direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C22⋊Q16, (C2×C14)⋊7Q16, Q8.6(C7×D4), (C2×Q16)⋊1C14, C4.25(D4×C14), (C7×Q8).40D4, C2.4(C14×Q16), C22⋊2(C7×Q16), Q8⋊C4⋊5C14, (C14×Q16)⋊15C2, (C2×C28).320D4, C28.386(C2×D4), C22⋊C8.3C14, C14.51(C2×Q16), C23.44(C7×D4), C22⋊Q8.2C14, C14.98C22≀C2, C22.81(D4×C14), (C22×Q8).6C14, (C2×C56).254C22, (C2×C28).916C23, (C22×C14).166D4, (Q8×C14).260C22, C14.132(C8.C22), (C22×C28).423C22, C4⋊C4.3(C2×C14), (C2×C8).2(C2×C14), (C2×C4).29(C7×D4), (Q8×C2×C14).16C2, C2.7(C7×C8.C22), (C7×Q8⋊C4)⋊28C2, C2.12(C7×C22≀C2), (C2×C14).637(C2×D4), (C7×C22⋊C8).12C2, (C2×Q8).45(C2×C14), (C7×C22⋊Q8).12C2, (C7×C4⋊C4).225C22, (C22×C4).41(C2×C14), (C2×C4).91(C22×C14), SmallGroup(448,859)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C22⋊Q16
G = < a,b,c,d,e | a7=b2=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 242 in 148 conjugacy classes, 62 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C56, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×C14, C22⋊Q16, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C7×Q16, C22×C28, C22×C28, Q8×C14, Q8×C14, Q8×C14, C7×C22⋊C8, C7×Q8⋊C4, C7×C22⋊Q8, C14×Q16, Q8×C2×C14, C7×C22⋊Q16
Quotients: C1, C2, C22, C7, D4, C23, C14, Q16, C2×D4, C2×C14, C22≀C2, C2×Q16, C8.C22, C7×D4, C22×C14, C22⋊Q16, C7×Q16, D4×C14, C7×C22≀C2, C14×Q16, C7×C8.C22, C7×C22⋊Q16
►On 224 points
Generators in S224
(1 70 116 62 133 76 125)(2 71 117 63 134 77 126)(3 72 118 64 135 78 127)(4 65 119 57 136 79 128)(5 66 120 58 129 80 121)(6 67 113 59 130 73 122)(7 68 114 60 131 74 123)(8 69 115 61 132 75 124)(9 161 193 25 185 17 177)(10 162 194 26 186 18 178)(11 163 195 27 187 19 179)(12 164 196 28 188 20 180)(13 165 197 29 189 21 181)(14 166 198 30 190 22 182)(15 167 199 31 191 23 183)(16 168 200 32 192 24 184)(33 174 217 49 209 41 201)(34 175 218 50 210 42 202)(35 176 219 51 211 43 203)(36 169 220 52 212 44 204)(37 170 221 53 213 45 205)(38 171 222 54 214 46 206)(39 172 223 55 215 47 207)(40 173 224 56 216 48 208)(81 107 156 99 148 91 143)(82 108 157 100 149 92 144)(83 109 158 101 150 93 137)(84 110 159 102 151 94 138)(85 111 160 103 152 95 139)(86 112 153 104 145 96 140)(87 105 154 97 146 89 141)(88 106 155 98 147 90 142)
(2 36)(4 38)(6 40)(8 34)(10 152)(12 146)(14 148)(16 150)(18 160)(20 154)(22 156)(24 158)(26 85)(28 87)(30 81)(32 83)(42 75)(44 77)(46 79)(48 73)(50 61)(52 63)(54 57)(56 59)(65 171)(67 173)(69 175)(71 169)(89 164)(91 166)(93 168)(95 162)(97 180)(99 182)(101 184)(103 178)(105 188)(107 190)(109 192)(111 186)(113 224)(115 218)(117 220)(119 222)(122 208)(124 202)(126 204)(128 206)(130 216)(132 210)(134 212)(136 214)(137 200)(139 194)(141 196)(143 198)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 151)(10 152)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 159)(18 160)(19 153)(20 154)(21 155)(22 156)(23 157)(24 158)(25 84)(26 85)(27 86)(28 87)(29 88)(30 81)(31 82)(32 83)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 73)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)(65 171)(66 172)(67 173)(68 174)(69 175)(70 176)(71 169)(72 170)(89 164)(90 165)(91 166)(92 167)(93 168)(94 161)(95 162)(96 163)(97 180)(98 181)(99 182)(100 183)(101 184)(102 177)(103 178)(104 179)(105 188)(106 189)(107 190)(108 191)(109 192)(110 185)(111 186)(112 187)(113 224)(114 217)(115 218)(116 219)(117 220)(118 221)(119 222)(120 223)(121 207)(122 208)(123 201)(124 202)(125 203)(126 204)(127 205)(128 206)(129 215)(130 216)(131 209)(132 210)(133 211)(134 212)(135 213)(136 214)(137 200)(138 193)(139 194)(140 195)(141 196)(142 197)(143 198)(144 199)
(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 92 5 96)(2 91 6 95)(3 90 7 94)(4 89 8 93)(9 205 13 201)(10 204 14 208)(11 203 15 207)(12 202 16 206)(17 213 21 209)(18 212 22 216)(19 211 23 215)(20 210 24 214)(25 221 29 217)(26 220 30 224)(27 219 31 223)(28 218 32 222)(33 161 37 165)(34 168 38 164)(35 167 39 163)(36 166 40 162)(41 177 45 181)(42 184 46 180)(43 183 47 179)(44 182 48 178)(49 185 53 189)(50 192 54 188)(51 191 55 187)(52 190 56 186)(57 105 61 109)(58 112 62 108)(59 111 63 107)(60 110 64 106)(65 141 69 137)(66 140 70 144)(67 139 71 143)(68 138 72 142)(73 103 77 99)(74 102 78 98)(75 101 79 97)(76 100 80 104)(81 113 85 117)(82 120 86 116)(83 119 87 115)(84 118 88 114)(121 145 125 149)(122 152 126 148)(123 151 127 147)(124 150 128 146)(129 153 133 157)(130 160 134 156)(131 159 135 155)(132 158 136 154)(169 198 173 194)(170 197 174 193)(171 196 175 200)(172 195 176 199)
G:=sub<Sym(224)| (1,70,116,62,133,76,125)(2,71,117,63,134,77,126)(3,72,118,64,135,78,127)(4,65,119,57,136,79,128)(5,66,120,58,129,80,121)(6,67,113,59,130,73,122)(7,68,114,60,131,74,123)(8,69,115,61,132,75,124)(9,161,193,25,185,17,177)(10,162,194,26,186,18,178)(11,163,195,27,187,19,179)(12,164,196,28,188,20,180)(13,165,197,29,189,21,181)(14,166,198,30,190,22,182)(15,167,199,31,191,23,183)(16,168,200,32,192,24,184)(33,174,217,49,209,41,201)(34,175,218,50,210,42,202)(35,176,219,51,211,43,203)(36,169,220,52,212,44,204)(37,170,221,53,213,45,205)(38,171,222,54,214,46,206)(39,172,223,55,215,47,207)(40,173,224,56,216,48,208)(81,107,156,99,148,91,143)(82,108,157,100,149,92,144)(83,109,158,101,150,93,137)(84,110,159,102,151,94,138)(85,111,160,103,152,95,139)(86,112,153,104,145,96,140)(87,105,154,97,146,89,141)(88,106,155,98,147,90,142), (2,36)(4,38)(6,40)(8,34)(10,152)(12,146)(14,148)(16,150)(18,160)(20,154)(22,156)(24,158)(26,85)(28,87)(30,81)(32,83)(42,75)(44,77)(46,79)(48,73)(50,61)(52,63)(54,57)(56,59)(65,171)(67,173)(69,175)(71,169)(89,164)(91,166)(93,168)(95,162)(97,180)(99,182)(101,184)(103,178)(105,188)(107,190)(109,192)(111,186)(113,224)(115,218)(117,220)(119,222)(122,208)(124,202)(126,204)(128,206)(130,216)(132,210)(134,212)(136,214)(137,200)(139,194)(141,196)(143,198), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,151)(10,152)(11,145)(12,146)(13,147)(14,148)(15,149)(16,150)(17,159)(18,160)(19,153)(20,154)(21,155)(22,156)(23,157)(24,158)(25,84)(26,85)(27,86)(28,87)(29,88)(30,81)(31,82)(32,83)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,73)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59)(65,171)(66,172)(67,173)(68,174)(69,175)(70,176)(71,169)(72,170)(89,164)(90,165)(91,166)(92,167)(93,168)(94,161)(95,162)(96,163)(97,180)(98,181)(99,182)(100,183)(101,184)(102,177)(103,178)(104,179)(105,188)(106,189)(107,190)(108,191)(109,192)(110,185)(111,186)(112,187)(113,224)(114,217)(115,218)(116,219)(117,220)(118,221)(119,222)(120,223)(121,207)(122,208)(123,201)(124,202)(125,203)(126,204)(127,205)(128,206)(129,215)(130,216)(131,209)(132,210)(133,211)(134,212)(135,213)(136,214)(137,200)(138,193)(139,194)(140,195)(141,196)(142,197)(143,198)(144,199), (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), (1,92,5,96)(2,91,6,95)(3,90,7,94)(4,89,8,93)(9,205,13,201)(10,204,14,208)(11,203,15,207)(12,202,16,206)(17,213,21,209)(18,212,22,216)(19,211,23,215)(20,210,24,214)(25,221,29,217)(26,220,30,224)(27,219,31,223)(28,218,32,222)(33,161,37,165)(34,168,38,164)(35,167,39,163)(36,166,40,162)(41,177,45,181)(42,184,46,180)(43,183,47,179)(44,182,48,178)(49,185,53,189)(50,192,54,188)(51,191,55,187)(52,190,56,186)(57,105,61,109)(58,112,62,108)(59,111,63,107)(60,110,64,106)(65,141,69,137)(66,140,70,144)(67,139,71,143)(68,138,72,142)(73,103,77,99)(74,102,78,98)(75,101,79,97)(76,100,80,104)(81,113,85,117)(82,120,86,116)(83,119,87,115)(84,118,88,114)(121,145,125,149)(122,152,126,148)(123,151,127,147)(124,150,128,146)(129,153,133,157)(130,160,134,156)(131,159,135,155)(132,158,136,154)(169,198,173,194)(170,197,174,193)(171,196,175,200)(172,195,176,199)>;
G:=Group( (1,70,116,62,133,76,125)(2,71,117,63,134,77,126)(3,72,118,64,135,78,127)(4,65,119,57,136,79,128)(5,66,120,58,129,80,121)(6,67,113,59,130,73,122)(7,68,114,60,131,74,123)(8,69,115,61,132,75,124)(9,161,193,25,185,17,177)(10,162,194,26,186,18,178)(11,163,195,27,187,19,179)(12,164,196,28,188,20,180)(13,165,197,29,189,21,181)(14,166,198,30,190,22,182)(15,167,199,31,191,23,183)(16,168,200,32,192,24,184)(33,174,217,49,209,41,201)(34,175,218,50,210,42,202)(35,176,219,51,211,43,203)(36,169,220,52,212,44,204)(37,170,221,53,213,45,205)(38,171,222,54,214,46,206)(39,172,223,55,215,47,207)(40,173,224,56,216,48,208)(81,107,156,99,148,91,143)(82,108,157,100,149,92,144)(83,109,158,101,150,93,137)(84,110,159,102,151,94,138)(85,111,160,103,152,95,139)(86,112,153,104,145,96,140)(87,105,154,97,146,89,141)(88,106,155,98,147,90,142), (2,36)(4,38)(6,40)(8,34)(10,152)(12,146)(14,148)(16,150)(18,160)(20,154)(22,156)(24,158)(26,85)(28,87)(30,81)(32,83)(42,75)(44,77)(46,79)(48,73)(50,61)(52,63)(54,57)(56,59)(65,171)(67,173)(69,175)(71,169)(89,164)(91,166)(93,168)(95,162)(97,180)(99,182)(101,184)(103,178)(105,188)(107,190)(109,192)(111,186)(113,224)(115,218)(117,220)(119,222)(122,208)(124,202)(126,204)(128,206)(130,216)(132,210)(134,212)(136,214)(137,200)(139,194)(141,196)(143,198), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,151)(10,152)(11,145)(12,146)(13,147)(14,148)(15,149)(16,150)(17,159)(18,160)(19,153)(20,154)(21,155)(22,156)(23,157)(24,158)(25,84)(26,85)(27,86)(28,87)(29,88)(30,81)(31,82)(32,83)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,73)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59)(65,171)(66,172)(67,173)(68,174)(69,175)(70,176)(71,169)(72,170)(89,164)(90,165)(91,166)(92,167)(93,168)(94,161)(95,162)(96,163)(97,180)(98,181)(99,182)(100,183)(101,184)(102,177)(103,178)(104,179)(105,188)(106,189)(107,190)(108,191)(109,192)(110,185)(111,186)(112,187)(113,224)(114,217)(115,218)(116,219)(117,220)(118,221)(119,222)(120,223)(121,207)(122,208)(123,201)(124,202)(125,203)(126,204)(127,205)(128,206)(129,215)(130,216)(131,209)(132,210)(133,211)(134,212)(135,213)(136,214)(137,200)(138,193)(139,194)(140,195)(141,196)(142,197)(143,198)(144,199), (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), (1,92,5,96)(2,91,6,95)(3,90,7,94)(4,89,8,93)(9,205,13,201)(10,204,14,208)(11,203,15,207)(12,202,16,206)(17,213,21,209)(18,212,22,216)(19,211,23,215)(20,210,24,214)(25,221,29,217)(26,220,30,224)(27,219,31,223)(28,218,32,222)(33,161,37,165)(34,168,38,164)(35,167,39,163)(36,166,40,162)(41,177,45,181)(42,184,46,180)(43,183,47,179)(44,182,48,178)(49,185,53,189)(50,192,54,188)(51,191,55,187)(52,190,56,186)(57,105,61,109)(58,112,62,108)(59,111,63,107)(60,110,64,106)(65,141,69,137)(66,140,70,144)(67,139,71,143)(68,138,72,142)(73,103,77,99)(74,102,78,98)(75,101,79,97)(76,100,80,104)(81,113,85,117)(82,120,86,116)(83,119,87,115)(84,118,88,114)(121,145,125,149)(122,152,126,148)(123,151,127,147)(124,150,128,146)(129,153,133,157)(130,160,134,156)(131,159,135,155)(132,158,136,154)(169,198,173,194)(170,197,174,193)(171,196,175,200)(172,195,176,199) );
G=PermutationGroup([[(1,70,116,62,133,76,125),(2,71,117,63,134,77,126),(3,72,118,64,135,78,127),(4,65,119,57,136,79,128),(5,66,120,58,129,80,121),(6,67,113,59,130,73,122),(7,68,114,60,131,74,123),(8,69,115,61,132,75,124),(9,161,193,25,185,17,177),(10,162,194,26,186,18,178),(11,163,195,27,187,19,179),(12,164,196,28,188,20,180),(13,165,197,29,189,21,181),(14,166,198,30,190,22,182),(15,167,199,31,191,23,183),(16,168,200,32,192,24,184),(33,174,217,49,209,41,201),(34,175,218,50,210,42,202),(35,176,219,51,211,43,203),(36,169,220,52,212,44,204),(37,170,221,53,213,45,205),(38,171,222,54,214,46,206),(39,172,223,55,215,47,207),(40,173,224,56,216,48,208),(81,107,156,99,148,91,143),(82,108,157,100,149,92,144),(83,109,158,101,150,93,137),(84,110,159,102,151,94,138),(85,111,160,103,152,95,139),(86,112,153,104,145,96,140),(87,105,154,97,146,89,141),(88,106,155,98,147,90,142)], [(2,36),(4,38),(6,40),(8,34),(10,152),(12,146),(14,148),(16,150),(18,160),(20,154),(22,156),(24,158),(26,85),(28,87),(30,81),(32,83),(42,75),(44,77),(46,79),(48,73),(50,61),(52,63),(54,57),(56,59),(65,171),(67,173),(69,175),(71,169),(89,164),(91,166),(93,168),(95,162),(97,180),(99,182),(101,184),(103,178),(105,188),(107,190),(109,192),(111,186),(113,224),(115,218),(117,220),(119,222),(122,208),(124,202),(126,204),(128,206),(130,216),(132,210),(134,212),(136,214),(137,200),(139,194),(141,196),(143,198)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(9,151),(10,152),(11,145),(12,146),(13,147),(14,148),(15,149),(16,150),(17,159),(18,160),(19,153),(20,154),(21,155),(22,156),(23,157),(24,158),(25,84),(26,85),(27,86),(28,87),(29,88),(30,81),(31,82),(32,83),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,73),(49,60),(50,61),(51,62),(52,63),(53,64),(54,57),(55,58),(56,59),(65,171),(66,172),(67,173),(68,174),(69,175),(70,176),(71,169),(72,170),(89,164),(90,165),(91,166),(92,167),(93,168),(94,161),(95,162),(96,163),(97,180),(98,181),(99,182),(100,183),(101,184),(102,177),(103,178),(104,179),(105,188),(106,189),(107,190),(108,191),(109,192),(110,185),(111,186),(112,187),(113,224),(114,217),(115,218),(116,219),(117,220),(118,221),(119,222),(120,223),(121,207),(122,208),(123,201),(124,202),(125,203),(126,204),(127,205),(128,206),(129,215),(130,216),(131,209),(132,210),(133,211),(134,212),(135,213),(136,214),(137,200),(138,193),(139,194),(140,195),(141,196),(142,197),(143,198),(144,199)], [(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)], [(1,92,5,96),(2,91,6,95),(3,90,7,94),(4,89,8,93),(9,205,13,201),(10,204,14,208),(11,203,15,207),(12,202,16,206),(17,213,21,209),(18,212,22,216),(19,211,23,215),(20,210,24,214),(25,221,29,217),(26,220,30,224),(27,219,31,223),(28,218,32,222),(33,161,37,165),(34,168,38,164),(35,167,39,163),(36,166,40,162),(41,177,45,181),(42,184,46,180),(43,183,47,179),(44,182,48,178),(49,185,53,189),(50,192,54,188),(51,191,55,187),(52,190,56,186),(57,105,61,109),(58,112,62,108),(59,111,63,107),(60,110,64,106),(65,141,69,137),(66,140,70,144),(67,139,71,143),(68,138,72,142),(73,103,77,99),(74,102,78,98),(75,101,79,97),(76,100,80,104),(81,113,85,117),(82,120,86,116),(83,119,87,115),(84,118,88,114),(121,145,125,149),(122,152,126,148),(123,151,127,147),(124,150,128,146),(129,153,133,157),(130,160,134,156),(131,159,135,155),(132,158,136,154),(169,198,173,194),(170,197,174,193),(171,196,175,200),(172,195,176,199)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28L | 28M | ··· | 28AP | 28AQ | ··· | 28BB | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | D4 | Q16 | C7×D4 | C7×D4 | C7×D4 | C7×Q16 | C8.C22 | C7×C8.C22 |
| kernel | C7×C22⋊Q16 | C7×C22⋊C8 | C7×Q8⋊C4 | C7×C22⋊Q8 | C14×Q16 | Q8×C2×C14 | C22⋊Q16 | C22⋊C8 | Q8⋊C4 | C22⋊Q8 | C2×Q16 | C22×Q8 | C2×C28 | C7×Q8 | C22×C14 | C2×C14 | C2×C4 | Q8 | C23 | C22 | C14 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 6 | 6 | 12 | 6 | 12 | 6 | 1 | 4 | 1 | 4 | 6 | 24 | 6 | 24 | 1 | 6 |
Matrix representation of C7×C22⋊Q16 ►in GL4(𝔽113) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 109 | 0 |
| 0 | 0 | 0 | 109 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 18 | 102 |
| 0 | 0 | 18 | 33 |
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 95 |
| 0 | 0 | 94 | 83 |
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,109,0,0,0,0,109],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,112,0,0,0,0,0,18,18,0,0,102,33],[112,0,0,0,0,1,0,0,0,0,30,94,0,0,95,83] >;
C7×C22⋊Q16 in GAP, Magma, Sage, TeX
C_7\times C_2^2\rtimes Q_{16} % in TeX
G:=Group("C7xC2^2:Q16"); // GroupNames label
G:=SmallGroup(448,859);
// by ID
G=gap.SmallGroup(448,859);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,1576,2438,9804,4911,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations