(36,208,50,222)(37,196,51,182)(38,210,52,224)(39,170,53,184)(40,212,54,198)(41,172,55,186)(42,214,56,200)(57,223,71,209)(58,183,72,169)(59,197,73,211)(60,185,74,171)(61,199,75,213)(62,187,76,173)(63,201,77,215)(64,189,78,175)(65,203,79,217)(66,191,80,177)(67,205,81,219)(68,193,82,179)(69,207,83,221)(70,195,84,181)(86,163,100,149)(88,165,102,151)(90,167,104,153)(92,141,106,155)(94,143,108,157)(96,145,110,159)(98,147,112,161)(114,158,128,144)(116,160,130,146)(118,162,132,148)(120,164,134,150)(122,166,136,152)(124,168,138,154)(126,142,140,156), (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)(29,82,43,68)(30,67,44,81)(31,80,45,66)(32,65,46,79)(33,78,47,64)(34,63,48,77)(35,76,49,62)(36,61,50,75)(37,74,51,60)(38,59,52,73)(39,72,53,58)(40,57,54,71)(41,70,55,84)(42,83,56,69)(85,121,99,135)(86,134,100,120)(87,119,101,133)(88,132,102,118)(89,117,103,131)(90,130,104,116)(91,115,105,129)(92,128,106,114)(93,113,107,127)(94,126,108,140)(95,139,109,125)(96,124,110,138)(97,137,111,123)(98,122,112,136)(141,158,155,144)(142,143,156,157)(145,154,159,168)(146,167,160,153)(147,152,161,166)(148,165,162,151)(149,150,163,164)(169,170,183,184)(171,196,185,182)(172,181,186,195)(173,194,187,180)(174,179,188,193)(175,192,189,178)(176,177,190,191)(197,210,211,224)(198,223,212,209)(199,208,213,222)(200,221,214,207)(201,206,215,220)(202,219,216,205)(203,204,217,218)>;
G:=Group( (1,177,143,204)(2,192,144,219)(3,179,145,206)(4,194,146,221)(5,181,147,208)(6,196,148,223)(7,183,149,210)(8,170,150,197)(9,185,151,212)(10,172,152,199)(11,187,153,214)(12,174,154,201)(13,189,155,216)(14,176,156,203)(15,191,157,218)(16,178,158,205)(17,193,159,220)(18,180,160,207)(19,195,161,222)(20,182,162,209)(21,169,163,224)(22,184,164,211)(23,171,165,198)(24,186,166,213)(25,173,167,200)(26,188,168,215)(27,175,141,202)(28,190,142,217)(29,124,77,105)(30,139,78,92)(31,126,79,107)(32,113,80,94)(33,128,81,109)(34,115,82,96)(35,130,83,111)(36,117,84,98)(37,132,57,85)(38,119,58,100)(39,134,59,87)(40,121,60,102)(41,136,61,89)(42,123,62,104)(43,138,63,91)(44,125,64,106)(45,140,65,93)(46,127,66,108)(47,114,67,95)(48,129,68,110)(49,116,69,97)(50,131,70,112)(51,118,71,99)(52,133,72,86)(53,120,73,101)(54,135,74,88)(55,122,75,103)(56,137,76,90), (1,127,15,113)(2,95,16,109)(3,129,17,115)(4,97,18,111)(5,131,19,117)(6,99,20,85)(7,133,21,119)(8,101,22,87)(9,135,23,121)(10,103,24,89)(11,137,25,123)(12,105,26,91)(13,139,27,125)(14,107,28,93)(29,188,43,174)(30,202,44,216)(31,190,45,176)(32,204,46,218)(33,192,47,178)(34,206,48,220)(35,194,49,180)(36,208,50,222)(37,196,51,182)(38,210,52,224)(39,170,53,184)(40,212,54,198)(41,172,55,186)(42,214,56,200)(57,223,71,209)(58,183,72,169)(59,197,73,211)(60,185,74,171)(61,199,75,213)(62,187,76,173)(63,201,77,215)(64,189,78,175)(65,203,79,217)(66,191,80,177)(67,205,81,219)(68,193,82,179)(69,207,83,221)(70,195,84,181)(86,163,100,149)(88,165,102,151)(90,167,104,153)(92,141,106,155)(94,143,108,157)(96,145,110,159)(98,147,112,161)(114,158,128,144)(116,160,130,146)(118,162,132,148)(120,164,134,150)(122,166,136,152)(124,168,138,154)(126,142,140,156), (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)(29,82,43,68)(30,67,44,81)(31,80,45,66)(32,65,46,79)(33,78,47,64)(34,63,48,77)(35,76,49,62)(36,61,50,75)(37,74,51,60)(38,59,52,73)(39,72,53,58)(40,57,54,71)(41,70,55,84)(42,83,56,69)(85,121,99,135)(86,134,100,120)(87,119,101,133)(88,132,102,118)(89,117,103,131)(90,130,104,116)(91,115,105,129)(92,128,106,114)(93,113,107,127)(94,126,108,140)(95,139,109,125)(96,124,110,138)(97,137,111,123)(98,122,112,136)(141,158,155,144)(142,143,156,157)(145,154,159,168)(146,167,160,153)(147,152,161,166)(148,165,162,151)(149,150,163,164)(169,170,183,184)(171,196,185,182)(172,181,186,195)(173,194,187,180)(174,179,188,193)(175,192,189,178)(176,177,190,191)(197,210,211,224)(198,223,212,209)(199,208,213,222)(200,221,214,207)(201,206,215,220)(202,219,216,205)(203,204,217,218) );
G=PermutationGroup([(1,177,143,204),(2,192,144,219),(3,179,145,206),(4,194,146,221),(5,181,147,208),(6,196,148,223),(7,183,149,210),(8,170,150,197),(9,185,151,212),(10,172,152,199),(11,187,153,214),(12,174,154,201),(13,189,155,216),(14,176,156,203),(15,191,157,218),(16,178,158,205),(17,193,159,220),(18,180,160,207),(19,195,161,222),(20,182,162,209),(21,169,163,224),(22,184,164,211),(23,171,165,198),(24,186,166,213),(25,173,167,200),(26,188,168,215),(27,175,141,202),(28,190,142,217),(29,124,77,105),(30,139,78,92),(31,126,79,107),(32,113,80,94),(33,128,81,109),(34,115,82,96),(35,130,83,111),(36,117,84,98),(37,132,57,85),(38,119,58,100),(39,134,59,87),(40,121,60,102),(41,136,61,89),(42,123,62,104),(43,138,63,91),(44,125,64,106),(45,140,65,93),(46,127,66,108),(47,114,67,95),(48,129,68,110),(49,116,69,97),(50,131,70,112),(51,118,71,99),(52,133,72,86),(53,120,73,101),(54,135,74,88),(55,122,75,103),(56,137,76,90)], [(1,127,15,113),(2,95,16,109),(3,129,17,115),(4,97,18,111),(5,131,19,117),(6,99,20,85),(7,133,21,119),(8,101,22,87),(9,135,23,121),(10,103,24,89),(11,137,25,123),(12,105,26,91),(13,139,27,125),(14,107,28,93),(29,188,43,174),(30,202,44,216),(31,190,45,176),(32,204,46,218),(33,192,47,178),(34,206,48,220),(35,194,49,180),(36,208,50,222),(37,196,51,182),(38,210,52,224),(39,170,53,184),(40,212,54,198),(41,172,55,186),(42,214,56,200),(57,223,71,209),(58,183,72,169),(59,197,73,211),(60,185,74,171),(61,199,75,213),(62,187,76,173),(63,201,77,215),(64,189,78,175),(65,203,79,217),(66,191,80,177),(67,205,81,219),(68,193,82,179),(69,207,83,221),(70,195,84,181),(86,163,100,149),(88,165,102,151),(90,167,104,153),(92,141,106,155),(94,143,108,157),(96,145,110,159),(98,147,112,161),(114,158,128,144),(116,160,130,146),(118,162,132,148),(120,164,134,150),(122,166,136,152),(124,168,138,154),(126,142,140,156)], [(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),(29,82,43,68),(30,67,44,81),(31,80,45,66),(32,65,46,79),(33,78,47,64),(34,63,48,77),(35,76,49,62),(36,61,50,75),(37,74,51,60),(38,59,52,73),(39,72,53,58),(40,57,54,71),(41,70,55,84),(42,83,56,69),(85,121,99,135),(86,134,100,120),(87,119,101,133),(88,132,102,118),(89,117,103,131),(90,130,104,116),(91,115,105,129),(92,128,106,114),(93,113,107,127),(94,126,108,140),(95,139,109,125),(96,124,110,138),(97,137,111,123),(98,122,112,136),(141,158,155,144),(142,143,156,157),(145,154,159,168),(146,167,160,153),(147,152,161,166),(148,165,162,151),(149,150,163,164),(169,170,183,184),(171,196,185,182),(172,181,186,195),(173,194,187,180),(174,179,188,193),(175,192,189,178),(176,177,190,191),(197,210,211,224),(198,223,212,209),(199,208,213,222),(200,221,214,207),(201,206,215,220),(202,219,216,205),(203,204,217,218)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 12 | 11 | 0 | 0 | 0 | 0 |
| 16 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 1 | 13 | 0 | 0 | 0 | 0 |
| 11 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 22 | 8 | 0 | 0 | 0 | 0 |
| 1 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 8 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 5 | 28 |
| 22 | 8 | 0 | 0 | 0 | 0 |
| 1 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 26 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [12,16,0,0,0,0,11,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[1,11,0,0,0,0,13,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,12,1],[22,1,0,0,0,0,8,7,0,0,0,0,0,0,21,8,0,0,0,0,21,26,0,0,0,0,0,0,1,5,0,0,0,0,0,28],[22,1,0,0,0,0,8,7,0,0,0,0,0,0,21,26,0,0,0,0,21,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 4Q | 4R | 7A | 7B | 7C | 14A | ··· | 14I | 14J | 14K | 14L | 28A | ··· | 28R | 28S | ··· | 28AA |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | 2- (1+4) | D7×C4○D4 | D4.10D14 |
| kernel | C42.162D14 | C4×Dic14 | C42⋊D7 | C23.11D14 | C22⋊Dic14 | C23.D14 | Dic7⋊4D4 | D14.D4 | Dic7.Q8 | C4.Dic14 | D7×C4⋊C4 | C4⋊C4⋊7D7 | D14⋊Q8 | C7×C42⋊2C2 | C42⋊2C2 | Dic7 | D14 | C42 | C22⋊C4 | C4⋊C4 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 3 | 9 | 9 | 1 | 12 | 6 |
In GAP, Magma, Sage, TeX
C_4^2._{162}D_{14} % in TeX
G:=Group("C4^2.162D14"); // GroupNames label
G:=SmallGroup(448,1161);
// by ID
G=gap.SmallGroup(448,1161);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,268,1571,570,80,18822]);
// 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=b^-1,d*c*d^-1=c^13>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀