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## G = C7×Q8⋊C4order 224 = 25·7

### Direct product of C7 and Q8⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×Q8⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C28 — C7×C4⋊C4 — C7×Q8⋊C4
 Lower central C1 — C2 — C4 — C7×Q8⋊C4
 Upper central C1 — C2×C14 — C2×C28 — C7×Q8⋊C4

Generators and relations for C7×Q8⋊C4
G = < a,b,c,d | a7=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >

Smallest permutation representation of C7×Q8⋊C4
Regular action on 224 points
Generators in S224
(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 50 78 59)(2 51 79 60)(3 52 80 61)(4 53 81 62)(5 54 82 63)(6 55 83 57)(7 56 84 58)(8 37 222 35)(9 38 223 29)(10 39 224 30)(11 40 218 31)(12 41 219 32)(13 42 220 33)(14 36 221 34)(15 211 24 43)(16 212 25 44)(17 213 26 45)(18 214 27 46)(19 215 28 47)(20 216 22 48)(21 217 23 49)(64 91 73 92)(65 85 74 93)(66 86 75 94)(67 87 76 95)(68 88 77 96)(69 89 71 97)(70 90 72 98)(99 125 153 127)(100 126 154 128)(101 120 148 129)(102 121 149 130)(103 122 150 131)(104 123 151 132)(105 124 152 133)(106 143 115 134)(107 144 116 135)(108 145 117 136)(109 146 118 137)(110 147 119 138)(111 141 113 139)(112 142 114 140)(155 181 209 183)(156 182 210 184)(157 176 204 185)(158 177 205 186)(159 178 206 187)(160 179 207 188)(161 180 208 189)(162 199 171 190)(163 200 172 191)(164 201 173 192)(165 202 174 193)(166 203 175 194)(167 197 169 195)(168 198 170 196)
(1 123 78 132)(2 124 79 133)(3 125 80 127)(4 126 81 128)(5 120 82 129)(6 121 83 130)(7 122 84 131)(8 185 222 176)(9 186 223 177)(10 187 224 178)(11 188 218 179)(12 189 219 180)(13 183 220 181)(14 184 221 182)(15 173 24 164)(16 174 25 165)(17 175 26 166)(18 169 27 167)(19 170 28 168)(20 171 22 162)(21 172 23 163)(29 158 38 205)(30 159 39 206)(31 160 40 207)(32 161 41 208)(33 155 42 209)(34 156 36 210)(35 157 37 204)(43 192 211 201)(44 193 212 202)(45 194 213 203)(46 195 214 197)(47 196 215 198)(48 190 216 199)(49 191 217 200)(50 104 59 151)(51 105 60 152)(52 99 61 153)(53 100 62 154)(54 101 63 148)(55 102 57 149)(56 103 58 150)(64 147 73 138)(65 141 74 139)(66 142 75 140)(67 143 76 134)(68 144 77 135)(69 145 71 136)(70 146 72 137)(85 111 93 113)(86 112 94 114)(87 106 95 115)(88 107 96 116)(89 108 97 117)(90 109 98 118)(91 110 92 119)
(1 188 76 199)(2 189 77 200)(3 183 71 201)(4 184 72 202)(5 185 73 203)(6 186 74 197)(7 187 75 198)(8 119 213 101)(9 113 214 102)(10 114 215 103)(11 115 216 104)(12 116 217 105)(13 117 211 99)(14 118 212 100)(15 125 33 136)(16 126 34 137)(17 120 35 138)(18 121 29 139)(19 122 30 140)(20 123 31 134)(21 124 32 135)(22 132 40 143)(23 133 41 144)(24 127 42 145)(25 128 36 146)(26 129 37 147)(27 130 38 141)(28 131 39 142)(43 153 220 108)(44 154 221 109)(45 148 222 110)(46 149 223 111)(47 150 224 112)(48 151 218 106)(49 152 219 107)(50 207 95 162)(51 208 96 163)(52 209 97 164)(53 210 98 165)(54 204 92 166)(55 205 93 167)(56 206 94 168)(57 158 85 169)(58 159 86 170)(59 160 87 171)(60 161 88 172)(61 155 89 173)(62 156 90 174)(63 157 91 175)(64 194 82 176)(65 195 83 177)(66 196 84 178)(67 190 78 179)(68 191 79 180)(69 192 80 181)(70 193 81 182)

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

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

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

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 1 1 2 2 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C4 C7 C14 C14 C14 C28 D4 D4 SD16 Q16 C7×D4 C7×D4 C7×SD16 C7×Q16 kernel C7×Q8⋊C4 C7×C4⋊C4 C2×C56 Q8×C14 C7×Q8 Q8⋊C4 C4⋊C4 C2×C8 C2×Q8 Q8 C28 C2×C14 C14 C14 C4 C22 C2 C2 # reps 1 1 1 1 4 6 6 6 6 24 1 1 2 2 6 6 12 12

Matrix representation of C7×Q8⋊C4 in GL3(𝔽113) generated by

 1 0 0 0 30 0 0 0 30
,
 1 0 0 0 0 1 0 112 0
,
 112 0 0 0 55 108 0 108 58
,
 98 0 0 0 28 102 0 102 85
G:=sub<GL(3,GF(113))| [1,0,0,0,30,0,0,0,30],[1,0,0,0,0,112,0,1,0],[112,0,0,0,55,108,0,108,58],[98,0,0,0,28,102,0,102,85] >;

C7×Q8⋊C4 in GAP, Magma, Sage, TeX

C_7\times Q_8\rtimes C_4
% in TeX

G:=Group("C7xQ8:C4");
// GroupNames label

G:=SmallGroup(224,52);
// by ID

G=gap.SmallGroup(224,52);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,679,3363,1689,117]);
// Polycyclic

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

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