direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×C22⋊Q8, C52.62D4, C4⋊C4⋊3C26, (C2×C26)⋊2Q8, C22⋊(Q8×C13), (C2×Q8)⋊1C26, (Q8×C26)⋊8C2, C2.6(D4×C26), C2.3(Q8×C26), C26.69(C2×D4), C4.13(D4×C13), C26.20(C2×Q8), C22⋊C4.1C26, (C22×C4).5C26, C23.9(C2×C26), C26.42(C4○D4), (C22×C52).15C2, (C2×C26).77C23, (C2×C52).124C22, (C22×C26).28C22, C22.12(C22×C26), (C13×C4⋊C4)⋊12C2, (C2×C4).4(C2×C26), C2.5(C13×C4○D4), (C13×C22⋊C4).4C2, SmallGroup(416,183)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C22⋊Q8
G = < a,b,c,d,e | a13=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 100 in 74 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, Q8, C23, C13, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C26, C26, C22⋊Q8, C52, C52, C2×C26, C2×C26, C2×C26, C2×C52, C2×C52, C2×C52, Q8×C13, C22×C26, C13×C22⋊C4, C13×C4⋊C4, C13×C4⋊C4, C22×C52, Q8×C26, C13×C22⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C13, C2×D4, C2×Q8, C4○D4, C26, C22⋊Q8, C2×C26, D4×C13, Q8×C13, C22×C26, D4×C26, Q8×C26, C13×C4○D4, C13×C22⋊Q8
►On 208 points
Generators in S208
(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)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 49)(15 50)(16 51)(17 52)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 79)(37 80)(38 81)(39 82)(66 192)(67 193)(68 194)(69 195)(70 183)(71 184)(72 185)(73 186)(74 187)(75 188)(76 189)(77 190)(78 191)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 105)(104 106)(118 175)(119 176)(120 177)(121 178)(122 179)(123 180)(124 181)(125 182)(126 170)(127 171)(128 172)(129 173)(130 174)(131 163)(132 164)(133 165)(134 166)(135 167)(136 168)(137 169)(138 157)(139 158)(140 159)(141 160)(142 161)(143 162)(144 207)(145 208)(146 196)(147 197)(148 198)(149 199)(150 200)(151 201)(152 202)(153 203)(154 204)(155 205)(156 206)
(1 187)(2 188)(3 189)(4 190)(5 191)(6 192)(7 193)(8 194)(9 195)(10 183)(11 184)(12 185)(13 186)(14 161)(15 162)(16 163)(17 164)(18 165)(19 166)(20 167)(21 168)(22 169)(23 157)(24 158)(25 159)(26 160)(27 151)(28 152)(29 153)(30 154)(31 155)(32 156)(33 144)(34 145)(35 146)(36 147)(37 148)(38 149)(39 150)(40 133)(41 134)(42 135)(43 136)(44 137)(45 138)(46 139)(47 140)(48 141)(49 142)(50 143)(51 131)(52 132)(53 68)(54 69)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 66)(65 67)(79 197)(80 198)(81 199)(82 200)(83 201)(84 202)(85 203)(86 204)(87 205)(88 206)(89 207)(90 208)(91 196)(92 128)(93 129)(94 130)(95 118)(96 119)(97 120)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(105 170)(106 171)(107 172)(108 173)(109 174)(110 175)(111 176)(112 177)(113 178)(114 179)(115 180)(116 181)(117 182)
(1 85 59 29)(2 86 60 30)(3 87 61 31)(4 88 62 32)(5 89 63 33)(6 90 64 34)(7 91 65 35)(8 79 53 36)(9 80 54 37)(10 81 55 38)(11 82 56 39)(12 83 57 27)(13 84 58 28)(14 102 142 182)(15 103 143 170)(16 104 131 171)(17 92 132 172)(18 93 133 173)(19 94 134 174)(20 95 135 175)(21 96 136 176)(22 97 137 177)(23 98 138 178)(24 99 139 179)(25 100 140 180)(26 101 141 181)(40 108 165 129)(41 109 166 130)(42 110 167 118)(43 111 168 119)(44 112 169 120)(45 113 157 121)(46 114 158 122)(47 115 159 123)(48 116 160 124)(49 117 161 125)(50 105 162 126)(51 106 163 127)(52 107 164 128)(66 145 192 208)(67 146 193 196)(68 147 194 197)(69 148 195 198)(70 149 183 199)(71 150 184 200)(72 151 185 201)(73 152 186 202)(74 153 187 203)(75 154 188 204)(76 155 189 205)(77 156 190 206)(78 144 191 207)
(1 46 59 158)(2 47 60 159)(3 48 61 160)(4 49 62 161)(5 50 63 162)(6 51 64 163)(7 52 65 164)(8 40 53 165)(9 41 54 166)(10 42 55 167)(11 43 56 168)(12 44 57 169)(13 45 58 157)(14 190 142 77)(15 191 143 78)(16 192 131 66)(17 193 132 67)(18 194 133 68)(19 195 134 69)(20 183 135 70)(21 184 136 71)(22 185 137 72)(23 186 138 73)(24 187 139 74)(25 188 140 75)(26 189 141 76)(27 112 83 120)(28 113 84 121)(29 114 85 122)(30 115 86 123)(31 116 87 124)(32 117 88 125)(33 105 89 126)(34 106 90 127)(35 107 91 128)(36 108 79 129)(37 109 80 130)(38 110 81 118)(39 111 82 119)(92 146 172 196)(93 147 173 197)(94 148 174 198)(95 149 175 199)(96 150 176 200)(97 151 177 201)(98 152 178 202)(99 153 179 203)(100 154 180 204)(101 155 181 205)(102 156 182 206)(103 144 170 207)(104 145 171 208)
G:=sub<Sym(208)| (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), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,49)(15,50)(16,51)(17,52)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,79)(37,80)(38,81)(39,82)(66,192)(67,193)(68,194)(69,195)(70,183)(71,184)(72,185)(73,186)(74,187)(75,188)(76,189)(77,190)(78,191)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,105)(104,106)(118,175)(119,176)(120,177)(121,178)(122,179)(123,180)(124,181)(125,182)(126,170)(127,171)(128,172)(129,173)(130,174)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)(137,169)(138,157)(139,158)(140,159)(141,160)(142,161)(143,162)(144,207)(145,208)(146,196)(147,197)(148,198)(149,199)(150,200)(151,201)(152,202)(153,203)(154,204)(155,205)(156,206), (1,187)(2,188)(3,189)(4,190)(5,191)(6,192)(7,193)(8,194)(9,195)(10,183)(11,184)(12,185)(13,186)(14,161)(15,162)(16,163)(17,164)(18,165)(19,166)(20,167)(21,168)(22,169)(23,157)(24,158)(25,159)(26,160)(27,151)(28,152)(29,153)(30,154)(31,155)(32,156)(33,144)(34,145)(35,146)(36,147)(37,148)(38,149)(39,150)(40,133)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,141)(49,142)(50,143)(51,131)(52,132)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,66)(65,67)(79,197)(80,198)(81,199)(82,200)(83,201)(84,202)(85,203)(86,204)(87,205)(88,206)(89,207)(90,208)(91,196)(92,128)(93,129)(94,130)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,177)(113,178)(114,179)(115,180)(116,181)(117,182), (1,85,59,29)(2,86,60,30)(3,87,61,31)(4,88,62,32)(5,89,63,33)(6,90,64,34)(7,91,65,35)(8,79,53,36)(9,80,54,37)(10,81,55,38)(11,82,56,39)(12,83,57,27)(13,84,58,28)(14,102,142,182)(15,103,143,170)(16,104,131,171)(17,92,132,172)(18,93,133,173)(19,94,134,174)(20,95,135,175)(21,96,136,176)(22,97,137,177)(23,98,138,178)(24,99,139,179)(25,100,140,180)(26,101,141,181)(40,108,165,129)(41,109,166,130)(42,110,167,118)(43,111,168,119)(44,112,169,120)(45,113,157,121)(46,114,158,122)(47,115,159,123)(48,116,160,124)(49,117,161,125)(50,105,162,126)(51,106,163,127)(52,107,164,128)(66,145,192,208)(67,146,193,196)(68,147,194,197)(69,148,195,198)(70,149,183,199)(71,150,184,200)(72,151,185,201)(73,152,186,202)(74,153,187,203)(75,154,188,204)(76,155,189,205)(77,156,190,206)(78,144,191,207), (1,46,59,158)(2,47,60,159)(3,48,61,160)(4,49,62,161)(5,50,63,162)(6,51,64,163)(7,52,65,164)(8,40,53,165)(9,41,54,166)(10,42,55,167)(11,43,56,168)(12,44,57,169)(13,45,58,157)(14,190,142,77)(15,191,143,78)(16,192,131,66)(17,193,132,67)(18,194,133,68)(19,195,134,69)(20,183,135,70)(21,184,136,71)(22,185,137,72)(23,186,138,73)(24,187,139,74)(25,188,140,75)(26,189,141,76)(27,112,83,120)(28,113,84,121)(29,114,85,122)(30,115,86,123)(31,116,87,124)(32,117,88,125)(33,105,89,126)(34,106,90,127)(35,107,91,128)(36,108,79,129)(37,109,80,130)(38,110,81,118)(39,111,82,119)(92,146,172,196)(93,147,173,197)(94,148,174,198)(95,149,175,199)(96,150,176,200)(97,151,177,201)(98,152,178,202)(99,153,179,203)(100,154,180,204)(101,155,181,205)(102,156,182,206)(103,144,170,207)(104,145,171,208)>;
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), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,49)(15,50)(16,51)(17,52)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,79)(37,80)(38,81)(39,82)(66,192)(67,193)(68,194)(69,195)(70,183)(71,184)(72,185)(73,186)(74,187)(75,188)(76,189)(77,190)(78,191)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,105)(104,106)(118,175)(119,176)(120,177)(121,178)(122,179)(123,180)(124,181)(125,182)(126,170)(127,171)(128,172)(129,173)(130,174)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)(137,169)(138,157)(139,158)(140,159)(141,160)(142,161)(143,162)(144,207)(145,208)(146,196)(147,197)(148,198)(149,199)(150,200)(151,201)(152,202)(153,203)(154,204)(155,205)(156,206), (1,187)(2,188)(3,189)(4,190)(5,191)(6,192)(7,193)(8,194)(9,195)(10,183)(11,184)(12,185)(13,186)(14,161)(15,162)(16,163)(17,164)(18,165)(19,166)(20,167)(21,168)(22,169)(23,157)(24,158)(25,159)(26,160)(27,151)(28,152)(29,153)(30,154)(31,155)(32,156)(33,144)(34,145)(35,146)(36,147)(37,148)(38,149)(39,150)(40,133)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,141)(49,142)(50,143)(51,131)(52,132)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,66)(65,67)(79,197)(80,198)(81,199)(82,200)(83,201)(84,202)(85,203)(86,204)(87,205)(88,206)(89,207)(90,208)(91,196)(92,128)(93,129)(94,130)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,177)(113,178)(114,179)(115,180)(116,181)(117,182), (1,85,59,29)(2,86,60,30)(3,87,61,31)(4,88,62,32)(5,89,63,33)(6,90,64,34)(7,91,65,35)(8,79,53,36)(9,80,54,37)(10,81,55,38)(11,82,56,39)(12,83,57,27)(13,84,58,28)(14,102,142,182)(15,103,143,170)(16,104,131,171)(17,92,132,172)(18,93,133,173)(19,94,134,174)(20,95,135,175)(21,96,136,176)(22,97,137,177)(23,98,138,178)(24,99,139,179)(25,100,140,180)(26,101,141,181)(40,108,165,129)(41,109,166,130)(42,110,167,118)(43,111,168,119)(44,112,169,120)(45,113,157,121)(46,114,158,122)(47,115,159,123)(48,116,160,124)(49,117,161,125)(50,105,162,126)(51,106,163,127)(52,107,164,128)(66,145,192,208)(67,146,193,196)(68,147,194,197)(69,148,195,198)(70,149,183,199)(71,150,184,200)(72,151,185,201)(73,152,186,202)(74,153,187,203)(75,154,188,204)(76,155,189,205)(77,156,190,206)(78,144,191,207), (1,46,59,158)(2,47,60,159)(3,48,61,160)(4,49,62,161)(5,50,63,162)(6,51,64,163)(7,52,65,164)(8,40,53,165)(9,41,54,166)(10,42,55,167)(11,43,56,168)(12,44,57,169)(13,45,58,157)(14,190,142,77)(15,191,143,78)(16,192,131,66)(17,193,132,67)(18,194,133,68)(19,195,134,69)(20,183,135,70)(21,184,136,71)(22,185,137,72)(23,186,138,73)(24,187,139,74)(25,188,140,75)(26,189,141,76)(27,112,83,120)(28,113,84,121)(29,114,85,122)(30,115,86,123)(31,116,87,124)(32,117,88,125)(33,105,89,126)(34,106,90,127)(35,107,91,128)(36,108,79,129)(37,109,80,130)(38,110,81,118)(39,111,82,119)(92,146,172,196)(93,147,173,197)(94,148,174,198)(95,149,175,199)(96,150,176,200)(97,151,177,201)(98,152,178,202)(99,153,179,203)(100,154,180,204)(101,155,181,205)(102,156,182,206)(103,144,170,207)(104,145,171,208) );
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)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,65),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,49),(15,50),(16,51),(17,52),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,79),(37,80),(38,81),(39,82),(66,192),(67,193),(68,194),(69,195),(70,183),(71,184),(72,185),(73,186),(74,187),(75,188),(76,189),(77,190),(78,191),(92,107),(93,108),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,115),(101,116),(102,117),(103,105),(104,106),(118,175),(119,176),(120,177),(121,178),(122,179),(123,180),(124,181),(125,182),(126,170),(127,171),(128,172),(129,173),(130,174),(131,163),(132,164),(133,165),(134,166),(135,167),(136,168),(137,169),(138,157),(139,158),(140,159),(141,160),(142,161),(143,162),(144,207),(145,208),(146,196),(147,197),(148,198),(149,199),(150,200),(151,201),(152,202),(153,203),(154,204),(155,205),(156,206)], [(1,187),(2,188),(3,189),(4,190),(5,191),(6,192),(7,193),(8,194),(9,195),(10,183),(11,184),(12,185),(13,186),(14,161),(15,162),(16,163),(17,164),(18,165),(19,166),(20,167),(21,168),(22,169),(23,157),(24,158),(25,159),(26,160),(27,151),(28,152),(29,153),(30,154),(31,155),(32,156),(33,144),(34,145),(35,146),(36,147),(37,148),(38,149),(39,150),(40,133),(41,134),(42,135),(43,136),(44,137),(45,138),(46,139),(47,140),(48,141),(49,142),(50,143),(51,131),(52,132),(53,68),(54,69),(55,70),(56,71),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,78),(64,66),(65,67),(79,197),(80,198),(81,199),(82,200),(83,201),(84,202),(85,203),(86,204),(87,205),(88,206),(89,207),(90,208),(91,196),(92,128),(93,129),(94,130),(95,118),(96,119),(97,120),(98,121),(99,122),(100,123),(101,124),(102,125),(103,126),(104,127),(105,170),(106,171),(107,172),(108,173),(109,174),(110,175),(111,176),(112,177),(113,178),(114,179),(115,180),(116,181),(117,182)], [(1,85,59,29),(2,86,60,30),(3,87,61,31),(4,88,62,32),(5,89,63,33),(6,90,64,34),(7,91,65,35),(8,79,53,36),(9,80,54,37),(10,81,55,38),(11,82,56,39),(12,83,57,27),(13,84,58,28),(14,102,142,182),(15,103,143,170),(16,104,131,171),(17,92,132,172),(18,93,133,173),(19,94,134,174),(20,95,135,175),(21,96,136,176),(22,97,137,177),(23,98,138,178),(24,99,139,179),(25,100,140,180),(26,101,141,181),(40,108,165,129),(41,109,166,130),(42,110,167,118),(43,111,168,119),(44,112,169,120),(45,113,157,121),(46,114,158,122),(47,115,159,123),(48,116,160,124),(49,117,161,125),(50,105,162,126),(51,106,163,127),(52,107,164,128),(66,145,192,208),(67,146,193,196),(68,147,194,197),(69,148,195,198),(70,149,183,199),(71,150,184,200),(72,151,185,201),(73,152,186,202),(74,153,187,203),(75,154,188,204),(76,155,189,205),(77,156,190,206),(78,144,191,207)], [(1,46,59,158),(2,47,60,159),(3,48,61,160),(4,49,62,161),(5,50,63,162),(6,51,64,163),(7,52,65,164),(8,40,53,165),(9,41,54,166),(10,42,55,167),(11,43,56,168),(12,44,57,169),(13,45,58,157),(14,190,142,77),(15,191,143,78),(16,192,131,66),(17,193,132,67),(18,194,133,68),(19,195,134,69),(20,183,135,70),(21,184,136,71),(22,185,137,72),(23,186,138,73),(24,187,139,74),(25,188,140,75),(26,189,141,76),(27,112,83,120),(28,113,84,121),(29,114,85,122),(30,115,86,123),(31,116,87,124),(32,117,88,125),(33,105,89,126),(34,106,90,127),(35,107,91,128),(36,108,79,129),(37,109,80,130),(38,110,81,118),(39,111,82,119),(92,146,172,196),(93,147,173,197),(94,148,174,198),(95,149,175,199),(96,150,176,200),(97,151,177,201),(98,152,178,202),(99,153,179,203),(100,154,180,204),(101,155,181,205),(102,156,182,206),(103,144,170,207),(104,145,171,208)]])
182 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 13A | ··· | 13L | 26A | ··· | 26AJ | 26AK | ··· | 26BH | 52A | ··· | 52AV | 52AW | ··· | 52CR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
182 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C13 | C26 | C26 | C26 | C26 | D4 | Q8 | C4○D4 | D4×C13 | Q8×C13 | C13×C4○D4 |
| kernel | C13×C22⋊Q8 | C13×C22⋊C4 | C13×C4⋊C4 | C22×C52 | Q8×C26 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C52 | C2×C26 | C26 | C4 | C22 | C2 |
| # reps | 1 | 2 | 3 | 1 | 1 | 12 | 24 | 36 | 12 | 12 | 2 | 2 | 2 | 24 | 24 | 24 |
Matrix representation of C13×C22⋊Q8 ►in GL4(𝔽53) generated by
| 42 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 52 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 52 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 30 | 39 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 4 | 18 | 0 | 0 |
| 2 | 49 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(53))| [42,0,0,0,0,42,0,0,0,0,16,0,0,0,0,16],[52,0,0,0,4,1,0,0,0,0,1,0,0,0,0,52],[52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[30,0,0,0,39,23,0,0,0,0,52,0,0,0,0,52],[4,2,0,0,18,49,0,0,0,0,0,1,0,0,1,0] >;
C13×C22⋊Q8 in GAP, Magma, Sage, TeX
C_{13}\times C_2^2\rtimes Q_8 % in TeX
G:=Group("C13xC2^2:Q8"); // GroupNames label
G:=SmallGroup(416,183);
// by ID
G=gap.SmallGroup(416,183);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,624,1273,631,3818]);
// Polycyclic
G:=Group<a,b,c,d,e|a^13=b^2=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations