C4^2.151D14 - GroupNames

G = C₄².₁₅₁D₁₄ order 448 = 2⁶·7

151^st non-split extension by C₄² of D₁₄ acting via D₁₄/C₇=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄².₁₅₁D₁₄, C₁₄.₂₉2_-¹⁺⁴, C₄².C₂⋊₇D₇, C₄⋊C₄.₁₁₂D₁₄, D₁₄⋊Q₈⋊₃₆C₂, C₄²⋊D₇⋊₃₇C₂, Dic₇.Q₈⋊₃₄C₂, (C₂×C₂₈).₈₉C₂³, Dic₇⋊₃Q₈⋊₃₆C₂, D₁₄.₂₄(C₄○D₄), D₂₈⋊C₄.₁₂C₂, (C₄×C₂₈).₂₄₀C₂², (C₂×C₁₄).₂₃₇C₂⁴, D₁₄.₅D₄.₂C₂, D₁₄⋊C₄.₁₃₇C₂², Dic₇.₃₀(C₄○D₄), (C₂×D₂₈).₁₆₅C₂², Dic₇⋊C₄.₅₃C₂², C₄⋊Dic₇.₂₄₂C₂², C₂².₂₅₈(C₂³×D₇), C₇⋊₉(C₂².₄₆C₂⁴), (C₂×Dic₇).₂₅₉C₂³, (C₄×Dic₇).₂₁₅C₂², (C₂²×D₇).₂₂₂C₂³, C₂.₃₀(Q₈.₁₀D₁₄), (C₂×Dic₁₄).₁₈₁C₂², (D₇×C₄⋊C₄)⋊₃₇C₂, C₂.₈₈(D₇×C₄○D₄), C₄⋊C₄⋊₇D₇⋊₃₆C₂, C₄⋊C₄⋊D₇⋊₃₅C₂, C₁₄.₁₉₉(C₂×C₄○D₄), (C₇×C₄².C₂)⋊₁₀C₂, (C₂×C₄×D₇).₁₂₇C₂², (C₇×C₄⋊C₄).₁₉₂C₂², (C₂×C₄).₂₀₄(C₂²×D₇), SmallGroup(448,1146)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — C₄².₁₅₁D₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×D₇ — C₂×C₄×D₇ — C₄²⋊D₇ — C₄².₁₅₁D₁₄

Lower central

C₇ — C₂×C₁₄ — C₄².₁₅₁D₁₄

Upper central

C₁ — C₂² — C₄².C₂

Generators and relations for C₄².₁₅₁D₁₄
G = < a,b,c,d | a⁴=b⁴=1, c¹⁴=d²=a², ab=ba, cac^-1=dad^-1=ab², cbc^-1=a²b, bd=db, dcd^-1=c¹³ >

Subgroups: 924 in 214 conjugacy classes, 95 normal (91 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, D₇, C₁₄, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, Dic₇, Dic₇, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²⋊Q₈, C₂².D₄, C₄².C₂, C₄².C₂, C₄²⋊₂C₂, Dic₁₄, C₄×D₇, D₂₈, C₂×Dic₇, C₂×C₂₈, C₂²×D₇, C₂².₄₆C₂⁴, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, D₁₄⋊C₄, C₄×C₂₈, C₇×C₄⋊C₄, C₂×Dic₁₄, C₂×C₄×D₇, C₂×D₂₈, C₄²⋊D₇, Dic₇⋊₃Q₈, Dic₇.Q₈, D₇×C₄⋊C₄, C₄⋊C₄⋊₇D₇, D₂₈⋊C₄, D₁₄.₅D₄, D₁₄⋊Q₈, C₄⋊C₄⋊D₇, C₇×C₄².C₂, C₄².₁₅₁D₁₄
Quotients: C₁, C₂, C₂², C₂³, D₇, C₄○D₄, C₂⁴, D₁₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂²×D₇, C₂².₄₆C₂⁴, C₂³×D₇, Q₈.₁₀D₁₄, D₇×C₄○D₄, C₄².₁₅₁D₁₄

Smallest permutation representation of C₄².₁₅₁D₁₄
►On 224 points

Generators in S₂₂₄

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

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

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

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

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

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

67 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	···	4I	4J	···	4O	4P	4Q	4R	7A	7B	7C	14A	···	14I	28A	···	28R	28S	···	28AD
order	1	2	2	2	2	2	2	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	28	2	2	2	2	4	···	4	14	···	14	28	28	28	2	2	2	2	···	2	4	···	4	8	···	8

67 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	C₄○D₄	D₁₄	D₁₄	2_-¹⁺⁴	Q₈.₁₀D₁₄	D₇×C₄○D₄
kernel	C₄².₁₅₁D₁₄	C₄²⋊D₇	Dic₇⋊₃Q₈	Dic₇.Q₈	D₇×C₄⋊C₄	C₄⋊C₄⋊₇D₇	D₂₈⋊C₄	D₁₄.₅D₄	D₁₄⋊Q₈	C₄⋊C₄⋊D₇	C₇×C₄².C₂	C₄².C₂	Dic₇	D₁₄	C₄²	C₄⋊C₄	C₁₄	C₂	C₂
# reps	1	2	1	2	1	1	1	2	2	2	1	3	4	4	3	18	1	6	12

Matrix representation of C₄².₁₅₁D₁₄ ►in GL₆(𝔽₂₉)

12	0	0	0	0	0
0	12	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	26	1
0	0	0	0	21	3

13	7	0	0	0	0
5	16	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	12	0
0	0	0	0	0	12

12	0	0	0	0	0
1	17	0	0	0	0
0	0	9	8	0	0
0	0	13	2	0	0
0	0	0	0	12	2
0	0	0	0	1	17

12	0	0	0	0	0
0	12	0	0	0	0
0	0	26	26	0	0
0	0	22	3	0	0
0	0	0	0	12	2
0	0	0	0	1	17

G:=sub<GL(6,GF(29))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,26,21,0,0,0,0,1,3],[13,5,0,0,0,0,7,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,0,17,0,0,0,0,0,0,9,13,0,0,0,0,8,2,0,0,0,0,0,0,12,1,0,0,0,0,2,17],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,26,22,0,0,0,0,26,3,0,0,0,0,0,0,12,1,0,0,0,0,2,17] >;

C₄².₁₅₁D₁₄ in GAP, Magma, Sage, TeX

C_4^2._{151}D_{14}

% in TeX

G:=Group("C4^2.151D14");

// GroupNames label

G:=SmallGroup(448,1146);

// by ID

G=gap.SmallGroup(448,1146);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,100,346,297,136,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^14=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,b*d=d*b,d*c*d^-1=c^13>;

// generators/relations

G = C42.151D14 order 448 = 26·7

151st non-split extension by C42 of D14 acting via D14/C7=C22

G = C₄².₁₅₁D₁₄ order 448 = 2⁶·7

151^st non-split extension by C₄² of D₁₄ acting via D₁₄/C₇=C₂²