direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D52, C52⋊5D4, C42⋊4D13, C13⋊2(C4×D4), (C4×C52)⋊7C2, C52⋊7(C2×C4), C4⋊2(C4×D13), D26⋊2(C2×C4), C2.1(C2×D52), C26.2(C2×D4), C52⋊3C4⋊16C2, (C2×C4).75D26, (C2×D52).11C2, C26.4(C4○D4), D26⋊C4⋊17C2, (C2×C26).14C23, (C2×C52).86C22, C26.17(C22×C4), C2.3(D52⋊5C2), C22.11(C22×D13), (C2×Dic13).28C22, (C22×D13).18C22, (C2×C4×D13)⋊7C2, C2.6(C2×C4×D13), SmallGroup(416,94)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D52
G = < a,b,c | a4=b52=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 688 in 94 conjugacy classes, 45 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, C23, C13, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D13, C26, C4×D4, Dic13, C52, C52, D26, D26, C2×C26, C4×D13, D52, C2×Dic13, C2×C52, C22×D13, C52⋊3C4, D26⋊C4, C4×C52, C2×C4×D13, C2×D52, C4×D52
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, D13, C4×D4, D26, C4×D13, D52, C22×D13, C2×C4×D13, C2×D52, D52⋊5C2, C4×D52
►On 208 points
Generators in S208
(1 142 76 179)(2 143 77 180)(3 144 78 181)(4 145 79 182)(5 146 80 183)(6 147 81 184)(7 148 82 185)(8 149 83 186)(9 150 84 187)(10 151 85 188)(11 152 86 189)(12 153 87 190)(13 154 88 191)(14 155 89 192)(15 156 90 193)(16 105 91 194)(17 106 92 195)(18 107 93 196)(19 108 94 197)(20 109 95 198)(21 110 96 199)(22 111 97 200)(23 112 98 201)(24 113 99 202)(25 114 100 203)(26 115 101 204)(27 116 102 205)(28 117 103 206)(29 118 104 207)(30 119 53 208)(31 120 54 157)(32 121 55 158)(33 122 56 159)(34 123 57 160)(35 124 58 161)(36 125 59 162)(37 126 60 163)(38 127 61 164)(39 128 62 165)(40 129 63 166)(41 130 64 167)(42 131 65 168)(43 132 66 169)(44 133 67 170)(45 134 68 171)(46 135 69 172)(47 136 70 173)(48 137 71 174)(49 138 72 175)(50 139 73 176)(51 140 74 177)(52 141 75 178)
(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(53 85)(54 84)(55 83)(56 82)(57 81)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(105 113)(106 112)(107 111)(108 110)(114 156)(115 155)(116 154)(117 153)(118 152)(119 151)(120 150)(121 149)(122 148)(123 147)(124 146)(125 145)(126 144)(127 143)(128 142)(129 141)(130 140)(131 139)(132 138)(133 137)(134 136)(157 187)(158 186)(159 185)(160 184)(161 183)(162 182)(163 181)(164 180)(165 179)(166 178)(167 177)(168 176)(169 175)(170 174)(171 173)(188 208)(189 207)(190 206)(191 205)(192 204)(193 203)(194 202)(195 201)(196 200)(197 199)
G:=sub<Sym(208)| (1,142,76,179)(2,143,77,180)(3,144,78,181)(4,145,79,182)(5,146,80,183)(6,147,81,184)(7,148,82,185)(8,149,83,186)(9,150,84,187)(10,151,85,188)(11,152,86,189)(12,153,87,190)(13,154,88,191)(14,155,89,192)(15,156,90,193)(16,105,91,194)(17,106,92,195)(18,107,93,196)(19,108,94,197)(20,109,95,198)(21,110,96,199)(22,111,97,200)(23,112,98,201)(24,113,99,202)(25,114,100,203)(26,115,101,204)(27,116,102,205)(28,117,103,206)(29,118,104,207)(30,119,53,208)(31,120,54,157)(32,121,55,158)(33,122,56,159)(34,123,57,160)(35,124,58,161)(36,125,59,162)(37,126,60,163)(38,127,61,164)(39,128,62,165)(40,129,63,166)(41,130,64,167)(42,131,65,168)(43,132,66,169)(44,133,67,170)(45,134,68,171)(46,135,69,172)(47,136,70,173)(48,137,71,174)(49,138,72,175)(50,139,73,176)(51,140,74,177)(52,141,75,178), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(105,113)(106,112)(107,111)(108,110)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150)(121,149)(122,148)(123,147)(124,146)(125,145)(126,144)(127,143)(128,142)(129,141)(130,140)(131,139)(132,138)(133,137)(134,136)(157,187)(158,186)(159,185)(160,184)(161,183)(162,182)(163,181)(164,180)(165,179)(166,178)(167,177)(168,176)(169,175)(170,174)(171,173)(188,208)(189,207)(190,206)(191,205)(192,204)(193,203)(194,202)(195,201)(196,200)(197,199)>;
G:=Group( (1,142,76,179)(2,143,77,180)(3,144,78,181)(4,145,79,182)(5,146,80,183)(6,147,81,184)(7,148,82,185)(8,149,83,186)(9,150,84,187)(10,151,85,188)(11,152,86,189)(12,153,87,190)(13,154,88,191)(14,155,89,192)(15,156,90,193)(16,105,91,194)(17,106,92,195)(18,107,93,196)(19,108,94,197)(20,109,95,198)(21,110,96,199)(22,111,97,200)(23,112,98,201)(24,113,99,202)(25,114,100,203)(26,115,101,204)(27,116,102,205)(28,117,103,206)(29,118,104,207)(30,119,53,208)(31,120,54,157)(32,121,55,158)(33,122,56,159)(34,123,57,160)(35,124,58,161)(36,125,59,162)(37,126,60,163)(38,127,61,164)(39,128,62,165)(40,129,63,166)(41,130,64,167)(42,131,65,168)(43,132,66,169)(44,133,67,170)(45,134,68,171)(46,135,69,172)(47,136,70,173)(48,137,71,174)(49,138,72,175)(50,139,73,176)(51,140,74,177)(52,141,75,178), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(105,113)(106,112)(107,111)(108,110)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150)(121,149)(122,148)(123,147)(124,146)(125,145)(126,144)(127,143)(128,142)(129,141)(130,140)(131,139)(132,138)(133,137)(134,136)(157,187)(158,186)(159,185)(160,184)(161,183)(162,182)(163,181)(164,180)(165,179)(166,178)(167,177)(168,176)(169,175)(170,174)(171,173)(188,208)(189,207)(190,206)(191,205)(192,204)(193,203)(194,202)(195,201)(196,200)(197,199) );
G=PermutationGroup([[(1,142,76,179),(2,143,77,180),(3,144,78,181),(4,145,79,182),(5,146,80,183),(6,147,81,184),(7,148,82,185),(8,149,83,186),(9,150,84,187),(10,151,85,188),(11,152,86,189),(12,153,87,190),(13,154,88,191),(14,155,89,192),(15,156,90,193),(16,105,91,194),(17,106,92,195),(18,107,93,196),(19,108,94,197),(20,109,95,198),(21,110,96,199),(22,111,97,200),(23,112,98,201),(24,113,99,202),(25,114,100,203),(26,115,101,204),(27,116,102,205),(28,117,103,206),(29,118,104,207),(30,119,53,208),(31,120,54,157),(32,121,55,158),(33,122,56,159),(34,123,57,160),(35,124,58,161),(36,125,59,162),(37,126,60,163),(38,127,61,164),(39,128,62,165),(40,129,63,166),(41,130,64,167),(42,131,65,168),(43,132,66,169),(44,133,67,170),(45,134,68,171),(46,135,69,172),(47,136,70,173),(48,137,71,174),(49,138,72,175),(50,139,73,176),(51,140,74,177),(52,141,75,178)], [(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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,24),(17,23),(18,22),(19,21),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(53,85),(54,84),(55,83),(56,82),(57,81),(58,80),(59,79),(60,78),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(86,104),(87,103),(88,102),(89,101),(90,100),(91,99),(92,98),(93,97),(94,96),(105,113),(106,112),(107,111),(108,110),(114,156),(115,155),(116,154),(117,153),(118,152),(119,151),(120,150),(121,149),(122,148),(123,147),(124,146),(125,145),(126,144),(127,143),(128,142),(129,141),(130,140),(131,139),(132,138),(133,137),(134,136),(157,187),(158,186),(159,185),(160,184),(161,183),(162,182),(163,181),(164,180),(165,179),(166,178),(167,177),(168,176),(169,175),(170,174),(171,173),(188,208),(189,207),(190,206),(191,205),(192,204),(193,203),(194,202),(195,201),(196,200),(197,199)]])
116 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 26 | 26 | 26 | 26 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | C4○D4 | D13 | D26 | C4×D13 | D52 | D52⋊5C2 |
| kernel | C4×D52 | C52⋊3C4 | D26⋊C4 | C4×C52 | C2×C4×D13 | C2×D52 | D52 | C52 | C26 | C42 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 2 | 2 | 6 | 18 | 24 | 24 | 24 |
Matrix representation of C4×D52 ►in GL3(𝔽53) generated by
| 30 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 52 | 0 | 0 |
| 0 | 49 | 2 |
| 0 | 51 | 14 |
| 52 | 0 | 0 |
| 0 | 9 | 26 |
| 0 | 1 | 44 |
G:=sub<GL(3,GF(53))| [30,0,0,0,1,0,0,0,1],[52,0,0,0,49,51,0,2,14],[52,0,0,0,9,1,0,26,44] >;
C4×D52 in GAP, Magma, Sage, TeX
C_4\times D_{52} % in TeX
G:=Group("C4xD52"); // GroupNames label
G:=SmallGroup(416,94);
// by ID
G=gap.SmallGroup(416,94);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,103,50,13829]);
// Polycyclic
G:=Group<a,b,c|a^4=b^52=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations