metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52.17D4, C23.7D26, (C2×D4).6D13, (D4×C26).5C2, (C2×C4).50D26, C26.48(C2×D4), (C4×Dic13)⋊5C2, C13⋊3(C4.4D4), C4.7(C13⋊D4), C23.D13⋊9C2, (C2×Dic26)⋊10C2, C26.30(C4○D4), (C2×C52).33C22, (C2×C26).51C23, C2.16(D4⋊2D13), (C22×C26).19C22, C22.58(C22×D13), (C2×Dic13).18C22, C2.12(C2×C13⋊D4), SmallGroup(416,157)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.17D4
G = < a,b,c | a52=b4=1, c2=a26, bab-1=a25, cac-1=a-1, cbc-1=a26b-1 >
Subgroups: 392 in 76 conjugacy classes, 33 normal (13 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, Dic13, C52, C2×C26, C2×C26, Dic26, C2×Dic13, C2×C52, D4×C13, C22×C26, C4×Dic13, C23.D13, C2×Dic26, D4×C26, C52.17D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D13, C4.4D4, D26, C13⋊D4, C22×D13, D4⋊2D13, C2×C13⋊D4, C52.17D4
►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)
(1 153 164 97)(2 126 165 70)(3 151 166 95)(4 124 167 68)(5 149 168 93)(6 122 169 66)(7 147 170 91)(8 120 171 64)(9 145 172 89)(10 118 173 62)(11 143 174 87)(12 116 175 60)(13 141 176 85)(14 114 177 58)(15 139 178 83)(16 112 179 56)(17 137 180 81)(18 110 181 54)(19 135 182 79)(20 108 183 104)(21 133 184 77)(22 106 185 102)(23 131 186 75)(24 156 187 100)(25 129 188 73)(26 154 189 98)(27 127 190 71)(28 152 191 96)(29 125 192 69)(30 150 193 94)(31 123 194 67)(32 148 195 92)(33 121 196 65)(34 146 197 90)(35 119 198 63)(36 144 199 88)(37 117 200 61)(38 142 201 86)(39 115 202 59)(40 140 203 84)(41 113 204 57)(42 138 205 82)(43 111 206 55)(44 136 207 80)(45 109 208 53)(46 134 157 78)(47 107 158 103)(48 132 159 76)(49 105 160 101)(50 130 161 74)(51 155 162 99)(52 128 163 72)
(1 140 27 114)(2 139 28 113)(3 138 29 112)(4 137 30 111)(5 136 31 110)(6 135 32 109)(7 134 33 108)(8 133 34 107)(9 132 35 106)(10 131 36 105)(11 130 37 156)(12 129 38 155)(13 128 39 154)(14 127 40 153)(15 126 41 152)(16 125 42 151)(17 124 43 150)(18 123 44 149)(19 122 45 148)(20 121 46 147)(21 120 47 146)(22 119 48 145)(23 118 49 144)(24 117 50 143)(25 116 51 142)(26 115 52 141)(53 169 79 195)(54 168 80 194)(55 167 81 193)(56 166 82 192)(57 165 83 191)(58 164 84 190)(59 163 85 189)(60 162 86 188)(61 161 87 187)(62 160 88 186)(63 159 89 185)(64 158 90 184)(65 157 91 183)(66 208 92 182)(67 207 93 181)(68 206 94 180)(69 205 95 179)(70 204 96 178)(71 203 97 177)(72 202 98 176)(73 201 99 175)(74 200 100 174)(75 199 101 173)(76 198 102 172)(77 197 103 171)(78 196 104 170)
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), (1,153,164,97)(2,126,165,70)(3,151,166,95)(4,124,167,68)(5,149,168,93)(6,122,169,66)(7,147,170,91)(8,120,171,64)(9,145,172,89)(10,118,173,62)(11,143,174,87)(12,116,175,60)(13,141,176,85)(14,114,177,58)(15,139,178,83)(16,112,179,56)(17,137,180,81)(18,110,181,54)(19,135,182,79)(20,108,183,104)(21,133,184,77)(22,106,185,102)(23,131,186,75)(24,156,187,100)(25,129,188,73)(26,154,189,98)(27,127,190,71)(28,152,191,96)(29,125,192,69)(30,150,193,94)(31,123,194,67)(32,148,195,92)(33,121,196,65)(34,146,197,90)(35,119,198,63)(36,144,199,88)(37,117,200,61)(38,142,201,86)(39,115,202,59)(40,140,203,84)(41,113,204,57)(42,138,205,82)(43,111,206,55)(44,136,207,80)(45,109,208,53)(46,134,157,78)(47,107,158,103)(48,132,159,76)(49,105,160,101)(50,130,161,74)(51,155,162,99)(52,128,163,72), (1,140,27,114)(2,139,28,113)(3,138,29,112)(4,137,30,111)(5,136,31,110)(6,135,32,109)(7,134,33,108)(8,133,34,107)(9,132,35,106)(10,131,36,105)(11,130,37,156)(12,129,38,155)(13,128,39,154)(14,127,40,153)(15,126,41,152)(16,125,42,151)(17,124,43,150)(18,123,44,149)(19,122,45,148)(20,121,46,147)(21,120,47,146)(22,119,48,145)(23,118,49,144)(24,117,50,143)(25,116,51,142)(26,115,52,141)(53,169,79,195)(54,168,80,194)(55,167,81,193)(56,166,82,192)(57,165,83,191)(58,164,84,190)(59,163,85,189)(60,162,86,188)(61,161,87,187)(62,160,88,186)(63,159,89,185)(64,158,90,184)(65,157,91,183)(66,208,92,182)(67,207,93,181)(68,206,94,180)(69,205,95,179)(70,204,96,178)(71,203,97,177)(72,202,98,176)(73,201,99,175)(74,200,100,174)(75,199,101,173)(76,198,102,172)(77,197,103,171)(78,196,104,170)>;
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), (1,153,164,97)(2,126,165,70)(3,151,166,95)(4,124,167,68)(5,149,168,93)(6,122,169,66)(7,147,170,91)(8,120,171,64)(9,145,172,89)(10,118,173,62)(11,143,174,87)(12,116,175,60)(13,141,176,85)(14,114,177,58)(15,139,178,83)(16,112,179,56)(17,137,180,81)(18,110,181,54)(19,135,182,79)(20,108,183,104)(21,133,184,77)(22,106,185,102)(23,131,186,75)(24,156,187,100)(25,129,188,73)(26,154,189,98)(27,127,190,71)(28,152,191,96)(29,125,192,69)(30,150,193,94)(31,123,194,67)(32,148,195,92)(33,121,196,65)(34,146,197,90)(35,119,198,63)(36,144,199,88)(37,117,200,61)(38,142,201,86)(39,115,202,59)(40,140,203,84)(41,113,204,57)(42,138,205,82)(43,111,206,55)(44,136,207,80)(45,109,208,53)(46,134,157,78)(47,107,158,103)(48,132,159,76)(49,105,160,101)(50,130,161,74)(51,155,162,99)(52,128,163,72), (1,140,27,114)(2,139,28,113)(3,138,29,112)(4,137,30,111)(5,136,31,110)(6,135,32,109)(7,134,33,108)(8,133,34,107)(9,132,35,106)(10,131,36,105)(11,130,37,156)(12,129,38,155)(13,128,39,154)(14,127,40,153)(15,126,41,152)(16,125,42,151)(17,124,43,150)(18,123,44,149)(19,122,45,148)(20,121,46,147)(21,120,47,146)(22,119,48,145)(23,118,49,144)(24,117,50,143)(25,116,51,142)(26,115,52,141)(53,169,79,195)(54,168,80,194)(55,167,81,193)(56,166,82,192)(57,165,83,191)(58,164,84,190)(59,163,85,189)(60,162,86,188)(61,161,87,187)(62,160,88,186)(63,159,89,185)(64,158,90,184)(65,157,91,183)(66,208,92,182)(67,207,93,181)(68,206,94,180)(69,205,95,179)(70,204,96,178)(71,203,97,177)(72,202,98,176)(73,201,99,175)(74,200,100,174)(75,199,101,173)(76,198,102,172)(77,197,103,171)(78,196,104,170) );
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)], [(1,153,164,97),(2,126,165,70),(3,151,166,95),(4,124,167,68),(5,149,168,93),(6,122,169,66),(7,147,170,91),(8,120,171,64),(9,145,172,89),(10,118,173,62),(11,143,174,87),(12,116,175,60),(13,141,176,85),(14,114,177,58),(15,139,178,83),(16,112,179,56),(17,137,180,81),(18,110,181,54),(19,135,182,79),(20,108,183,104),(21,133,184,77),(22,106,185,102),(23,131,186,75),(24,156,187,100),(25,129,188,73),(26,154,189,98),(27,127,190,71),(28,152,191,96),(29,125,192,69),(30,150,193,94),(31,123,194,67),(32,148,195,92),(33,121,196,65),(34,146,197,90),(35,119,198,63),(36,144,199,88),(37,117,200,61),(38,142,201,86),(39,115,202,59),(40,140,203,84),(41,113,204,57),(42,138,205,82),(43,111,206,55),(44,136,207,80),(45,109,208,53),(46,134,157,78),(47,107,158,103),(48,132,159,76),(49,105,160,101),(50,130,161,74),(51,155,162,99),(52,128,163,72)], [(1,140,27,114),(2,139,28,113),(3,138,29,112),(4,137,30,111),(5,136,31,110),(6,135,32,109),(7,134,33,108),(8,133,34,107),(9,132,35,106),(10,131,36,105),(11,130,37,156),(12,129,38,155),(13,128,39,154),(14,127,40,153),(15,126,41,152),(16,125,42,151),(17,124,43,150),(18,123,44,149),(19,122,45,148),(20,121,46,147),(21,120,47,146),(22,119,48,145),(23,118,49,144),(24,117,50,143),(25,116,51,142),(26,115,52,141),(53,169,79,195),(54,168,80,194),(55,167,81,193),(56,166,82,192),(57,165,83,191),(58,164,84,190),(59,163,85,189),(60,162,86,188),(61,161,87,187),(62,160,88,186),(63,159,89,185),(64,158,90,184),(65,157,91,183),(66,208,92,182),(67,207,93,181),(68,206,94,180),(69,205,95,179),(70,204,96,178),(71,203,97,177),(72,202,98,176),(73,201,99,175),(74,200,100,174),(75,199,101,173),(76,198,102,172),(77,197,103,171),(78,196,104,170)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 13A | ··· | 13F | 26A | ··· | 26R | 26S | ··· | 26AP | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 26 | 26 | 26 | 26 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | C4○D4 | D13 | D26 | D26 | C13⋊D4 | D4⋊2D13 |
| kernel | C52.17D4 | C4×Dic13 | C23.D13 | C2×Dic26 | D4×C26 | C52 | C26 | C2×D4 | C2×C4 | C23 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 6 | 6 | 12 | 24 | 12 |
Matrix representation of C52.17D4 ►in GL4(𝔽53) generated by
| 37 | 0 | 0 | 0 |
| 0 | 43 | 0 | 0 |
| 0 | 0 | 52 | 51 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 52 | 0 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 23 | 46 |
| 0 | 0 | 0 | 30 |
G:=sub<GL(4,GF(53))| [37,0,0,0,0,43,0,0,0,0,52,1,0,0,51,1],[0,52,0,0,1,0,0,0,0,0,30,0,0,0,0,30],[0,1,0,0,1,0,0,0,0,0,23,0,0,0,46,30] >;
C52.17D4 in GAP, Magma, Sage, TeX
C_{52}._{17}D_4 % in TeX
G:=Group("C52.17D4"); // GroupNames label
G:=SmallGroup(416,157);
// by ID
G=gap.SmallGroup(416,157);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,55,506,116,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=b^4=1,c^2=a^26,b*a*b^-1=a^25,c*a*c^-1=a^-1,c*b*c^-1=a^26*b^-1>;
// generators/relations