direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C4.Q8, C8⋊2C28, C56⋊6C4, C28.9Q8, C14.11SD16, C4⋊C4.2C14, C4.1(C7×Q8), C4.6(C2×C28), (C2×C8).6C14, C28.43(C2×C4), (C2×C56).16C2, (C2×C14).48D4, C14.12(C4⋊C4), C2.3(C7×SD16), C22.10(C7×D4), (C2×C28).117C22, C2.3(C7×C4⋊C4), (C7×C4⋊C4).9C2, (C2×C4).20(C2×C14), SmallGroup(224,55)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4.Q8
G = < a,b,c,d | a7=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Smallest permutation representation of C7×C4.Q8
►Regular action on 224 points
Generators in S224
(1 70 23 11 87 31 79)(2 71 24 12 88 32 80)(3 72 17 13 81 25 73)(4 65 18 14 82 26 74)(5 66 19 15 83 27 75)(6 67 20 16 84 28 76)(7 68 21 9 85 29 77)(8 69 22 10 86 30 78)(33 117 105 49 97 41 89)(34 118 106 50 98 42 90)(35 119 107 51 99 43 91)(36 120 108 52 100 44 92)(37 113 109 53 101 45 93)(38 114 110 54 102 46 94)(39 115 111 55 103 47 95)(40 116 112 56 104 48 96)(57 219 167 211 159 203 151)(58 220 168 212 160 204 152)(59 221 161 213 153 205 145)(60 222 162 214 154 206 146)(61 223 163 215 155 207 147)(62 224 164 216 156 208 148)(63 217 165 209 157 201 149)(64 218 166 210 158 202 150)(121 170 197 137 189 129 181)(122 171 198 138 190 130 182)(123 172 199 139 191 131 183)(124 173 200 140 192 132 184)(125 174 193 141 185 133 177)(126 175 194 142 186 134 178)(127 176 195 143 187 135 179)(128 169 196 144 188 136 180)
(1 33 5 37)(2 34 6 38)(3 35 7 39)(4 36 8 40)(9 55 13 51)(10 56 14 52)(11 49 15 53)(12 50 16 54)(17 107 21 111)(18 108 22 112)(19 109 23 105)(20 110 24 106)(25 43 29 47)(26 44 30 48)(27 45 31 41)(28 46 32 42)(57 172 61 176)(58 173 62 169)(59 174 63 170)(60 175 64 171)(65 120 69 116)(66 113 70 117)(67 114 71 118)(68 115 72 119)(73 91 77 95)(74 92 78 96)(75 93 79 89)(76 94 80 90)(81 99 85 103)(82 100 86 104)(83 101 87 97)(84 102 88 98)(121 145 125 149)(122 146 126 150)(123 147 127 151)(124 148 128 152)(129 153 133 157)(130 154 134 158)(131 155 135 159)(132 156 136 160)(137 161 141 165)(138 162 142 166)(139 163 143 167)(140 164 144 168)(177 201 181 205)(178 202 182 206)(179 203 183 207)(180 204 184 208)(185 209 189 213)(186 210 190 214)(187 211 191 215)(188 212 192 216)(193 217 197 221)(194 218 198 222)(195 219 199 223)(196 220 200 224)
(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)
(1 145 39 123)(2 148 40 126)(3 151 33 121)(4 146 34 124)(5 149 35 127)(6 152 36 122)(7 147 37 125)(8 150 38 128)(9 163 53 141)(10 166 54 144)(11 161 55 139)(12 164 56 142)(13 167 49 137)(14 162 50 140)(15 165 51 143)(16 168 52 138)(17 219 105 197)(18 222 106 200)(19 217 107 195)(20 220 108 198)(21 223 109 193)(22 218 110 196)(23 221 111 199)(24 224 112 194)(25 159 41 129)(26 154 42 132)(27 157 43 135)(28 160 44 130)(29 155 45 133)(30 158 46 136)(31 153 47 131)(32 156 48 134)(57 117 170 72)(58 120 171 67)(59 115 172 70)(60 118 173 65)(61 113 174 68)(62 116 175 71)(63 119 176 66)(64 114 169 69)(73 203 89 181)(74 206 90 184)(75 201 91 179)(76 204 92 182)(77 207 93 177)(78 202 94 180)(79 205 95 183)(80 208 96 178)(81 211 97 189)(82 214 98 192)(83 209 99 187)(84 212 100 190)(85 215 101 185)(86 210 102 188)(87 213 103 191)(88 216 104 186)
G:=sub<Sym(224)| (1,70,23,11,87,31,79)(2,71,24,12,88,32,80)(3,72,17,13,81,25,73)(4,65,18,14,82,26,74)(5,66,19,15,83,27,75)(6,67,20,16,84,28,76)(7,68,21,9,85,29,77)(8,69,22,10,86,30,78)(33,117,105,49,97,41,89)(34,118,106,50,98,42,90)(35,119,107,51,99,43,91)(36,120,108,52,100,44,92)(37,113,109,53,101,45,93)(38,114,110,54,102,46,94)(39,115,111,55,103,47,95)(40,116,112,56,104,48,96)(57,219,167,211,159,203,151)(58,220,168,212,160,204,152)(59,221,161,213,153,205,145)(60,222,162,214,154,206,146)(61,223,163,215,155,207,147)(62,224,164,216,156,208,148)(63,217,165,209,157,201,149)(64,218,166,210,158,202,150)(121,170,197,137,189,129,181)(122,171,198,138,190,130,182)(123,172,199,139,191,131,183)(124,173,200,140,192,132,184)(125,174,193,141,185,133,177)(126,175,194,142,186,134,178)(127,176,195,143,187,135,179)(128,169,196,144,188,136,180), (1,33,5,37)(2,34,6,38)(3,35,7,39)(4,36,8,40)(9,55,13,51)(10,56,14,52)(11,49,15,53)(12,50,16,54)(17,107,21,111)(18,108,22,112)(19,109,23,105)(20,110,24,106)(25,43,29,47)(26,44,30,48)(27,45,31,41)(28,46,32,42)(57,172,61,176)(58,173,62,169)(59,174,63,170)(60,175,64,171)(65,120,69,116)(66,113,70,117)(67,114,71,118)(68,115,72,119)(73,91,77,95)(74,92,78,96)(75,93,79,89)(76,94,80,90)(81,99,85,103)(82,100,86,104)(83,101,87,97)(84,102,88,98)(121,145,125,149)(122,146,126,150)(123,147,127,151)(124,148,128,152)(129,153,133,157)(130,154,134,158)(131,155,135,159)(132,156,136,160)(137,161,141,165)(138,162,142,166)(139,163,143,167)(140,164,144,168)(177,201,181,205)(178,202,182,206)(179,203,183,207)(180,204,184,208)(185,209,189,213)(186,210,190,214)(187,211,191,215)(188,212,192,216)(193,217,197,221)(194,218,198,222)(195,219,199,223)(196,220,200,224), (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), (1,145,39,123)(2,148,40,126)(3,151,33,121)(4,146,34,124)(5,149,35,127)(6,152,36,122)(7,147,37,125)(8,150,38,128)(9,163,53,141)(10,166,54,144)(11,161,55,139)(12,164,56,142)(13,167,49,137)(14,162,50,140)(15,165,51,143)(16,168,52,138)(17,219,105,197)(18,222,106,200)(19,217,107,195)(20,220,108,198)(21,223,109,193)(22,218,110,196)(23,221,111,199)(24,224,112,194)(25,159,41,129)(26,154,42,132)(27,157,43,135)(28,160,44,130)(29,155,45,133)(30,158,46,136)(31,153,47,131)(32,156,48,134)(57,117,170,72)(58,120,171,67)(59,115,172,70)(60,118,173,65)(61,113,174,68)(62,116,175,71)(63,119,176,66)(64,114,169,69)(73,203,89,181)(74,206,90,184)(75,201,91,179)(76,204,92,182)(77,207,93,177)(78,202,94,180)(79,205,95,183)(80,208,96,178)(81,211,97,189)(82,214,98,192)(83,209,99,187)(84,212,100,190)(85,215,101,185)(86,210,102,188)(87,213,103,191)(88,216,104,186)>;
G:=Group( (1,70,23,11,87,31,79)(2,71,24,12,88,32,80)(3,72,17,13,81,25,73)(4,65,18,14,82,26,74)(5,66,19,15,83,27,75)(6,67,20,16,84,28,76)(7,68,21,9,85,29,77)(8,69,22,10,86,30,78)(33,117,105,49,97,41,89)(34,118,106,50,98,42,90)(35,119,107,51,99,43,91)(36,120,108,52,100,44,92)(37,113,109,53,101,45,93)(38,114,110,54,102,46,94)(39,115,111,55,103,47,95)(40,116,112,56,104,48,96)(57,219,167,211,159,203,151)(58,220,168,212,160,204,152)(59,221,161,213,153,205,145)(60,222,162,214,154,206,146)(61,223,163,215,155,207,147)(62,224,164,216,156,208,148)(63,217,165,209,157,201,149)(64,218,166,210,158,202,150)(121,170,197,137,189,129,181)(122,171,198,138,190,130,182)(123,172,199,139,191,131,183)(124,173,200,140,192,132,184)(125,174,193,141,185,133,177)(126,175,194,142,186,134,178)(127,176,195,143,187,135,179)(128,169,196,144,188,136,180), (1,33,5,37)(2,34,6,38)(3,35,7,39)(4,36,8,40)(9,55,13,51)(10,56,14,52)(11,49,15,53)(12,50,16,54)(17,107,21,111)(18,108,22,112)(19,109,23,105)(20,110,24,106)(25,43,29,47)(26,44,30,48)(27,45,31,41)(28,46,32,42)(57,172,61,176)(58,173,62,169)(59,174,63,170)(60,175,64,171)(65,120,69,116)(66,113,70,117)(67,114,71,118)(68,115,72,119)(73,91,77,95)(74,92,78,96)(75,93,79,89)(76,94,80,90)(81,99,85,103)(82,100,86,104)(83,101,87,97)(84,102,88,98)(121,145,125,149)(122,146,126,150)(123,147,127,151)(124,148,128,152)(129,153,133,157)(130,154,134,158)(131,155,135,159)(132,156,136,160)(137,161,141,165)(138,162,142,166)(139,163,143,167)(140,164,144,168)(177,201,181,205)(178,202,182,206)(179,203,183,207)(180,204,184,208)(185,209,189,213)(186,210,190,214)(187,211,191,215)(188,212,192,216)(193,217,197,221)(194,218,198,222)(195,219,199,223)(196,220,200,224), (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), (1,145,39,123)(2,148,40,126)(3,151,33,121)(4,146,34,124)(5,149,35,127)(6,152,36,122)(7,147,37,125)(8,150,38,128)(9,163,53,141)(10,166,54,144)(11,161,55,139)(12,164,56,142)(13,167,49,137)(14,162,50,140)(15,165,51,143)(16,168,52,138)(17,219,105,197)(18,222,106,200)(19,217,107,195)(20,220,108,198)(21,223,109,193)(22,218,110,196)(23,221,111,199)(24,224,112,194)(25,159,41,129)(26,154,42,132)(27,157,43,135)(28,160,44,130)(29,155,45,133)(30,158,46,136)(31,153,47,131)(32,156,48,134)(57,117,170,72)(58,120,171,67)(59,115,172,70)(60,118,173,65)(61,113,174,68)(62,116,175,71)(63,119,176,66)(64,114,169,69)(73,203,89,181)(74,206,90,184)(75,201,91,179)(76,204,92,182)(77,207,93,177)(78,202,94,180)(79,205,95,183)(80,208,96,178)(81,211,97,189)(82,214,98,192)(83,209,99,187)(84,212,100,190)(85,215,101,185)(86,210,102,188)(87,213,103,191)(88,216,104,186) );
G=PermutationGroup([[(1,70,23,11,87,31,79),(2,71,24,12,88,32,80),(3,72,17,13,81,25,73),(4,65,18,14,82,26,74),(5,66,19,15,83,27,75),(6,67,20,16,84,28,76),(7,68,21,9,85,29,77),(8,69,22,10,86,30,78),(33,117,105,49,97,41,89),(34,118,106,50,98,42,90),(35,119,107,51,99,43,91),(36,120,108,52,100,44,92),(37,113,109,53,101,45,93),(38,114,110,54,102,46,94),(39,115,111,55,103,47,95),(40,116,112,56,104,48,96),(57,219,167,211,159,203,151),(58,220,168,212,160,204,152),(59,221,161,213,153,205,145),(60,222,162,214,154,206,146),(61,223,163,215,155,207,147),(62,224,164,216,156,208,148),(63,217,165,209,157,201,149),(64,218,166,210,158,202,150),(121,170,197,137,189,129,181),(122,171,198,138,190,130,182),(123,172,199,139,191,131,183),(124,173,200,140,192,132,184),(125,174,193,141,185,133,177),(126,175,194,142,186,134,178),(127,176,195,143,187,135,179),(128,169,196,144,188,136,180)], [(1,33,5,37),(2,34,6,38),(3,35,7,39),(4,36,8,40),(9,55,13,51),(10,56,14,52),(11,49,15,53),(12,50,16,54),(17,107,21,111),(18,108,22,112),(19,109,23,105),(20,110,24,106),(25,43,29,47),(26,44,30,48),(27,45,31,41),(28,46,32,42),(57,172,61,176),(58,173,62,169),(59,174,63,170),(60,175,64,171),(65,120,69,116),(66,113,70,117),(67,114,71,118),(68,115,72,119),(73,91,77,95),(74,92,78,96),(75,93,79,89),(76,94,80,90),(81,99,85,103),(82,100,86,104),(83,101,87,97),(84,102,88,98),(121,145,125,149),(122,146,126,150),(123,147,127,151),(124,148,128,152),(129,153,133,157),(130,154,134,158),(131,155,135,159),(132,156,136,160),(137,161,141,165),(138,162,142,166),(139,163,143,167),(140,164,144,168),(177,201,181,205),(178,202,182,206),(179,203,183,207),(180,204,184,208),(185,209,189,213),(186,210,190,214),(187,211,191,215),(188,212,192,216),(193,217,197,221),(194,218,198,222),(195,219,199,223),(196,220,200,224)], [(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)], [(1,145,39,123),(2,148,40,126),(3,151,33,121),(4,146,34,124),(5,149,35,127),(6,152,36,122),(7,147,37,125),(8,150,38,128),(9,163,53,141),(10,166,54,144),(11,161,55,139),(12,164,56,142),(13,167,49,137),(14,162,50,140),(15,165,51,143),(16,168,52,138),(17,219,105,197),(18,222,106,200),(19,217,107,195),(20,220,108,198),(21,223,109,193),(22,218,110,196),(23,221,111,199),(24,224,112,194),(25,159,41,129),(26,154,42,132),(27,157,43,135),(28,160,44,130),(29,155,45,133),(30,158,46,136),(31,153,47,131),(32,156,48,134),(57,117,170,72),(58,120,171,67),(59,115,172,70),(60,118,173,65),(61,113,174,68),(62,116,175,71),(63,119,176,66),(64,114,169,69),(73,203,89,181),(74,206,90,184),(75,201,91,179),(76,204,92,182),(77,207,93,177),(78,202,94,180),(79,205,95,183),(80,208,96,178),(81,211,97,189),(82,214,98,192),(83,209,99,187),(84,212,100,190),(85,215,101,185),(86,210,102,188),(87,213,103,191),(88,216,104,186)]])
C7×C4.Q8 is a maximal subgroup of
C56.Q8 D56⋊8C4 Dic7⋊8SD16 Dic28⋊9C4 Dic14⋊Q8 C56⋊5Q8 C56⋊3Q8 Dic14.Q8 C56.8Q8 (C8×D7)⋊C4 C8⋊(C4×D7) D14.2SD16 D14.4SD16 C8⋊8D28 C56⋊7D4 C4.Q8⋊D7 C28.(C4○D4) C8.2D28 D56⋊9C4 D28⋊Q8 D28.Q8 SD16×C28
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 28A | ··· | 28L | 28M | ··· | 28AJ | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | Q8 | D4 | SD16 | C7×Q8 | C7×D4 | C7×SD16 |
| kernel | C7×C4.Q8 | C7×C4⋊C4 | C2×C56 | C56 | C4.Q8 | C4⋊C4 | C2×C8 | C8 | C28 | C2×C14 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 | 1 | 1 | 4 | 6 | 6 | 24 |
Matrix representation of C7×C4.Q8 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 30 | 0 |
| 0 | 0 | 30 |
| 112 | 0 | 0 |
| 0 | 112 | 111 |
| 0 | 1 | 1 |
| 112 | 0 | 0 |
| 0 | 26 | 26 |
| 0 | 100 | 0 |
| 15 | 0 | 0 |
| 0 | 17 | 56 |
| 0 | 11 | 96 |
G:=sub<GL(3,GF(113))| [1,0,0,0,30,0,0,0,30],[112,0,0,0,112,1,0,111,1],[112,0,0,0,26,100,0,26,0],[15,0,0,0,17,11,0,56,96] >;
C7×C4.Q8 in GAP, Magma, Sage, TeX
C_7\times C_4.Q_8
% in TeX
G:=Group("C7xC4.Q8"); // GroupNames label
G:=SmallGroup(224,55);
// by ID
G=gap.SmallGroup(224,55);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,175,3363,117]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
Export