C4^2.162D14 - GroupNames

G = C₄².₁₆₂D₁₄ order 448 = 2⁶·7

162^nd non-split extension by C₄² of D₁₄ acting via D₁₄/C₇=C₂²

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, C₂₈.₃Q₈⋊₄₀C₂, D₁₄.₂₇(C₄○D₄), (C₂×C₂₈).₆₀₄C₂³, (C₂×C₁₄).₂₅₂C₂⁴, 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

Derived series

C₁ — C₂×C₁₄ — C₄².₁₆₂D₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×D₇ — C₂×C₄×D₇ — D₇×C₄⋊C₄ — C₄².₁₆₂D₁₄

Lower central

C₇ — C₂×C₁₄ — C₄².₁₆₂D₁₄

Upper central

C₁ — C₂² — C₄²⋊₂C₂

Generators and relations for C₄².₁₆₂D₁₄
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 >

Subgroups: 876 in 214 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, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²⋊Q₈, C₂².D₄, C₄².C₂, C₄²⋊₂C₂, C₄²⋊₂C₂, Dic₁₄, C₄×D₇, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂²×D₇, C₂²×C₁₄, C₂².₄₆C₂⁴, 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, D₁₄.D₄, Dic₇.Q₈, C₂₈.₃Q₈, D₇×C₄⋊C₄, C₄⋊C₄⋊₇D₇, D₁₄⋊Q₈, C₇×C₄²⋊₂C₂, C₄².₁₆₂D₁₄
Quotients: C₁, C₂, C₂², C₂³, D₇, C₄○D₄, C₂⁴, D₁₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂²×D₇, C₂².₄₆C₂⁴, C₂³×D₇, D₇×C₄○D₄, D₄.₁₀D₁₄, C₄².₁₆₂D₁₄

Smallest permutation representation of C₄².₁₆₂D₁₄
►On 224 points

Generators in S₂₂₄

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

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

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

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

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

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

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	C₂₈.₃Q₈	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

Matrix representation of C₄².₁₆₂D₁₄ ►in 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] >;

C₄².₁₆₂D₁₄ 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

G = C42.162D14 order 448 = 26·7

162nd non-split extension by C42 of D14 acting via D14/C7=C22

G = C₄².₁₆₂D₁₄ order 448 = 2⁶·7

162^nd non-split extension by C₄² of D₁₄ acting via D₁₄/C₇=C₂²