C4^2.241D14 - GroupNames

G = C₄².₂₄₁D₁₄ order 448 = 2⁶·7

61^st non-split extension by C₄² of D₁₄ acting via D₁₄/D₇=C₂

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, C₂₈.₃Q₈⋊₄₂C₂, Dic₇.₁₇(C₂×Q₈), (D₇×C₄²).₁₀C₂, Dic₇⋊₃Q₈⋊₄₂C₂, C₂₈.₁₃₆(C₄○D₄), C₄.₄₁(D₄⋊₂D₇), C₁₄.₄₈(C₂²×Q₈), (C₂×C₁₄).₂₇₂C₂⁴, (C₂×C₂₈).₁₀₅C₂³, (C₄×C₂₈).₂₁₃C₂², D₁₄⋊C₄.₅₁C₂², D₁₄⋊₃Q₈.₁₁C₂, D₁₄⋊₂Q₈.₁₃C₂, 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

Derived series

C₁ — C₂×C₁₄ — C₄².₂₄₁D₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×D₇ — C₂×C₄×D₇ — D₇×C₄² — C₄².₂₄₁D₁₄

Lower central

C₇ — C₂×C₁₄ — C₄².₂₄₁D₁₄

Upper central

C₁ — C₂² — C₄⋊Q₈

Generators and relations for C₄².₂₄₁D₁₄
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 >

Subgroups: 844 in 222 conjugacy classes, 111 normal (33 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, D₇, C₁₄, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₄², C₄²⋊C₂, C₄×Q₈, C₂²⋊Q₈, C₄².C₂, C₄⋊Q₈, C₄⋊Q₈, Dic₁₄, C₄×D₇, C₄×D₇, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, C₂²×D₇, C₂³.₃₇C₂³, C₄×Dic₇, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, D₁₄⋊C₄, C₄×C₂₈, C₇×C₄⋊C₄, C₂×Dic₁₄, C₂×C₄×D₇, Q₈×C₁₄, C₂₈⋊₂Q₈, D₇×C₄², Dic₇⋊₃Q₈, C₂₈.₃Q₈, C₄⋊C₄⋊₇D₇, D₁₄⋊₂Q₈, Q₈×Dic₇, D₁₄⋊₃Q₈, C₇×C₄⋊Q₈, C₄².₂₄₁D₁₄
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₇, C₂×Q₈, C₄○D₄, C₂⁴, D₁₄, C₂²×Q₈, C₂×C₄○D₄, C₂²×D₇, C₂³.₃₇C₂³, D₄⋊₂D₇, Q₈×D₇, Q₈⋊₂D₇, C₂³×D₇, C₂×D₄⋊₂D₇, C₂×Q₈×D₇, C₂×Q₈⋊₂D₇, C₄².₂₄₁D₁₄

Smallest permutation representation of C₄².₂₄₁D₁₄
►On 224 points

Generators in S₂₂₄

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

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

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

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

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

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

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	C₂₈.₃Q₈	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

Matrix representation of C₄².₂₄₁D₁₄ ►in 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] >;

C₄².₂₄₁D₁₄ 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

G = C42.241D14 order 448 = 26·7

61st non-split extension by C42 of D14 acting via D14/D7=C2

G = C₄².₂₄₁D₁₄ order 448 = 2⁶·7

61^st non-split extension by C₄² of D₁₄ acting via D₁₄/D₇=C₂