C4^2.236D14 - GroupNames

G = C₄².₂₃₆D₁₄ order 448 = 2⁶·7

56^th non-split extension by C₄² of D₁₄ acting via D₁₄/D₇=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄².₂₃₆D₁₄, (C₄×D₇)⋊₃Q₈, C₂₈⋊Q₈⋊₃₄C₂, C₄.₃₈(Q₈×D₇), D₁₄.₃(C₂×Q₈), C₂₈.₄₉(C₂×Q₈), C₄⋊C₄.₂₀₄D₁₄, C₄².C₂⋊₁₅D₇, (D₇×C₄²).₈C₂, (C₂×C₂₈).₈₆C₂³, D₁₄⋊Q₈.₁C₂, C₂₈.₆Q₈⋊₂₂C₂, Dic₇.₁₆(C₂×Q₈), Dic₇⋊₃Q₈⋊₃₄C₂, C₁₄.₄₁(C₂²×Q₈), (C₄×C₂₈).₁₉₂C₂², (C₂×C₁₄).₂₃₂C₂⁴, D₁₄⋊C₄.₃₈C₂², Dic₇.₁₂(C₄○D₄), Dic₇⋊C₄.₅₀C₂², C₄⋊Dic₇.₂₃₉C₂², C₂².₂₅₃(C₂³×D₇), C₇⋊₅(C₂³.₃₇C₂³), (C₂×Dic₇).₃₁₁C₂³, (C₄×Dic₇).₁₃₉C₂², (C₂²×D₇).₂₁₉C₂³, (C₂×Dic₁₄).₁₇₈C₂², C₂.₂₄(C₂×Q₈×D₇), C₂.₈₄(D₇×C₄○D₄), (C₇×C₄².C₂)⋊₅C₂, C₄⋊C₄⋊₇D₇.₁₁C₂, C₁₄.₁₉₅(C₂×C₄○D₄), (C₂×C₄×D₇).₂₄₉C₂², (C₂×C₄).₇₇(C₂²×D₇), (C₇×C₄⋊C₄).₁₈₇C₂², SmallGroup(448,1141)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — C₄².₂₃₆D₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×D₇ — C₂×C₄×D₇ — D₇×C₄² — C₄².₂₃₆D₁₄

Lower central

C₇ — C₂×C₁₄ — C₄².₂₃₆D₁₄

Upper central

C₁ — C₂² — C₄².C₂

Generators and relations for C₄².₂₃₆D₁₄
G = < a,b,c,d | a⁴=b⁴=1, c¹⁴=b², d²=a²b², ab=ba, cac^-1=dad^-1=a^-1, cbc^-1=dbd^-1=a²b^-1, dcd^-1=a²c¹³ >

Subgroups: 892 in 222 conjugacy classes, 107 normal (19 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, D₇, C₁₄, C₁₄, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×Q₈, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₄², C₄²⋊C₂, C₄×Q₈, C₂²⋊Q₈, C₄².C₂, C₄².C₂, C₄⋊Q₈, Dic₁₄, C₄×D₇, C₄×D₇, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₂²×D₇, C₂³.₃₇C₂³, C₄×Dic₇, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, D₁₄⋊C₄, C₄×C₂₈, C₇×C₄⋊C₄, C₂×Dic₁₄, C₂×C₄×D₇, C₂×C₄×D₇, C₂₈.₆Q₈, D₇×C₄², Dic₇⋊₃Q₈, C₂₈⋊Q₈, C₄⋊C₄⋊₇D₇, D₁₄⋊Q₈, C₇×C₄².C₂, C₄².₂₃₆D₁₄
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₇, C₂×Q₈, C₄○D₄, C₂⁴, D₁₄, C₂²×Q₈, C₂×C₄○D₄, C₂²×D₇, C₂³.₃₇C₂³, Q₈×D₇, C₂³×D₇, C₂×Q₈×D₇, D₇×C₄○D₄, C₄².₂₃₆D₁₄

Smallest permutation representation of C₄².₂₃₆D₁₄
►On 224 points

Generators in S₂₂₄

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

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

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

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

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

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

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	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+	-	+		+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	D₇	C₄○D₄	D₁₄	D₁₄	Q₈×D₇	D₇×C₄○D₄
kernel	C₄².₂₃₆D₁₄	C₂₈.₆Q₈	D₇×C₄²	Dic₇⋊₃Q₈	C₂₈⋊Q₈	C₄⋊C₄⋊₇D₇	D₁₄⋊Q₈	C₇×C₄².C₂	C₄×D₇	C₄².C₂	Dic₇	C₄²	C₄⋊C₄	C₄	C₂
# reps	1	1	1	4	2	2	4	1	4	3	8	3	18	6	12

Matrix representation of C₄².₂₃₆D₁₄ ►in GL₆(𝔽₂₉)

12	0	0	0	0	0
28	17	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

28	0	0	0	0	0
17	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	2	16
0	0	0	0	16	27

1	24	0	0	0	0
0	28	0	0	0	0
0	0	19	19	0	0
0	0	10	7	0	0
0	0	0	0	0	1
0	0	0	0	28	0

28	5	0	0	0	0
17	1	0	0	0	0
0	0	19	19	0	0
0	0	7	10	0	0
0	0	0	0	0	28
0	0	0	0	1	0

G:=sub<GL(6,GF(29))| [12,28,0,0,0,0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,17,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,16,0,0,0,0,16,27],[1,0,0,0,0,0,24,28,0,0,0,0,0,0,19,10,0,0,0,0,19,7,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[28,17,0,0,0,0,5,1,0,0,0,0,0,0,19,7,0,0,0,0,19,10,0,0,0,0,0,0,0,1,0,0,0,0,28,0] >;

C₄².₂₃₆D₁₄ in GAP, Magma, Sage, TeX

C_4^2._{236}D_{14}

% in TeX

G:=Group("C4^2.236D14");

// GroupNames label

G:=SmallGroup(448,1141);

// by ID

G=gap.SmallGroup(448,1141);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,100,1123,570,409,80,18822]);

// 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=a^2*b^-1,d*c*d^-1=a^2*c^13>;

// generators/relations

G = C42.236D14 order 448 = 26·7

56th non-split extension by C42 of D14 acting via D14/D7=C2

G = C₄².₂₃₆D₁₄ order 448 = 2⁶·7

56^th non-split extension by C₄² of D₁₄ acting via D₁₄/D₇=C₂