direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×C4.4D4, C42⋊5C26, C52.39D4, (C4×C52)⋊12C2, (Q8×C26)⋊9C2, (C2×Q8)⋊2C26, C2.8(D4×C26), C4.4(D4×C13), C22⋊C4⋊5C26, (C2×D4).5C26, C26.71(C2×D4), (D4×C26).12C2, C23.2(C2×C26), C26.44(C4○D4), (C2×C26).79C23, (C2×C52).66C22, (C22×C26).2C22, C22.14(C22×C26), (C2×C4).6(C2×C26), C2.7(C13×C4○D4), (C13×C22⋊C4)⋊13C2, SmallGroup(416,185)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C4.4D4
G = < a,b,c,d | a13=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 116 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C13, C42, C22⋊C4, C2×D4, C2×Q8, C26, C26, C26, C4.4D4, C52, C52, C2×C26, C2×C26, C2×C52, C2×C52, D4×C13, Q8×C13, C22×C26, C4×C52, C13×C22⋊C4, D4×C26, Q8×C26, C13×C4.4D4
Quotients: C1, C2, C22, D4, C23, C13, C2×D4, C4○D4, C26, C4.4D4, C2×C26, D4×C13, C22×C26, D4×C26, C13×C4○D4, C13×C4.4D4
►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 82 42 62)(2 83 43 63)(3 84 44 64)(4 85 45 65)(5 86 46 53)(6 87 47 54)(7 88 48 55)(8 89 49 56)(9 90 50 57)(10 91 51 58)(11 79 52 59)(12 80 40 60)(13 81 41 61)(14 195 103 202)(15 183 104 203)(16 184 92 204)(17 185 93 205)(18 186 94 206)(19 187 95 207)(20 188 96 208)(21 189 97 196)(22 190 98 197)(23 191 99 198)(24 192 100 199)(25 193 101 200)(26 194 102 201)(27 171 163 137)(28 172 164 138)(29 173 165 139)(30 174 166 140)(31 175 167 141)(32 176 168 142)(33 177 169 143)(34 178 157 131)(35 179 158 132)(36 180 159 133)(37 181 160 134)(38 182 161 135)(39 170 162 136)(66 130 107 144)(67 118 108 145)(68 119 109 146)(69 120 110 147)(70 121 111 148)(71 122 112 149)(72 123 113 150)(73 124 114 151)(74 125 115 152)(75 126 116 153)(76 127 117 154)(77 128 105 155)(78 129 106 156)
(1 166 125 22)(2 167 126 23)(3 168 127 24)(4 169 128 25)(5 157 129 26)(6 158 130 14)(7 159 118 15)(8 160 119 16)(9 161 120 17)(10 162 121 18)(11 163 122 19)(12 164 123 20)(13 165 124 21)(27 149 95 52)(28 150 96 40)(29 151 97 41)(30 152 98 42)(31 153 99 43)(32 154 100 44)(33 155 101 45)(34 156 102 46)(35 144 103 47)(36 145 104 48)(37 146 92 49)(38 147 93 50)(39 148 94 51)(53 178 78 201)(54 179 66 202)(55 180 67 203)(56 181 68 204)(57 182 69 205)(58 170 70 206)(59 171 71 207)(60 172 72 208)(61 173 73 196)(62 174 74 197)(63 175 75 198)(64 176 76 199)(65 177 77 200)(79 137 112 187)(80 138 113 188)(81 139 114 189)(82 140 115 190)(83 141 116 191)(84 142 117 192)(85 143 105 193)(86 131 106 194)(87 132 107 195)(88 133 108 183)(89 134 109 184)(90 135 110 185)(91 136 111 186)
(1 190 42 197)(2 191 43 198)(3 192 44 199)(4 193 45 200)(5 194 46 201)(6 195 47 202)(7 183 48 203)(8 184 49 204)(9 185 50 205)(10 186 51 206)(11 187 52 207)(12 188 40 208)(13 189 41 196)(14 54 103 87)(15 55 104 88)(16 56 92 89)(17 57 93 90)(18 58 94 91)(19 59 95 79)(20 60 96 80)(21 61 97 81)(22 62 98 82)(23 63 99 83)(24 64 100 84)(25 65 101 85)(26 53 102 86)(27 112 163 71)(28 113 164 72)(29 114 165 73)(30 115 166 74)(31 116 167 75)(32 117 168 76)(33 105 169 77)(34 106 157 78)(35 107 158 66)(36 108 159 67)(37 109 160 68)(38 110 161 69)(39 111 162 70)(118 133 145 180)(119 134 146 181)(120 135 147 182)(121 136 148 170)(122 137 149 171)(123 138 150 172)(124 139 151 173)(125 140 152 174)(126 141 153 175)(127 142 154 176)(128 143 155 177)(129 131 156 178)(130 132 144 179)
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), (1,82,42,62)(2,83,43,63)(3,84,44,64)(4,85,45,65)(5,86,46,53)(6,87,47,54)(7,88,48,55)(8,89,49,56)(9,90,50,57)(10,91,51,58)(11,79,52,59)(12,80,40,60)(13,81,41,61)(14,195,103,202)(15,183,104,203)(16,184,92,204)(17,185,93,205)(18,186,94,206)(19,187,95,207)(20,188,96,208)(21,189,97,196)(22,190,98,197)(23,191,99,198)(24,192,100,199)(25,193,101,200)(26,194,102,201)(27,171,163,137)(28,172,164,138)(29,173,165,139)(30,174,166,140)(31,175,167,141)(32,176,168,142)(33,177,169,143)(34,178,157,131)(35,179,158,132)(36,180,159,133)(37,181,160,134)(38,182,161,135)(39,170,162,136)(66,130,107,144)(67,118,108,145)(68,119,109,146)(69,120,110,147)(70,121,111,148)(71,122,112,149)(72,123,113,150)(73,124,114,151)(74,125,115,152)(75,126,116,153)(76,127,117,154)(77,128,105,155)(78,129,106,156), (1,166,125,22)(2,167,126,23)(3,168,127,24)(4,169,128,25)(5,157,129,26)(6,158,130,14)(7,159,118,15)(8,160,119,16)(9,161,120,17)(10,162,121,18)(11,163,122,19)(12,164,123,20)(13,165,124,21)(27,149,95,52)(28,150,96,40)(29,151,97,41)(30,152,98,42)(31,153,99,43)(32,154,100,44)(33,155,101,45)(34,156,102,46)(35,144,103,47)(36,145,104,48)(37,146,92,49)(38,147,93,50)(39,148,94,51)(53,178,78,201)(54,179,66,202)(55,180,67,203)(56,181,68,204)(57,182,69,205)(58,170,70,206)(59,171,71,207)(60,172,72,208)(61,173,73,196)(62,174,74,197)(63,175,75,198)(64,176,76,199)(65,177,77,200)(79,137,112,187)(80,138,113,188)(81,139,114,189)(82,140,115,190)(83,141,116,191)(84,142,117,192)(85,143,105,193)(86,131,106,194)(87,132,107,195)(88,133,108,183)(89,134,109,184)(90,135,110,185)(91,136,111,186), (1,190,42,197)(2,191,43,198)(3,192,44,199)(4,193,45,200)(5,194,46,201)(6,195,47,202)(7,183,48,203)(8,184,49,204)(9,185,50,205)(10,186,51,206)(11,187,52,207)(12,188,40,208)(13,189,41,196)(14,54,103,87)(15,55,104,88)(16,56,92,89)(17,57,93,90)(18,58,94,91)(19,59,95,79)(20,60,96,80)(21,61,97,81)(22,62,98,82)(23,63,99,83)(24,64,100,84)(25,65,101,85)(26,53,102,86)(27,112,163,71)(28,113,164,72)(29,114,165,73)(30,115,166,74)(31,116,167,75)(32,117,168,76)(33,105,169,77)(34,106,157,78)(35,107,158,66)(36,108,159,67)(37,109,160,68)(38,110,161,69)(39,111,162,70)(118,133,145,180)(119,134,146,181)(120,135,147,182)(121,136,148,170)(122,137,149,171)(123,138,150,172)(124,139,151,173)(125,140,152,174)(126,141,153,175)(127,142,154,176)(128,143,155,177)(129,131,156,178)(130,132,144,179)>;
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), (1,82,42,62)(2,83,43,63)(3,84,44,64)(4,85,45,65)(5,86,46,53)(6,87,47,54)(7,88,48,55)(8,89,49,56)(9,90,50,57)(10,91,51,58)(11,79,52,59)(12,80,40,60)(13,81,41,61)(14,195,103,202)(15,183,104,203)(16,184,92,204)(17,185,93,205)(18,186,94,206)(19,187,95,207)(20,188,96,208)(21,189,97,196)(22,190,98,197)(23,191,99,198)(24,192,100,199)(25,193,101,200)(26,194,102,201)(27,171,163,137)(28,172,164,138)(29,173,165,139)(30,174,166,140)(31,175,167,141)(32,176,168,142)(33,177,169,143)(34,178,157,131)(35,179,158,132)(36,180,159,133)(37,181,160,134)(38,182,161,135)(39,170,162,136)(66,130,107,144)(67,118,108,145)(68,119,109,146)(69,120,110,147)(70,121,111,148)(71,122,112,149)(72,123,113,150)(73,124,114,151)(74,125,115,152)(75,126,116,153)(76,127,117,154)(77,128,105,155)(78,129,106,156), (1,166,125,22)(2,167,126,23)(3,168,127,24)(4,169,128,25)(5,157,129,26)(6,158,130,14)(7,159,118,15)(8,160,119,16)(9,161,120,17)(10,162,121,18)(11,163,122,19)(12,164,123,20)(13,165,124,21)(27,149,95,52)(28,150,96,40)(29,151,97,41)(30,152,98,42)(31,153,99,43)(32,154,100,44)(33,155,101,45)(34,156,102,46)(35,144,103,47)(36,145,104,48)(37,146,92,49)(38,147,93,50)(39,148,94,51)(53,178,78,201)(54,179,66,202)(55,180,67,203)(56,181,68,204)(57,182,69,205)(58,170,70,206)(59,171,71,207)(60,172,72,208)(61,173,73,196)(62,174,74,197)(63,175,75,198)(64,176,76,199)(65,177,77,200)(79,137,112,187)(80,138,113,188)(81,139,114,189)(82,140,115,190)(83,141,116,191)(84,142,117,192)(85,143,105,193)(86,131,106,194)(87,132,107,195)(88,133,108,183)(89,134,109,184)(90,135,110,185)(91,136,111,186), (1,190,42,197)(2,191,43,198)(3,192,44,199)(4,193,45,200)(5,194,46,201)(6,195,47,202)(7,183,48,203)(8,184,49,204)(9,185,50,205)(10,186,51,206)(11,187,52,207)(12,188,40,208)(13,189,41,196)(14,54,103,87)(15,55,104,88)(16,56,92,89)(17,57,93,90)(18,58,94,91)(19,59,95,79)(20,60,96,80)(21,61,97,81)(22,62,98,82)(23,63,99,83)(24,64,100,84)(25,65,101,85)(26,53,102,86)(27,112,163,71)(28,113,164,72)(29,114,165,73)(30,115,166,74)(31,116,167,75)(32,117,168,76)(33,105,169,77)(34,106,157,78)(35,107,158,66)(36,108,159,67)(37,109,160,68)(38,110,161,69)(39,111,162,70)(118,133,145,180)(119,134,146,181)(120,135,147,182)(121,136,148,170)(122,137,149,171)(123,138,150,172)(124,139,151,173)(125,140,152,174)(126,141,153,175)(127,142,154,176)(128,143,155,177)(129,131,156,178)(130,132,144,179) );
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)], [(1,82,42,62),(2,83,43,63),(3,84,44,64),(4,85,45,65),(5,86,46,53),(6,87,47,54),(7,88,48,55),(8,89,49,56),(9,90,50,57),(10,91,51,58),(11,79,52,59),(12,80,40,60),(13,81,41,61),(14,195,103,202),(15,183,104,203),(16,184,92,204),(17,185,93,205),(18,186,94,206),(19,187,95,207),(20,188,96,208),(21,189,97,196),(22,190,98,197),(23,191,99,198),(24,192,100,199),(25,193,101,200),(26,194,102,201),(27,171,163,137),(28,172,164,138),(29,173,165,139),(30,174,166,140),(31,175,167,141),(32,176,168,142),(33,177,169,143),(34,178,157,131),(35,179,158,132),(36,180,159,133),(37,181,160,134),(38,182,161,135),(39,170,162,136),(66,130,107,144),(67,118,108,145),(68,119,109,146),(69,120,110,147),(70,121,111,148),(71,122,112,149),(72,123,113,150),(73,124,114,151),(74,125,115,152),(75,126,116,153),(76,127,117,154),(77,128,105,155),(78,129,106,156)], [(1,166,125,22),(2,167,126,23),(3,168,127,24),(4,169,128,25),(5,157,129,26),(6,158,130,14),(7,159,118,15),(8,160,119,16),(9,161,120,17),(10,162,121,18),(11,163,122,19),(12,164,123,20),(13,165,124,21),(27,149,95,52),(28,150,96,40),(29,151,97,41),(30,152,98,42),(31,153,99,43),(32,154,100,44),(33,155,101,45),(34,156,102,46),(35,144,103,47),(36,145,104,48),(37,146,92,49),(38,147,93,50),(39,148,94,51),(53,178,78,201),(54,179,66,202),(55,180,67,203),(56,181,68,204),(57,182,69,205),(58,170,70,206),(59,171,71,207),(60,172,72,208),(61,173,73,196),(62,174,74,197),(63,175,75,198),(64,176,76,199),(65,177,77,200),(79,137,112,187),(80,138,113,188),(81,139,114,189),(82,140,115,190),(83,141,116,191),(84,142,117,192),(85,143,105,193),(86,131,106,194),(87,132,107,195),(88,133,108,183),(89,134,109,184),(90,135,110,185),(91,136,111,186)], [(1,190,42,197),(2,191,43,198),(3,192,44,199),(4,193,45,200),(5,194,46,201),(6,195,47,202),(7,183,48,203),(8,184,49,204),(9,185,50,205),(10,186,51,206),(11,187,52,207),(12,188,40,208),(13,189,41,196),(14,54,103,87),(15,55,104,88),(16,56,92,89),(17,57,93,90),(18,58,94,91),(19,59,95,79),(20,60,96,80),(21,61,97,81),(22,62,98,82),(23,63,99,83),(24,64,100,84),(25,65,101,85),(26,53,102,86),(27,112,163,71),(28,113,164,72),(29,114,165,73),(30,115,166,74),(31,116,167,75),(32,117,168,76),(33,105,169,77),(34,106,157,78),(35,107,158,66),(36,108,159,67),(37,109,160,68),(38,110,161,69),(39,111,162,70),(118,133,145,180),(119,134,146,181),(120,135,147,182),(121,136,148,170),(122,137,149,171),(123,138,150,172),(124,139,151,173),(125,140,152,174),(126,141,153,175),(127,142,154,176),(128,143,155,177),(129,131,156,178),(130,132,144,179)]])
182 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 13A | ··· | 13L | 26A | ··· | 26AJ | 26AK | ··· | 26BH | 52A | ··· | 52BT | 52BU | ··· | 52CR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
182 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C13 | C26 | C26 | C26 | C26 | D4 | C4○D4 | D4×C13 | C13×C4○D4 |
| kernel | C13×C4.4D4 | C4×C52 | C13×C22⋊C4 | D4×C26 | Q8×C26 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C52 | C26 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 12 | 12 | 48 | 12 | 12 | 2 | 4 | 24 | 48 |
Matrix representation of C13×C4.4D4 ►in GL4(𝔽53) generated by
| 42 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 10 | 44 | 0 | 0 |
| 23 | 43 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 35 | 48 | 0 | 0 |
| 1 | 18 | 0 | 0 |
| 0 | 0 | 13 | 51 |
| 0 | 0 | 32 | 40 |
| 23 | 33 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 13 | 51 |
| 0 | 0 | 31 | 40 |
G:=sub<GL(4,GF(53))| [42,0,0,0,0,42,0,0,0,0,36,0,0,0,0,36],[10,23,0,0,44,43,0,0,0,0,1,0,0,0,0,1],[35,1,0,0,48,18,0,0,0,0,13,32,0,0,51,40],[23,0,0,0,33,30,0,0,0,0,13,31,0,0,51,40] >;
C13×C4.4D4 in GAP, Magma, Sage, TeX
C_{13}\times C_4._4D_4 % in TeX
G:=Group("C13xC4.4D4"); // GroupNames label
G:=SmallGroup(416,185);
// by ID
G=gap.SmallGroup(416,185);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,1255,3818,482]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^4=c^4=1,d^2=b^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=b^2*c^-1>;
// generators/relations