metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.5D52, C52.28D4, C42⋊6D13, (C4×C52)⋊5C2, C2.6(C2×D52), C26.4(C2×D4), (C2×D52).3C2, (C2×C4).77D26, D26⋊C4⋊1C2, (C2×Dic26)⋊1C2, C26.5(C4○D4), C13⋊1(C4.4D4), (C2×C52).74C22, (C2×C26).16C23, C2.7(D52⋊5C2), (C2×Dic13).3C22, (C22×D13).2C22, C22.37(C22×D13), SmallGroup(416,96)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.D52
G = < a,b,c | a4=b52=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >
Subgroups: 632 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C13, C42, C22⋊C4, C2×D4, C2×Q8, D13, C26, C26, C4.4D4, Dic13, C52, C52, D26, C2×C26, Dic26, D52, C2×Dic13, C2×C52, C2×C52, C22×D13, D26⋊C4, C4×C52, C2×Dic26, C2×D52, C4.D52
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D13, C4.4D4, D26, D52, C22×D13, C2×D52, D52⋊5C2, C4.D52
►On 208 points
Generators in S208
(1 174 135 57)(2 175 136 58)(3 176 137 59)(4 177 138 60)(5 178 139 61)(6 179 140 62)(7 180 141 63)(8 181 142 64)(9 182 143 65)(10 183 144 66)(11 184 145 67)(12 185 146 68)(13 186 147 69)(14 187 148 70)(15 188 149 71)(16 189 150 72)(17 190 151 73)(18 191 152 74)(19 192 153 75)(20 193 154 76)(21 194 155 77)(22 195 156 78)(23 196 105 79)(24 197 106 80)(25 198 107 81)(26 199 108 82)(27 200 109 83)(28 201 110 84)(29 202 111 85)(30 203 112 86)(31 204 113 87)(32 205 114 88)(33 206 115 89)(34 207 116 90)(35 208 117 91)(36 157 118 92)(37 158 119 93)(38 159 120 94)(39 160 121 95)(40 161 122 96)(41 162 123 97)(42 163 124 98)(43 164 125 99)(44 165 126 100)(45 166 127 101)(46 167 128 102)(47 168 129 103)(48 169 130 104)(49 170 131 53)(50 171 132 54)(51 172 133 55)(52 173 134 56)
(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 82 135 199)(2 198 136 81)(3 80 137 197)(4 196 138 79)(5 78 139 195)(6 194 140 77)(7 76 141 193)(8 192 142 75)(9 74 143 191)(10 190 144 73)(11 72 145 189)(12 188 146 71)(13 70 147 187)(14 186 148 69)(15 68 149 185)(16 184 150 67)(17 66 151 183)(18 182 152 65)(19 64 153 181)(20 180 154 63)(21 62 155 179)(22 178 156 61)(23 60 105 177)(24 176 106 59)(25 58 107 175)(26 174 108 57)(27 56 109 173)(28 172 110 55)(29 54 111 171)(30 170 112 53)(31 104 113 169)(32 168 114 103)(33 102 115 167)(34 166 116 101)(35 100 117 165)(36 164 118 99)(37 98 119 163)(38 162 120 97)(39 96 121 161)(40 160 122 95)(41 94 123 159)(42 158 124 93)(43 92 125 157)(44 208 126 91)(45 90 127 207)(46 206 128 89)(47 88 129 205)(48 204 130 87)(49 86 131 203)(50 202 132 85)(51 84 133 201)(52 200 134 83)
G:=sub<Sym(208)| (1,174,135,57)(2,175,136,58)(3,176,137,59)(4,177,138,60)(5,178,139,61)(6,179,140,62)(7,180,141,63)(8,181,142,64)(9,182,143,65)(10,183,144,66)(11,184,145,67)(12,185,146,68)(13,186,147,69)(14,187,148,70)(15,188,149,71)(16,189,150,72)(17,190,151,73)(18,191,152,74)(19,192,153,75)(20,193,154,76)(21,194,155,77)(22,195,156,78)(23,196,105,79)(24,197,106,80)(25,198,107,81)(26,199,108,82)(27,200,109,83)(28,201,110,84)(29,202,111,85)(30,203,112,86)(31,204,113,87)(32,205,114,88)(33,206,115,89)(34,207,116,90)(35,208,117,91)(36,157,118,92)(37,158,119,93)(38,159,120,94)(39,160,121,95)(40,161,122,96)(41,162,123,97)(42,163,124,98)(43,164,125,99)(44,165,126,100)(45,166,127,101)(46,167,128,102)(47,168,129,103)(48,169,130,104)(49,170,131,53)(50,171,132,54)(51,172,133,55)(52,173,134,56), (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), (1,82,135,199)(2,198,136,81)(3,80,137,197)(4,196,138,79)(5,78,139,195)(6,194,140,77)(7,76,141,193)(8,192,142,75)(9,74,143,191)(10,190,144,73)(11,72,145,189)(12,188,146,71)(13,70,147,187)(14,186,148,69)(15,68,149,185)(16,184,150,67)(17,66,151,183)(18,182,152,65)(19,64,153,181)(20,180,154,63)(21,62,155,179)(22,178,156,61)(23,60,105,177)(24,176,106,59)(25,58,107,175)(26,174,108,57)(27,56,109,173)(28,172,110,55)(29,54,111,171)(30,170,112,53)(31,104,113,169)(32,168,114,103)(33,102,115,167)(34,166,116,101)(35,100,117,165)(36,164,118,99)(37,98,119,163)(38,162,120,97)(39,96,121,161)(40,160,122,95)(41,94,123,159)(42,158,124,93)(43,92,125,157)(44,208,126,91)(45,90,127,207)(46,206,128,89)(47,88,129,205)(48,204,130,87)(49,86,131,203)(50,202,132,85)(51,84,133,201)(52,200,134,83)>;
G:=Group( (1,174,135,57)(2,175,136,58)(3,176,137,59)(4,177,138,60)(5,178,139,61)(6,179,140,62)(7,180,141,63)(8,181,142,64)(9,182,143,65)(10,183,144,66)(11,184,145,67)(12,185,146,68)(13,186,147,69)(14,187,148,70)(15,188,149,71)(16,189,150,72)(17,190,151,73)(18,191,152,74)(19,192,153,75)(20,193,154,76)(21,194,155,77)(22,195,156,78)(23,196,105,79)(24,197,106,80)(25,198,107,81)(26,199,108,82)(27,200,109,83)(28,201,110,84)(29,202,111,85)(30,203,112,86)(31,204,113,87)(32,205,114,88)(33,206,115,89)(34,207,116,90)(35,208,117,91)(36,157,118,92)(37,158,119,93)(38,159,120,94)(39,160,121,95)(40,161,122,96)(41,162,123,97)(42,163,124,98)(43,164,125,99)(44,165,126,100)(45,166,127,101)(46,167,128,102)(47,168,129,103)(48,169,130,104)(49,170,131,53)(50,171,132,54)(51,172,133,55)(52,173,134,56), (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), (1,82,135,199)(2,198,136,81)(3,80,137,197)(4,196,138,79)(5,78,139,195)(6,194,140,77)(7,76,141,193)(8,192,142,75)(9,74,143,191)(10,190,144,73)(11,72,145,189)(12,188,146,71)(13,70,147,187)(14,186,148,69)(15,68,149,185)(16,184,150,67)(17,66,151,183)(18,182,152,65)(19,64,153,181)(20,180,154,63)(21,62,155,179)(22,178,156,61)(23,60,105,177)(24,176,106,59)(25,58,107,175)(26,174,108,57)(27,56,109,173)(28,172,110,55)(29,54,111,171)(30,170,112,53)(31,104,113,169)(32,168,114,103)(33,102,115,167)(34,166,116,101)(35,100,117,165)(36,164,118,99)(37,98,119,163)(38,162,120,97)(39,96,121,161)(40,160,122,95)(41,94,123,159)(42,158,124,93)(43,92,125,157)(44,208,126,91)(45,90,127,207)(46,206,128,89)(47,88,129,205)(48,204,130,87)(49,86,131,203)(50,202,132,85)(51,84,133,201)(52,200,134,83) );
G=PermutationGroup([[(1,174,135,57),(2,175,136,58),(3,176,137,59),(4,177,138,60),(5,178,139,61),(6,179,140,62),(7,180,141,63),(8,181,142,64),(9,182,143,65),(10,183,144,66),(11,184,145,67),(12,185,146,68),(13,186,147,69),(14,187,148,70),(15,188,149,71),(16,189,150,72),(17,190,151,73),(18,191,152,74),(19,192,153,75),(20,193,154,76),(21,194,155,77),(22,195,156,78),(23,196,105,79),(24,197,106,80),(25,198,107,81),(26,199,108,82),(27,200,109,83),(28,201,110,84),(29,202,111,85),(30,203,112,86),(31,204,113,87),(32,205,114,88),(33,206,115,89),(34,207,116,90),(35,208,117,91),(36,157,118,92),(37,158,119,93),(38,159,120,94),(39,160,121,95),(40,161,122,96),(41,162,123,97),(42,163,124,98),(43,164,125,99),(44,165,126,100),(45,166,127,101),(46,167,128,102),(47,168,129,103),(48,169,130,104),(49,170,131,53),(50,171,132,54),(51,172,133,55),(52,173,134,56)], [(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)], [(1,82,135,199),(2,198,136,81),(3,80,137,197),(4,196,138,79),(5,78,139,195),(6,194,140,77),(7,76,141,193),(8,192,142,75),(9,74,143,191),(10,190,144,73),(11,72,145,189),(12,188,146,71),(13,70,147,187),(14,186,148,69),(15,68,149,185),(16,184,150,67),(17,66,151,183),(18,182,152,65),(19,64,153,181),(20,180,154,63),(21,62,155,179),(22,178,156,61),(23,60,105,177),(24,176,106,59),(25,58,107,175),(26,174,108,57),(27,56,109,173),(28,172,110,55),(29,54,111,171),(30,170,112,53),(31,104,113,169),(32,168,114,103),(33,102,115,167),(34,166,116,101),(35,100,117,165),(36,164,118,99),(37,98,119,163),(38,162,120,97),(39,96,121,161),(40,160,122,95),(41,94,123,159),(42,158,124,93),(43,92,125,157),(44,208,126,91),(45,90,127,207),(46,206,128,89),(47,88,129,205),(48,204,130,87),(49,86,131,203),(50,202,132,85),(51,84,133,201),(52,200,134,83)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 52 | 52 | 2 | ··· | 2 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | C4○D4 | D13 | D26 | D52 | D52⋊5C2 |
| kernel | C4.D52 | D26⋊C4 | C4×C52 | C2×Dic26 | C2×D52 | C52 | C26 | C42 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 6 | 18 | 24 | 48 |
Matrix representation of C4.D52 ►in GL4(𝔽53) generated by
| 15 | 4 | 0 | 0 |
| 23 | 38 | 0 | 0 |
| 0 | 0 | 10 | 19 |
| 0 | 0 | 17 | 43 |
| 27 | 39 | 0 | 0 |
| 52 | 26 | 0 | 0 |
| 0 | 0 | 11 | 4 |
| 0 | 0 | 51 | 4 |
| 23 | 23 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 51 | 48 |
| 0 | 0 | 1 | 2 |
G:=sub<GL(4,GF(53))| [15,23,0,0,4,38,0,0,0,0,10,17,0,0,19,43],[27,52,0,0,39,26,0,0,0,0,11,51,0,0,4,4],[23,0,0,0,23,30,0,0,0,0,51,1,0,0,48,2] >;
C4.D52 in GAP, Magma, Sage, TeX
C_4.D_{52} % in TeX
G:=Group("C4.D52"); // GroupNames label
G:=SmallGroup(416,96);
// by ID
G=gap.SmallGroup(416,96);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,55,218,86,13829]);
// Polycyclic
G:=Group<a,b,c|a^4=b^52=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^-1>;
// generators/relations