C2xC7:Q16 - GroupNames

G = C₂×C₇⋊Q₁₆ order 224 = 2⁵·7

Direct product of C₂ and C₇⋊Q₁₆

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂×C₇⋊Q₁₆, C₁₄⋊₂Q₁₆, C₂₈.₂₀D₄, Q₈.₇D₁₄, C₂₈.₁₆C₂³, Dic₁₄.₁₀C₂², C₇⋊₃(C₂×Q₁₆), C₇⋊C₈.₉C₂², (C₂×Q₈).₃D₇, C₁₄.₅₅(C₂×D₄), (C₂×C₄).₅₄D₁₄, (C₂×C₁₄).₄₃D₄, C₄.₉(C₇⋊D₄), (Q₈×C₁₄).₃C₂, C₄.₁₆(C₂²×D₇), (C₇×Q₈).₇C₂², (C₂×C₂₈).₃₈C₂², (C₂×Dic₁₄).₈C₂, C₂².₂₄(C₇⋊D₄), (C₂×C₇⋊C₈).₆C₂, C₂.₁₉(C₂×C₇⋊D₄), SmallGroup(224,138)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₈ — C₂×C₇⋊Q₁₆

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — Dic₁₄ — C₂×Dic₁₄ — C₂×C₇⋊Q₁₆

Lower central

C₇ — C₁₄ — C₂₈ — C₂×C₇⋊Q₁₆

Upper central

C₁ — C₂² — C₂×C₄ — C₂×Q₈

Generators and relations for C₂×C₇⋊Q₁₆
G = < a,b,c,d | a²=b⁷=c⁸=1, d²=c⁴, ab=ba, ac=ca, ad=da, cbc^-1=b^-1, bd=db, dcd^-1=c^-1 >

Subgroups: 190 in 60 conjugacy classes, 33 normal (17 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₇, C₈, C₂×C₄, C₂×C₄, Q₈, Q₈, C₁₄, C₁₄, C₂×C₈, Q₁₆, C₂×Q₈, C₂×Q₈, Dic₇, C₂₈, C₂₈, C₂×C₁₄, C₂×Q₁₆, C₇⋊C₈, Dic₁₄, Dic₁₄, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, C₇×Q₈, C₂×C₇⋊C₈, C₇⋊Q₁₆, C₂×Dic₁₄, Q₈×C₁₄, C₂×C₇⋊Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, Q₁₆, C₂×D₄, D₁₄, C₂×Q₁₆, C₇⋊D₄, C₂²×D₇, C₇⋊Q₁₆, C₂×C₇⋊D₄, C₂×C₇⋊Q₁₆

Smallest permutation representation of C₂×C₇⋊Q₁₆
►Regular action on 224 points

Generators in S₂₂₄

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

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

(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)

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

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

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

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

C₂×C₇⋊Q₁₆ is a maximal subgroup of
D₂₈.₇D₄ C₇⋊Q₁₆⋊C₄ Dic₇⋊₄Q₁₆ Dic₇⋊Q₁₆ Dic₁₄.₁₁D₄ D₁₄⋊₄Q₁₆ Q₈.D₂₈ D₁₄⋊Q₁₆ C₇⋊C₈.D₄ Q₈.₁D₂₈ C₄².₅₉D₁₄ C₂₈⋊₇Q₁₆ D₂₈.₃₇D₄ Dic₁₄.₃₇D₄ C₇⋊C₈.₂₉D₄ C₇⋊C₈.₆D₄ C₄².₆₁D₁₄ C₄².₂₁₄D₁₄ C₄².₆₅D₁₄ C₄².₈₀D₁₄ C₂₈⋊Q₁₆ C₂₈⋊₃Q₁₆ (C₇×Q₈).D₄ C₅₆.₃₁D₄ C₅₆.₄₃D₄ Dic₁₄.₁₆D₄ C₅₆.₂₆D₄ Dic₇⋊₃Q₁₆ D₁₄⋊₅Q₁₆ C₅₆.₃₇D₄ M₄(2).₁₆D₁₄ (C₂×C₁₄)⋊₈Q₁₆ (C₇×D₄).₃₂D₄ C₂×D₇×Q₁₆ D₂₈.₄₄D₄ D₂₈.₃₅C₂³
C₂×C₇⋊Q₁₆ is a maximal quotient of
C₄⋊C₄.₂₃₀D₁₄ C₂₈.₂₃Q₁₆ C₂₈⋊₇Q₁₆ (C₂×C₁₄).Q₁₆ Dic₁₄.₃₇D₄ C₇⋊C₈.₂₉D₄ C₂₈.₁₇D₈ C₂₈.Q₁₆ C₂₈⋊Q₁₆ Dic₁₄⋊₅Q₈ C₂₈⋊₃Q₁₆ C₂₈.₁₁Q₁₆ (C₂×C₁₄)⋊₈Q₁₆

44 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	7A	7B	7C	8A	8B	8C	8D	14A	···	14I	28A	···	28R
order	1	2	2	2	4	4	4	4	4	4	7	7	7	8	8	8	8	14	···	14	28	···	28
size	1	1	1	1	2	2	4	4	28	28	2	2	2	14	14	14	14	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	2	2	4
type	+	+	+	+	+	+	+	+	-	+	+			-
image	C₁	C₂	C₂	C₂	C₂	D₄	D₄	D₇	Q₁₆	D₁₄	D₁₄	C₇⋊D₄	C₇⋊D₄	C₇⋊Q₁₆
kernel	C₂×C₇⋊Q₁₆	C₂×C₇⋊C₈	C₇⋊Q₁₆	C₂×Dic₁₄	Q₈×C₁₄	C₂₈	C₂×C₁₄	C₂×Q₈	C₁₄	C₂×C₄	Q₈	C₄	C₂²	C₂
# reps	1	1	4	1	1	1	1	3	4	3	6	6	6	6

Matrix representation of C₂×C₇⋊Q₁₆ ►in GL₆(𝔽₁₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	112	0
0	0	0	0	0	112

24	112	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	1	0
0	0	0	0	0	1

30	31	0	0	0	0
73	83	0	0	0	0
0	0	78	14	0	0
0	0	90	35	0	0
0	0	0	0	51	109
0	0	0	0	85	0

29	7	0	0	0	0
106	84	0	0	0	0
0	0	25	1	0	0
0	0	54	88	0	0
0	0	0	0	103	78
0	0	0	0	61	10

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[24,1,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,73,0,0,0,0,31,83,0,0,0,0,0,0,78,90,0,0,0,0,14,35,0,0,0,0,0,0,51,85,0,0,0,0,109,0],[29,106,0,0,0,0,7,84,0,0,0,0,0,0,25,54,0,0,0,0,1,88,0,0,0,0,0,0,103,61,0,0,0,0,78,10] >;

C₂×C₇⋊Q₁₆ in GAP, Magma, Sage, TeX

C_2\times C_7\rtimes Q_{16}

% in TeX

G:=Group("C2xC7:Q16");

// GroupNames label

G:=SmallGroup(224,138);

// by ID

G=gap.SmallGroup(224,138);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,218,86,579,159,69,6917]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^7=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;

// generators/relations

G = C2×C7⋊Q16 order 224 = 25·7

Direct product of C2 and C7⋊Q16

G = C₂×C₇⋊Q₁₆ order 224 = 2⁵·7

Direct product of C₂ and C₇⋊Q₁₆