C4^2.159D14 - GroupNames

G = C₄².₁₅₉D₁₄ order 448 = 2⁶·7

159^th non-split extension by C₄² of D₁₄ acting via D₁₄/C₇=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄².₁₅₉D₁₄, C₁₄.₉₈2_-¹⁺⁴, C₂₈⋊Q₈⋊₃₉C₂, C₄²⋊₂C₂.D₇, C₄⋊C₄.₁₁₆D₁₄, C₂₈.₆Q₈⋊₈C₂, (C₄×Dic₁₄)⋊₁₃C₂, Dic₇.Q₈⋊₃₆C₂, (C₂×C₂₈).₉₃C₂³, (C₄×C₂₈).₃₁C₂², C₂²⋊C₄.₃₉D₁₄, C₂₈.₃Q₈⋊₃₈C₂, Dic₇⋊₃Q₈⋊₃₉C₂, (C₂×C₁₄).₂₄₅C₂⁴, C₄⋊Dic₇.₅₃C₂², C₂³.₅₁(C₂²×D₇), Dic₇.₁₄(C₄○D₄), (C₂²×C₁₄).₅₉C₂³, C₂²⋊Dic₁₄.₄C₂, C₂³.D₁₄.₃C₂, C₂².₂₆₆(C₂³×D₇), C₂³.D₇.₆₁C₂², Dic₇⋊C₄.₁₂₆C₂², C₇⋊₆(C₂².₃₅C₂⁴), (C₄×Dic₇).₂₁₇C₂², (C₂×Dic₇).₁₂₇C₂³, C₂³.₁₁D₁₄.₃C₂, C₂.₆₂(D₄.₁₀D₁₄), (C₂×Dic₁₄).₂₅₃C₂², (C₂²×Dic₇).₁₄₈C₂², C₂.₉₂(D₇×C₄○D₄), C₁₄.₂₀₃(C₂×C₄○D₄), (C₇×C₄⋊C₄).₂₀₀C₂², (C₇×C₄²⋊₂C₂).₁C₂, (C₂×C₄).₃₀₂(C₂²×D₇), (C₇×C₂²⋊C₄).₇₀C₂², SmallGroup(448,1154)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — C₄².₁₅₉D₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×Dic₇ — C₄×Dic₇ — Dic₇⋊₃Q₈ — C₄².₁₅₉D₁₄

Lower central

C₇ — C₂×C₁₄ — C₄².₁₅₉D₁₄

Upper central

C₁ — C₂² — C₄²⋊₂C₂

Generators and relations for C₄².₁₅₉D₁₄
G = < a,b,c,d | a⁴=b⁴=1, c¹⁴=b², d²=a², ab=ba, cac^-1=dad^-1=ab², cbc^-1=a²b^-1, dbd^-1=a²b, dcd^-1=c¹³ >

Subgroups: 716 in 192 conjugacy classes, 93 normal (91 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₄, C₁₄, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×Q₈, Dic₇, Dic₇, C₂₈, C₂×C₁₄, C₂×C₁₄, C₄²⋊C₂, C₄×Q₈, C₂²⋊Q₈, C₄².C₂, C₄²⋊₂C₂, C₄²⋊₂C₂, C₄⋊Q₈, Dic₁₄, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂²×C₁₄, C₂².₃₅C₂⁴, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, C₂³.D₇, C₄×C₂₈, C₇×C₂²⋊C₄, C₇×C₄⋊C₄, C₂×Dic₁₄, C₂²×Dic₇, C₄×Dic₁₄, C₂₈.₆Q₈, C₂³.₁₁D₁₄, C₂²⋊Dic₁₄, C₂³.D₁₄, Dic₇⋊₃Q₈, C₂₈⋊Q₈, Dic₇.Q₈, C₂₈.₃Q₈, C₇×C₄²⋊₂C₂, C₄².₁₅₉D₁₄
Quotients: C₁, C₂, C₂², C₂³, D₇, C₄○D₄, C₂⁴, D₁₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂²×D₇, C₂².₃₅C₂⁴, C₂³×D₇, D₇×C₄○D₄, D₄.₁₀D₁₄, C₄².₁₅₉D₁₄

Smallest permutation representation of C₄².₁₅₉D₁₄
►On 224 points

Generators in S₂₂₄

(1 181 220 160)(2 196 221 147)(3 183 222 162)(4 170 223 149)(5 185 224 164)(6 172 197 151)(7 187 198 166)(8 174 199 153)(9 189 200 168)(10 176 201 155)(11 191 202 142)(12 178 203 157)(13 193 204 144)(14 180 205 159)(15 195 206 146)(16 182 207 161)(17 169 208 148)(18 184 209 163)(19 171 210 150)(20 186 211 165)(21 173 212 152)(22 188 213 167)(23 175 214 154)(24 190 215 141)(25 177 216 156)(26 192 217 143)(27 179 218 158)(28 194 219 145)(29 57 135 89)(30 72 136 104)(31 59 137 91)(32 74 138 106)(33 61 139 93)(34 76 140 108)(35 63 113 95)(36 78 114 110)(37 65 115 97)(38 80 116 112)(39 67 117 99)(40 82 118 86)(41 69 119 101)(42 84 120 88)(43 71 121 103)(44 58 122 90)(45 73 123 105)(46 60 124 92)(47 75 125 107)(48 62 126 94)(49 77 127 109)(50 64 128 96)(51 79 129 111)(52 66 130 98)(53 81 131 85)(54 68 132 100)(55 83 133 87)(56 70 134 102)

(1 174 15 188)(2 168 16 154)(3 176 17 190)(4 142 18 156)(5 178 19 192)(6 144 20 158)(7 180 21 194)(8 146 22 160)(9 182 23 196)(10 148 24 162)(11 184 25 170)(12 150 26 164)(13 186 27 172)(14 152 28 166)(29 110 43 96)(30 65 44 79)(31 112 45 98)(32 67 46 81)(33 86 47 100)(34 69 48 83)(35 88 49 102)(36 71 50 57)(37 90 51 104)(38 73 52 59)(39 92 53 106)(40 75 54 61)(41 94 55 108)(42 77 56 63)(58 129 72 115)(60 131 74 117)(62 133 76 119)(64 135 78 121)(66 137 80 123)(68 139 82 125)(70 113 84 127)(85 138 99 124)(87 140 101 126)(89 114 103 128)(91 116 105 130)(93 118 107 132)(95 120 109 134)(97 122 111 136)(141 222 155 208)(143 224 157 210)(145 198 159 212)(147 200 161 214)(149 202 163 216)(151 204 165 218)(153 206 167 220)(169 215 183 201)(171 217 185 203)(173 219 187 205)(175 221 189 207)(177 223 191 209)(179 197 193 211)(181 199 195 213)

(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 71 220 103)(2 84 221 88)(3 69 222 101)(4 82 223 86)(5 67 224 99)(6 80 197 112)(7 65 198 97)(8 78 199 110)(9 63 200 95)(10 76 201 108)(11 61 202 93)(12 74 203 106)(13 59 204 91)(14 72 205 104)(15 57 206 89)(16 70 207 102)(17 83 208 87)(18 68 209 100)(19 81 210 85)(20 66 211 98)(21 79 212 111)(22 64 213 96)(23 77 214 109)(24 62 215 94)(25 75 216 107)(26 60 217 92)(27 73 218 105)(28 58 219 90)(29 181 135 160)(30 194 136 145)(31 179 137 158)(32 192 138 143)(33 177 139 156)(34 190 140 141)(35 175 113 154)(36 188 114 167)(37 173 115 152)(38 186 116 165)(39 171 117 150)(40 184 118 163)(41 169 119 148)(42 182 120 161)(43 195 121 146)(44 180 122 159)(45 193 123 144)(46 178 124 157)(47 191 125 142)(48 176 126 155)(49 189 127 168)(50 174 128 153)(51 187 129 166)(52 172 130 151)(53 185 131 164)(54 170 132 149)(55 183 133 162)(56 196 134 147)

G:=sub<Sym(224)| (1,181,220,160)(2,196,221,147)(3,183,222,162)(4,170,223,149)(5,185,224,164)(6,172,197,151)(7,187,198,166)(8,174,199,153)(9,189,200,168)(10,176,201,155)(11,191,202,142)(12,178,203,157)(13,193,204,144)(14,180,205,159)(15,195,206,146)(16,182,207,161)(17,169,208,148)(18,184,209,163)(19,171,210,150)(20,186,211,165)(21,173,212,152)(22,188,213,167)(23,175,214,154)(24,190,215,141)(25,177,216,156)(26,192,217,143)(27,179,218,158)(28,194,219,145)(29,57,135,89)(30,72,136,104)(31,59,137,91)(32,74,138,106)(33,61,139,93)(34,76,140,108)(35,63,113,95)(36,78,114,110)(37,65,115,97)(38,80,116,112)(39,67,117,99)(40,82,118,86)(41,69,119,101)(42,84,120,88)(43,71,121,103)(44,58,122,90)(45,73,123,105)(46,60,124,92)(47,75,125,107)(48,62,126,94)(49,77,127,109)(50,64,128,96)(51,79,129,111)(52,66,130,98)(53,81,131,85)(54,68,132,100)(55,83,133,87)(56,70,134,102), (1,174,15,188)(2,168,16,154)(3,176,17,190)(4,142,18,156)(5,178,19,192)(6,144,20,158)(7,180,21,194)(8,146,22,160)(9,182,23,196)(10,148,24,162)(11,184,25,170)(12,150,26,164)(13,186,27,172)(14,152,28,166)(29,110,43,96)(30,65,44,79)(31,112,45,98)(32,67,46,81)(33,86,47,100)(34,69,48,83)(35,88,49,102)(36,71,50,57)(37,90,51,104)(38,73,52,59)(39,92,53,106)(40,75,54,61)(41,94,55,108)(42,77,56,63)(58,129,72,115)(60,131,74,117)(62,133,76,119)(64,135,78,121)(66,137,80,123)(68,139,82,125)(70,113,84,127)(85,138,99,124)(87,140,101,126)(89,114,103,128)(91,116,105,130)(93,118,107,132)(95,120,109,134)(97,122,111,136)(141,222,155,208)(143,224,157,210)(145,198,159,212)(147,200,161,214)(149,202,163,216)(151,204,165,218)(153,206,167,220)(169,215,183,201)(171,217,185,203)(173,219,187,205)(175,221,189,207)(177,223,191,209)(179,197,193,211)(181,199,195,213), (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), (1,71,220,103)(2,84,221,88)(3,69,222,101)(4,82,223,86)(5,67,224,99)(6,80,197,112)(7,65,198,97)(8,78,199,110)(9,63,200,95)(10,76,201,108)(11,61,202,93)(12,74,203,106)(13,59,204,91)(14,72,205,104)(15,57,206,89)(16,70,207,102)(17,83,208,87)(18,68,209,100)(19,81,210,85)(20,66,211,98)(21,79,212,111)(22,64,213,96)(23,77,214,109)(24,62,215,94)(25,75,216,107)(26,60,217,92)(27,73,218,105)(28,58,219,90)(29,181,135,160)(30,194,136,145)(31,179,137,158)(32,192,138,143)(33,177,139,156)(34,190,140,141)(35,175,113,154)(36,188,114,167)(37,173,115,152)(38,186,116,165)(39,171,117,150)(40,184,118,163)(41,169,119,148)(42,182,120,161)(43,195,121,146)(44,180,122,159)(45,193,123,144)(46,178,124,157)(47,191,125,142)(48,176,126,155)(49,189,127,168)(50,174,128,153)(51,187,129,166)(52,172,130,151)(53,185,131,164)(54,170,132,149)(55,183,133,162)(56,196,134,147)>;

G:=Group( (1,181,220,160)(2,196,221,147)(3,183,222,162)(4,170,223,149)(5,185,224,164)(6,172,197,151)(7,187,198,166)(8,174,199,153)(9,189,200,168)(10,176,201,155)(11,191,202,142)(12,178,203,157)(13,193,204,144)(14,180,205,159)(15,195,206,146)(16,182,207,161)(17,169,208,148)(18,184,209,163)(19,171,210,150)(20,186,211,165)(21,173,212,152)(22,188,213,167)(23,175,214,154)(24,190,215,141)(25,177,216,156)(26,192,217,143)(27,179,218,158)(28,194,219,145)(29,57,135,89)(30,72,136,104)(31,59,137,91)(32,74,138,106)(33,61,139,93)(34,76,140,108)(35,63,113,95)(36,78,114,110)(37,65,115,97)(38,80,116,112)(39,67,117,99)(40,82,118,86)(41,69,119,101)(42,84,120,88)(43,71,121,103)(44,58,122,90)(45,73,123,105)(46,60,124,92)(47,75,125,107)(48,62,126,94)(49,77,127,109)(50,64,128,96)(51,79,129,111)(52,66,130,98)(53,81,131,85)(54,68,132,100)(55,83,133,87)(56,70,134,102), (1,174,15,188)(2,168,16,154)(3,176,17,190)(4,142,18,156)(5,178,19,192)(6,144,20,158)(7,180,21,194)(8,146,22,160)(9,182,23,196)(10,148,24,162)(11,184,25,170)(12,150,26,164)(13,186,27,172)(14,152,28,166)(29,110,43,96)(30,65,44,79)(31,112,45,98)(32,67,46,81)(33,86,47,100)(34,69,48,83)(35,88,49,102)(36,71,50,57)(37,90,51,104)(38,73,52,59)(39,92,53,106)(40,75,54,61)(41,94,55,108)(42,77,56,63)(58,129,72,115)(60,131,74,117)(62,133,76,119)(64,135,78,121)(66,137,80,123)(68,139,82,125)(70,113,84,127)(85,138,99,124)(87,140,101,126)(89,114,103,128)(91,116,105,130)(93,118,107,132)(95,120,109,134)(97,122,111,136)(141,222,155,208)(143,224,157,210)(145,198,159,212)(147,200,161,214)(149,202,163,216)(151,204,165,218)(153,206,167,220)(169,215,183,201)(171,217,185,203)(173,219,187,205)(175,221,189,207)(177,223,191,209)(179,197,193,211)(181,199,195,213), (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), (1,71,220,103)(2,84,221,88)(3,69,222,101)(4,82,223,86)(5,67,224,99)(6,80,197,112)(7,65,198,97)(8,78,199,110)(9,63,200,95)(10,76,201,108)(11,61,202,93)(12,74,203,106)(13,59,204,91)(14,72,205,104)(15,57,206,89)(16,70,207,102)(17,83,208,87)(18,68,209,100)(19,81,210,85)(20,66,211,98)(21,79,212,111)(22,64,213,96)(23,77,214,109)(24,62,215,94)(25,75,216,107)(26,60,217,92)(27,73,218,105)(28,58,219,90)(29,181,135,160)(30,194,136,145)(31,179,137,158)(32,192,138,143)(33,177,139,156)(34,190,140,141)(35,175,113,154)(36,188,114,167)(37,173,115,152)(38,186,116,165)(39,171,117,150)(40,184,118,163)(41,169,119,148)(42,182,120,161)(43,195,121,146)(44,180,122,159)(45,193,123,144)(46,178,124,157)(47,191,125,142)(48,176,126,155)(49,189,127,168)(50,174,128,153)(51,187,129,166)(52,172,130,151)(53,185,131,164)(54,170,132,149)(55,183,133,162)(56,196,134,147) );

G=PermutationGroup([[(1,181,220,160),(2,196,221,147),(3,183,222,162),(4,170,223,149),(5,185,224,164),(6,172,197,151),(7,187,198,166),(8,174,199,153),(9,189,200,168),(10,176,201,155),(11,191,202,142),(12,178,203,157),(13,193,204,144),(14,180,205,159),(15,195,206,146),(16,182,207,161),(17,169,208,148),(18,184,209,163),(19,171,210,150),(20,186,211,165),(21,173,212,152),(22,188,213,167),(23,175,214,154),(24,190,215,141),(25,177,216,156),(26,192,217,143),(27,179,218,158),(28,194,219,145),(29,57,135,89),(30,72,136,104),(31,59,137,91),(32,74,138,106),(33,61,139,93),(34,76,140,108),(35,63,113,95),(36,78,114,110),(37,65,115,97),(38,80,116,112),(39,67,117,99),(40,82,118,86),(41,69,119,101),(42,84,120,88),(43,71,121,103),(44,58,122,90),(45,73,123,105),(46,60,124,92),(47,75,125,107),(48,62,126,94),(49,77,127,109),(50,64,128,96),(51,79,129,111),(52,66,130,98),(53,81,131,85),(54,68,132,100),(55,83,133,87),(56,70,134,102)], [(1,174,15,188),(2,168,16,154),(3,176,17,190),(4,142,18,156),(5,178,19,192),(6,144,20,158),(7,180,21,194),(8,146,22,160),(9,182,23,196),(10,148,24,162),(11,184,25,170),(12,150,26,164),(13,186,27,172),(14,152,28,166),(29,110,43,96),(30,65,44,79),(31,112,45,98),(32,67,46,81),(33,86,47,100),(34,69,48,83),(35,88,49,102),(36,71,50,57),(37,90,51,104),(38,73,52,59),(39,92,53,106),(40,75,54,61),(41,94,55,108),(42,77,56,63),(58,129,72,115),(60,131,74,117),(62,133,76,119),(64,135,78,121),(66,137,80,123),(68,139,82,125),(70,113,84,127),(85,138,99,124),(87,140,101,126),(89,114,103,128),(91,116,105,130),(93,118,107,132),(95,120,109,134),(97,122,111,136),(141,222,155,208),(143,224,157,210),(145,198,159,212),(147,200,161,214),(149,202,163,216),(151,204,165,218),(153,206,167,220),(169,215,183,201),(171,217,185,203),(173,219,187,205),(175,221,189,207),(177,223,191,209),(179,197,193,211),(181,199,195,213)], [(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)], [(1,71,220,103),(2,84,221,88),(3,69,222,101),(4,82,223,86),(5,67,224,99),(6,80,197,112),(7,65,198,97),(8,78,199,110),(9,63,200,95),(10,76,201,108),(11,61,202,93),(12,74,203,106),(13,59,204,91),(14,72,205,104),(15,57,206,89),(16,70,207,102),(17,83,208,87),(18,68,209,100),(19,81,210,85),(20,66,211,98),(21,79,212,111),(22,64,213,96),(23,77,214,109),(24,62,215,94),(25,75,216,107),(26,60,217,92),(27,73,218,105),(28,58,219,90),(29,181,135,160),(30,194,136,145),(31,179,137,158),(32,192,138,143),(33,177,139,156),(34,190,140,141),(35,175,113,154),(36,188,114,167),(37,173,115,152),(38,186,116,165),(39,171,117,150),(40,184,118,163),(41,169,119,148),(42,182,120,161),(43,195,121,146),(44,180,122,159),(45,193,123,144),(46,178,124,157),(47,191,125,142),(48,176,126,155),(49,189,127,168),(50,174,128,153),(51,187,129,166),(52,172,130,151),(53,185,131,164),(54,170,132,149),(55,183,133,162),(56,196,134,147)]])

64 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	···	4G	4H	4I	4J	4K	4L	···	4Q	7A	7B	7C	14A	···	14I	14J	14K	14L	28A	···	28R	28S	···	28AA
order	1	2	2	2	2	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	2	2	4	···	4	14	14	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	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+		+	+	+	-		-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₇	C₄○D₄	D₁₄	D₁₄	D₁₄	2_-¹⁺⁴	D₇×C₄○D₄	D₄.₁₀D₁₄
kernel	C₄².₁₅₉D₁₄	C₄×Dic₁₄	C₂₈.₆Q₈	C₂³.₁₁D₁₄	C₂²⋊Dic₁₄	C₂³.D₁₄	Dic₇⋊₃Q₈	C₂₈⋊Q₈	Dic₇.Q₈	C₂₈.₃Q₈	C₇×C₄²⋊₂C₂	C₄²⋊₂C₂	Dic₇	C₄²	C₂²⋊C₄	C₄⋊C₄	C₁₄	C₂	C₂
# reps	1	1	1	1	2	3	1	1	3	1	1	3	4	3	9	9	2	6	12

Matrix representation of C₄².₁₅₉D₁₄ ►in GL₆(𝔽₂₉)

12	0	0	0	0	0
0	12	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	28	0	0	0
0	0	0	28	0	0

25	16	0	0	0	0
19	4	0	0	0	0
0	0	27	11	0	0
0	0	18	2	0	0
0	0	0	0	27	11
0	0	0	0	18	2

10	18	0	0	0	0
9	19	0	0	0	0
0	0	0	0	12	17
0	0	0	0	12	5
0	0	12	17	0	0
0	0	12	5	0	0

25	16	0	0	0	0
8	4	0	0	0	0
0	0	7	17	0	0
0	0	9	22	0	0
0	0	0	0	22	12
0	0	0	0	20	7

G:=sub<GL(6,GF(29))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,0,1,0,0],[25,19,0,0,0,0,16,4,0,0,0,0,0,0,27,18,0,0,0,0,11,2,0,0,0,0,0,0,27,18,0,0,0,0,11,2],[10,9,0,0,0,0,18,19,0,0,0,0,0,0,0,0,12,12,0,0,0,0,17,5,0,0,12,12,0,0,0,0,17,5,0,0],[25,8,0,0,0,0,16,4,0,0,0,0,0,0,7,9,0,0,0,0,17,22,0,0,0,0,0,0,22,20,0,0,0,0,12,7] >;

C₄².₁₅₉D₁₄ in GAP, Magma, Sage, TeX

C_4^2._{159}D_{14}

% in TeX

G:=Group("C4^2.159D14");

// GroupNames label

G:=SmallGroup(448,1154);

// by ID

G=gap.SmallGroup(448,1154);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,219,268,675,570,80,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^2,d^2=a^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

G = C42.159D14 order 448 = 26·7

159th non-split extension by C42 of D14 acting via D14/C7=C22

G = C₄².₁₅₉D₁₄ order 448 = 2⁶·7

159^th non-split extension by C₄² of D₁₄ acting via D₁₄/C₇=C₂²