C4^2.160D14 - GroupNames

G = C₄².₁₆₀D₁₄ order 448 = 2⁶·7

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄².₁₆₀D₁₄, C₁₄.₉₉2_-¹⁺⁴, C₂₈⋊Q₈⋊₄₀C₂, C₄⋊C₄.₁₁₇D₁₄, C₄²⋊₂C₂⋊₁D₇, C₄²⋊D₇⋊₅C₂, D₁₄⋊Q₈⋊₃₉C₂, (C₄×Dic₁₄)⋊₁₄C₂, (C₄×C₂₈).₃₂C₂², C₂²⋊C₄.₇₅D₁₄, Dic₇⋊₃Q₈⋊₄₀C₂, (C₂×C₁₄).₂₄₆C₂⁴, (C₂×C₂₈).₁₉₂C₂³, Dic₇⋊₄D₄.₄C₂, C₂³.₅₂(C₂²×D₇), D₁₄⋊C₄.₁₃₉C₂², Dic₇.₃₁(C₄○D₄), C₂²⋊Dic₁₄⋊₄₄C₂, C₂³.D₁₄⋊₄₃C₂, C₄⋊Dic₇.₃₁₇C₂², (C₂²×C₁₄).₆₀C₂³, Dic₇.D₄.₄C₂, C₂².₂₆₇(C₂³×D₇), C₂³.D₇.₆₂C₂², C₂³.₁₁D₁₄⋊₂₀C₂, Dic₇⋊C₄.₁₄₅C₂², C₇⋊₇(C₂².₅₀C₂⁴), (C₄×Dic₇).₁₄₉C₂², (C₂×Dic₇).₂₆₃C₂³, (C₂²×D₇).₁₁₀C₂³, C₂.₆₃(D₄.₁₀D₁₄), (C₂×Dic₁₄).₂₅₄C₂², (C₂²×Dic₇).₁₄₉C₂², C₂.₉₃(D₇×C₄○D₄), C₄⋊C₄⋊D₇⋊₃₉C₂, (C₇×C₄²⋊₂C₂)⋊₁C₂, C₁₄.₂₀₄(C₂×C₄○D₄), (C₂×C₄×D₇).₂₁₈C₂², (C₂×C₄).₈₃(C₂²×D₇), (C₇×C₄⋊C₄).₂₀₁C₂², (C₂×C₇⋊D₄).₆₇C₂², (C₇×C₂²⋊C₄).₇₁C₂², SmallGroup(448,1155)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — C₄².₁₆₀D₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×Dic₇ — C₂²×Dic₇ — C₂³.₁₁D₁₄ — C₄².₁₆₀D₁₄

Lower central

C₇ — C₂×C₁₄ — C₄².₁₆₀D₁₄

Upper central

C₁ — C₂² — C₄²⋊₂C₂

Generators and relations for C₄².₁₆₀D₁₄
G = < a,b,c,d | a⁴=b⁴=c¹⁴=1, d²=b², ab=ba, cac^-1=ab², ad=da, cbc^-1=dbd^-1=a²b^-1, dcd^-1=c^-1 >

Subgroups: 876 in 212 conjugacy classes, 95 normal (91 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, D₇, C₁₄, C₁₄, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, Dic₇, Dic₇, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²⋊Q₈, C₄.₄D₄, C₄²⋊₂C₂, C₄²⋊₂C₂, C₄⋊Q₈, Dic₁₄, C₄×D₇, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂²×D₇, C₂²×C₁₄, C₂².₅₀C₂⁴, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, D₁₄⋊C₄, C₂³.D₇, C₄×C₂₈, C₇×C₂²⋊C₄, C₇×C₄⋊C₄, C₂×Dic₁₄, C₂×C₄×D₇, C₂²×Dic₇, C₂×C₇⋊D₄, C₄×Dic₁₄, C₄²⋊D₇, C₂³.₁₁D₁₄, C₂²⋊Dic₁₄, C₂³.D₁₄, Dic₇⋊₄D₄, Dic₇.D₄, Dic₇⋊₃Q₈, C₂₈⋊Q₈, D₁₄⋊Q₈, C₄⋊C₄⋊D₇, C₇×C₄²⋊₂C₂, C₄².₁₆₀D₁₄
Quotients: C₁, C₂, C₂², C₂³, D₇, C₄○D₄, C₂⁴, D₁₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂²×D₇, C₂².₅₀C₂⁴, C₂³×D₇, D₇×C₄○D₄, D₄.₁₀D₁₄, C₄².₁₆₀D₁₄

Smallest permutation representation of C₄².₁₆₀D₁₄
►On 224 points

Generators in S₂₂₄

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

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

(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)

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

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

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

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

67 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	···	4P	4Q	4R	4S	7A	7B	7C	14A	···	14I	14J	14K	14L	28A	···	28R	28S	···	28AA
order	1	2	2	2	2	2	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	28	2	2	2	2	4	4	4	4	14	···	14	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	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+		+	+	+	-		-
image	C₁	C₂	C₂	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	C₄².₁₆₀D₁₄	C₄×Dic₁₄	C₄²⋊D₇	C₂³.₁₁D₁₄	C₂²⋊Dic₁₄	C₂³.D₁₄	Dic₇⋊₄D₄	Dic₇.D₄	Dic₇⋊₃Q₈	C₂₈⋊Q₈	D₁₄⋊Q₈	C₄⋊C₄⋊D₇	C₇×C₄²⋊₂C₂	C₄²⋊₂C₂	Dic₇	C₄²	C₂²⋊C₄	C₄⋊C₄	C₁₄	C₂	C₂
# reps	1	1	1	1	1	1	1	2	2	1	1	2	1	3	8	3	9	9	1	12	6

Matrix representation of C₄².₁₆₀D₁₄ ►in GL₆(𝔽₂₉)

0	28	0	0	0	0
1	0	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	17

17	0	0	0	0	0
0	17	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	1	0
0	0	0	0	15	28

28	0	0	0	0	0
0	1	0	0	0	0
0	0	22	10	0	0
0	0	19	10	0	0
0	0	0	0	6	5
0	0	0	0	22	23

17	0	0	0	0	0
0	17	0	0	0	0
0	0	0	28	0	0
0	0	28	0	0	0
0	0	0	0	6	5
0	0	0	0	22	23

G:=sub<GL(6,GF(29))| [0,1,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,15,0,0,0,0,0,28],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,22,19,0,0,0,0,10,10,0,0,0,0,0,0,6,22,0,0,0,0,5,23],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,6,22,0,0,0,0,5,23] >;

C₄².₁₆₀D₁₄ in GAP, Magma, Sage, TeX

C_4^2._{160}D_{14}

% in TeX

G:=Group("C4^2.160D14");

// GroupNames label

G:=SmallGroup(448,1155);

// by ID

G=gap.SmallGroup(448,1155);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,387,100,794,136,18822]);

// 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=a*b^2,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;

// generators/relations

G = C42.160D14 order 448 = 26·7

160th non-split extension by C42 of D14 acting via D14/C7=C22

G = C₄².₁₆₀D₁₄ order 448 = 2⁶·7

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