metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D52⋊5C4, C26.5D8, C52.45D4, C2.2D104, C26.3SD16, C22.10D52, (C2×C8)⋊2D13, (C2×C104)⋊2C2, C52⋊3C4⋊1C2, C4.8(C4×D13), C52.39(C2×C4), (C2×D52).1C2, (C2×C4).71D26, (C2×C26).15D4, C13⋊3(D4⋊C4), C2.3(C104⋊C2), C4.20(C13⋊D4), (C2×C52).83C22, C2.8(D26⋊C4), C26.18(C22⋊C4), SmallGroup(416,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D52⋊5C4
G = < a,b,c | a52=b2=c4=1, bab=cac-1=a-1, cbc-1=a11b >
52C2
52C2
26C22
26C22
52C22
52C22
52C4
4D13
4D13
2C8
13D4
13D4
26C2×C4
26C23
26D4
2D26
2D26
4D26
4D26
13C4⋊C4
13C2×D4
2C104
2D52
13D4⋊C4
Smallest permutation representation of D52⋊5C4
►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 76)(2 75)(3 74)(4 73)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 104)(26 103)(27 102)(28 101)(29 100)(30 99)(31 98)(32 97)(33 96)(34 95)(35 94)(36 93)(37 92)(38 91)(39 90)(40 89)(41 88)(42 87)(43 86)(44 85)(45 84)(46 83)(47 82)(48 81)(49 80)(50 79)(51 78)(52 77)(105 161)(106 160)(107 159)(108 158)(109 157)(110 208)(111 207)(112 206)(113 205)(114 204)(115 203)(116 202)(117 201)(118 200)(119 199)(120 198)(121 197)(122 196)(123 195)(124 194)(125 193)(126 192)(127 191)(128 190)(129 189)(130 188)(131 187)(132 186)(133 185)(134 184)(135 183)(136 182)(137 181)(138 180)(139 179)(140 178)(141 177)(142 176)(143 175)(144 174)(145 173)(146 172)(147 171)(148 170)(149 169)(150 168)(151 167)(152 166)(153 165)(154 164)(155 163)(156 162)
(1 145 103 187)(2 144 104 186)(3 143 53 185)(4 142 54 184)(5 141 55 183)(6 140 56 182)(7 139 57 181)(8 138 58 180)(9 137 59 179)(10 136 60 178)(11 135 61 177)(12 134 62 176)(13 133 63 175)(14 132 64 174)(15 131 65 173)(16 130 66 172)(17 129 67 171)(18 128 68 170)(19 127 69 169)(20 126 70 168)(21 125 71 167)(22 124 72 166)(23 123 73 165)(24 122 74 164)(25 121 75 163)(26 120 76 162)(27 119 77 161)(28 118 78 160)(29 117 79 159)(30 116 80 158)(31 115 81 157)(32 114 82 208)(33 113 83 207)(34 112 84 206)(35 111 85 205)(36 110 86 204)(37 109 87 203)(38 108 88 202)(39 107 89 201)(40 106 90 200)(41 105 91 199)(42 156 92 198)(43 155 93 197)(44 154 94 196)(45 153 95 195)(46 152 96 194)(47 151 97 193)(48 150 98 192)(49 149 99 191)(50 148 100 190)(51 147 101 189)(52 146 102 188)
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), (1,76)(2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,104)(26,103)(27,102)(28,101)(29,100)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(105,161)(106,160)(107,159)(108,158)(109,157)(110,208)(111,207)(112,206)(113,205)(114,204)(115,203)(116,202)(117,201)(118,200)(119,199)(120,198)(121,197)(122,196)(123,195)(124,194)(125,193)(126,192)(127,191)(128,190)(129,189)(130,188)(131,187)(132,186)(133,185)(134,184)(135,183)(136,182)(137,181)(138,180)(139,179)(140,178)(141,177)(142,176)(143,175)(144,174)(145,173)(146,172)(147,171)(148,170)(149,169)(150,168)(151,167)(152,166)(153,165)(154,164)(155,163)(156,162), (1,145,103,187)(2,144,104,186)(3,143,53,185)(4,142,54,184)(5,141,55,183)(6,140,56,182)(7,139,57,181)(8,138,58,180)(9,137,59,179)(10,136,60,178)(11,135,61,177)(12,134,62,176)(13,133,63,175)(14,132,64,174)(15,131,65,173)(16,130,66,172)(17,129,67,171)(18,128,68,170)(19,127,69,169)(20,126,70,168)(21,125,71,167)(22,124,72,166)(23,123,73,165)(24,122,74,164)(25,121,75,163)(26,120,76,162)(27,119,77,161)(28,118,78,160)(29,117,79,159)(30,116,80,158)(31,115,81,157)(32,114,82,208)(33,113,83,207)(34,112,84,206)(35,111,85,205)(36,110,86,204)(37,109,87,203)(38,108,88,202)(39,107,89,201)(40,106,90,200)(41,105,91,199)(42,156,92,198)(43,155,93,197)(44,154,94,196)(45,153,95,195)(46,152,96,194)(47,151,97,193)(48,150,98,192)(49,149,99,191)(50,148,100,190)(51,147,101,189)(52,146,102,188)>;
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), (1,76)(2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,104)(26,103)(27,102)(28,101)(29,100)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(105,161)(106,160)(107,159)(108,158)(109,157)(110,208)(111,207)(112,206)(113,205)(114,204)(115,203)(116,202)(117,201)(118,200)(119,199)(120,198)(121,197)(122,196)(123,195)(124,194)(125,193)(126,192)(127,191)(128,190)(129,189)(130,188)(131,187)(132,186)(133,185)(134,184)(135,183)(136,182)(137,181)(138,180)(139,179)(140,178)(141,177)(142,176)(143,175)(144,174)(145,173)(146,172)(147,171)(148,170)(149,169)(150,168)(151,167)(152,166)(153,165)(154,164)(155,163)(156,162), (1,145,103,187)(2,144,104,186)(3,143,53,185)(4,142,54,184)(5,141,55,183)(6,140,56,182)(7,139,57,181)(8,138,58,180)(9,137,59,179)(10,136,60,178)(11,135,61,177)(12,134,62,176)(13,133,63,175)(14,132,64,174)(15,131,65,173)(16,130,66,172)(17,129,67,171)(18,128,68,170)(19,127,69,169)(20,126,70,168)(21,125,71,167)(22,124,72,166)(23,123,73,165)(24,122,74,164)(25,121,75,163)(26,120,76,162)(27,119,77,161)(28,118,78,160)(29,117,79,159)(30,116,80,158)(31,115,81,157)(32,114,82,208)(33,113,83,207)(34,112,84,206)(35,111,85,205)(36,110,86,204)(37,109,87,203)(38,108,88,202)(39,107,89,201)(40,106,90,200)(41,105,91,199)(42,156,92,198)(43,155,93,197)(44,154,94,196)(45,153,95,195)(46,152,96,194)(47,151,97,193)(48,150,98,192)(49,149,99,191)(50,148,100,190)(51,147,101,189)(52,146,102,188) );
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)], [(1,76),(2,75),(3,74),(4,73),(5,72),(6,71),(7,70),(8,69),(9,68),(10,67),(11,66),(12,65),(13,64),(14,63),(15,62),(16,61),(17,60),(18,59),(19,58),(20,57),(21,56),(22,55),(23,54),(24,53),(25,104),(26,103),(27,102),(28,101),(29,100),(30,99),(31,98),(32,97),(33,96),(34,95),(35,94),(36,93),(37,92),(38,91),(39,90),(40,89),(41,88),(42,87),(43,86),(44,85),(45,84),(46,83),(47,82),(48,81),(49,80),(50,79),(51,78),(52,77),(105,161),(106,160),(107,159),(108,158),(109,157),(110,208),(111,207),(112,206),(113,205),(114,204),(115,203),(116,202),(117,201),(118,200),(119,199),(120,198),(121,197),(122,196),(123,195),(124,194),(125,193),(126,192),(127,191),(128,190),(129,189),(130,188),(131,187),(132,186),(133,185),(134,184),(135,183),(136,182),(137,181),(138,180),(139,179),(140,178),(141,177),(142,176),(143,175),(144,174),(145,173),(146,172),(147,171),(148,170),(149,169),(150,168),(151,167),(152,166),(153,165),(154,164),(155,163),(156,162)], [(1,145,103,187),(2,144,104,186),(3,143,53,185),(4,142,54,184),(5,141,55,183),(6,140,56,182),(7,139,57,181),(8,138,58,180),(9,137,59,179),(10,136,60,178),(11,135,61,177),(12,134,62,176),(13,133,63,175),(14,132,64,174),(15,131,65,173),(16,130,66,172),(17,129,67,171),(18,128,68,170),(19,127,69,169),(20,126,70,168),(21,125,71,167),(22,124,72,166),(23,123,73,165),(24,122,74,164),(25,121,75,163),(26,120,76,162),(27,119,77,161),(28,118,78,160),(29,117,79,159),(30,116,80,158),(31,115,81,157),(32,114,82,208),(33,113,83,207),(34,112,84,206),(35,111,85,205),(36,110,86,204),(37,109,87,203),(38,108,88,202),(39,107,89,201),(40,106,90,200),(41,105,91,199),(42,156,92,198),(43,155,93,197),(44,154,94,196),(45,153,95,195),(46,152,96,194),(47,151,97,193),(48,150,98,192),(49,149,99,191),(50,148,100,190),(51,147,101,189),(52,146,102,188)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 52 | 52 | 2 | 2 | 52 | 52 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D8 | SD16 | D13 | D26 | C4×D13 | C13⋊D4 | D52 | C104⋊C2 | D104 |
| kernel | D52⋊5C4 | C52⋊3C4 | C2×C104 | C2×D52 | D52 | C52 | C2×C26 | C26 | C26 | C2×C8 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 24 | 24 |
Matrix representation of D52⋊5C4 ►in GL3(𝔽313) generated by
| 1 | 0 | 0 |
| 0 | 155 | 69 |
| 0 | 172 | 121 |
| 1 | 0 | 0 |
| 0 | 204 | 202 |
| 0 | 217 | 109 |
| 25 | 0 | 0 |
| 0 | 116 | 272 |
| 0 | 290 | 197 |
G:=sub<GL(3,GF(313))| [1,0,0,0,155,172,0,69,121],[1,0,0,0,204,217,0,202,109],[25,0,0,0,116,290,0,272,197] >;
D52⋊5C4 in GAP, Magma, Sage, TeX
D_{52}\rtimes_5C_4 % in TeX
G:=Group("D52:5C4"); // GroupNames label
G:=SmallGroup(416,28);
// by ID
G=gap.SmallGroup(416,28);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,73,79,362,86,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=b^2=c^4=1,b*a*b=c*a*c^-1=a^-1,c*b*c^-1=a^11*b>;
// generators/relations
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