metabelian, supersoluble, monomial, A-group
Aliases: C52⋊4C8, C10.3F5, C5⋊1(C5⋊C8), (C5×C10).3C4, C2.(C5⋊F5), C52⋊6C4.3C2, SmallGroup(200,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52⋊4C8 |
Generators and relations for C52⋊4C8
G = < a,b,c | a5=b5=c8=1, ab=ba, cac-1=a3, cbc-1=b3 >
Character table of C52⋊4C8
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | |
| size | 1 | 1 | 25 | 25 | 4 | 4 | 4 | 4 | 4 | 4 | 25 | 25 | 25 | 25 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | ζ87 | ζ85 | ζ83 | ζ8 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ6 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | ζ83 | ζ8 | ζ87 | ζ85 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ7 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | ζ85 | ζ87 | ζ8 | ζ83 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ8 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | ζ8 | ζ83 | ζ85 | ζ87 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ9 | 4 | 4 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ10 | 4 | 4 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 4 | -1 | orthogonal lifted from F5 |
| ρ11 | 4 | 4 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 4 | 0 | 0 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ12 | 4 | 4 | 0 | 0 | -1 | -1 | -1 | -1 | 4 | -1 | 0 | 0 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ13 | 4 | 4 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 4 | orthogonal lifted from F5 |
| ρ14 | 4 | 4 | 0 | 0 | 4 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | orthogonal lifted from F5 |
| ρ15 | 4 | -4 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | -4 | 1 | 1 | 1 | 1 | 1 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ16 | 4 | -4 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -4 | 1 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ17 | 4 | -4 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | -4 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ18 | 4 | -4 | 0 | 0 | 4 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -4 | 1 | 1 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ19 | 4 | -4 | 0 | 0 | -1 | -1 | -1 | -1 | 4 | -1 | 0 | 0 | 0 | 0 | 1 | -4 | 1 | 1 | 1 | 1 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ20 | 4 | -4 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 4 | 0 | 0 | 0 | 0 | 1 | 1 | -4 | 1 | 1 | 1 | symplectic lifted from C5⋊C8, Schur index 2 |
►Regular action on 200 points
Generators in S200
(1 65 9 189 136)(2 190 66 129 10)(3 130 191 11 67)(4 12 131 68 192)(5 69 13 185 132)(6 186 70 133 14)(7 134 187 15 71)(8 16 135 72 188)(17 56 157 137 169)(18 138 49 170 158)(19 171 139 159 50)(20 160 172 51 140)(21 52 153 141 173)(22 142 53 174 154)(23 175 143 155 54)(24 156 176 55 144)(25 197 86 125 168)(26 126 198 161 87)(27 162 127 88 199)(28 81 163 200 128)(29 193 82 121 164)(30 122 194 165 83)(31 166 123 84 195)(32 85 167 196 124)(33 118 92 149 105)(34 150 119 106 93)(35 107 151 94 120)(36 95 108 113 152)(37 114 96 145 109)(38 146 115 110 89)(39 111 147 90 116)(40 91 112 117 148)(41 73 184 102 61)(42 103 74 62 177)(43 63 104 178 75)(44 179 64 76 97)(45 77 180 98 57)(46 99 78 58 181)(47 59 100 182 79)(48 183 60 80 101)
(1 23 168 89 104)(2 90 24 97 161)(3 98 91 162 17)(4 163 99 18 92)(5 19 164 93 100)(6 94 20 101 165)(7 102 95 166 21)(8 167 103 22 96)(9 143 197 146 75)(10 147 144 76 198)(11 77 148 199 137)(12 200 78 138 149)(13 139 193 150 79)(14 151 140 80 194)(15 73 152 195 141)(16 196 74 142 145)(25 38 178 65 175)(26 66 39 176 179)(27 169 67 180 40)(28 181 170 33 68)(29 34 182 69 171)(30 70 35 172 183)(31 173 71 184 36)(32 177 174 37 72)(41 113 84 153 187)(42 154 114 188 85)(43 189 155 86 115)(44 87 190 116 156)(45 117 88 157 191)(46 158 118 192 81)(47 185 159 82 119)(48 83 186 120 160)(49 105 131 128 58)(50 121 106 59 132)(51 60 122 133 107)(52 134 61 108 123)(53 109 135 124 62)(54 125 110 63 136)(55 64 126 129 111)(56 130 57 112 127)
(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)
G:=sub<Sym(200)| (1,65,9,189,136)(2,190,66,129,10)(3,130,191,11,67)(4,12,131,68,192)(5,69,13,185,132)(6,186,70,133,14)(7,134,187,15,71)(8,16,135,72,188)(17,56,157,137,169)(18,138,49,170,158)(19,171,139,159,50)(20,160,172,51,140)(21,52,153,141,173)(22,142,53,174,154)(23,175,143,155,54)(24,156,176,55,144)(25,197,86,125,168)(26,126,198,161,87)(27,162,127,88,199)(28,81,163,200,128)(29,193,82,121,164)(30,122,194,165,83)(31,166,123,84,195)(32,85,167,196,124)(33,118,92,149,105)(34,150,119,106,93)(35,107,151,94,120)(36,95,108,113,152)(37,114,96,145,109)(38,146,115,110,89)(39,111,147,90,116)(40,91,112,117,148)(41,73,184,102,61)(42,103,74,62,177)(43,63,104,178,75)(44,179,64,76,97)(45,77,180,98,57)(46,99,78,58,181)(47,59,100,182,79)(48,183,60,80,101), (1,23,168,89,104)(2,90,24,97,161)(3,98,91,162,17)(4,163,99,18,92)(5,19,164,93,100)(6,94,20,101,165)(7,102,95,166,21)(8,167,103,22,96)(9,143,197,146,75)(10,147,144,76,198)(11,77,148,199,137)(12,200,78,138,149)(13,139,193,150,79)(14,151,140,80,194)(15,73,152,195,141)(16,196,74,142,145)(25,38,178,65,175)(26,66,39,176,179)(27,169,67,180,40)(28,181,170,33,68)(29,34,182,69,171)(30,70,35,172,183)(31,173,71,184,36)(32,177,174,37,72)(41,113,84,153,187)(42,154,114,188,85)(43,189,155,86,115)(44,87,190,116,156)(45,117,88,157,191)(46,158,118,192,81)(47,185,159,82,119)(48,83,186,120,160)(49,105,131,128,58)(50,121,106,59,132)(51,60,122,133,107)(52,134,61,108,123)(53,109,135,124,62)(54,125,110,63,136)(55,64,126,129,111)(56,130,57,112,127), (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)>;
G:=Group( (1,65,9,189,136)(2,190,66,129,10)(3,130,191,11,67)(4,12,131,68,192)(5,69,13,185,132)(6,186,70,133,14)(7,134,187,15,71)(8,16,135,72,188)(17,56,157,137,169)(18,138,49,170,158)(19,171,139,159,50)(20,160,172,51,140)(21,52,153,141,173)(22,142,53,174,154)(23,175,143,155,54)(24,156,176,55,144)(25,197,86,125,168)(26,126,198,161,87)(27,162,127,88,199)(28,81,163,200,128)(29,193,82,121,164)(30,122,194,165,83)(31,166,123,84,195)(32,85,167,196,124)(33,118,92,149,105)(34,150,119,106,93)(35,107,151,94,120)(36,95,108,113,152)(37,114,96,145,109)(38,146,115,110,89)(39,111,147,90,116)(40,91,112,117,148)(41,73,184,102,61)(42,103,74,62,177)(43,63,104,178,75)(44,179,64,76,97)(45,77,180,98,57)(46,99,78,58,181)(47,59,100,182,79)(48,183,60,80,101), (1,23,168,89,104)(2,90,24,97,161)(3,98,91,162,17)(4,163,99,18,92)(5,19,164,93,100)(6,94,20,101,165)(7,102,95,166,21)(8,167,103,22,96)(9,143,197,146,75)(10,147,144,76,198)(11,77,148,199,137)(12,200,78,138,149)(13,139,193,150,79)(14,151,140,80,194)(15,73,152,195,141)(16,196,74,142,145)(25,38,178,65,175)(26,66,39,176,179)(27,169,67,180,40)(28,181,170,33,68)(29,34,182,69,171)(30,70,35,172,183)(31,173,71,184,36)(32,177,174,37,72)(41,113,84,153,187)(42,154,114,188,85)(43,189,155,86,115)(44,87,190,116,156)(45,117,88,157,191)(46,158,118,192,81)(47,185,159,82,119)(48,83,186,120,160)(49,105,131,128,58)(50,121,106,59,132)(51,60,122,133,107)(52,134,61,108,123)(53,109,135,124,62)(54,125,110,63,136)(55,64,126,129,111)(56,130,57,112,127), (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) );
G=PermutationGroup([[(1,65,9,189,136),(2,190,66,129,10),(3,130,191,11,67),(4,12,131,68,192),(5,69,13,185,132),(6,186,70,133,14),(7,134,187,15,71),(8,16,135,72,188),(17,56,157,137,169),(18,138,49,170,158),(19,171,139,159,50),(20,160,172,51,140),(21,52,153,141,173),(22,142,53,174,154),(23,175,143,155,54),(24,156,176,55,144),(25,197,86,125,168),(26,126,198,161,87),(27,162,127,88,199),(28,81,163,200,128),(29,193,82,121,164),(30,122,194,165,83),(31,166,123,84,195),(32,85,167,196,124),(33,118,92,149,105),(34,150,119,106,93),(35,107,151,94,120),(36,95,108,113,152),(37,114,96,145,109),(38,146,115,110,89),(39,111,147,90,116),(40,91,112,117,148),(41,73,184,102,61),(42,103,74,62,177),(43,63,104,178,75),(44,179,64,76,97),(45,77,180,98,57),(46,99,78,58,181),(47,59,100,182,79),(48,183,60,80,101)], [(1,23,168,89,104),(2,90,24,97,161),(3,98,91,162,17),(4,163,99,18,92),(5,19,164,93,100),(6,94,20,101,165),(7,102,95,166,21),(8,167,103,22,96),(9,143,197,146,75),(10,147,144,76,198),(11,77,148,199,137),(12,200,78,138,149),(13,139,193,150,79),(14,151,140,80,194),(15,73,152,195,141),(16,196,74,142,145),(25,38,178,65,175),(26,66,39,176,179),(27,169,67,180,40),(28,181,170,33,68),(29,34,182,69,171),(30,70,35,172,183),(31,173,71,184,36),(32,177,174,37,72),(41,113,84,153,187),(42,154,114,188,85),(43,189,155,86,115),(44,87,190,116,156),(45,117,88,157,191),(46,158,118,192,81),(47,185,159,82,119),(48,83,186,120,160),(49,105,131,128,58),(50,121,106,59,132),(51,60,122,133,107),(52,134,61,108,123),(53,109,135,124,62),(54,125,110,63,136),(55,64,126,129,111),(56,130,57,112,127)], [(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)]])
C52⋊4C8 is a maximal subgroup of
C52⋊C16 D10.2F5 C52⋊4M4(2) C20.F5 C52⋊7M4(2) C52⋊13M4(2)
C52⋊4C8 is a maximal quotient of C52⋊4C16
Matrix representation of C52⋊4C8 ►in GL8(𝔽41)
| 0 | 40 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 40 | 40 | 40 |
| 6 | 15 | 2 | 34 | 0 | 0 | 0 | 0 |
| 8 | 8 | 27 | 40 | 0 | 0 | 0 | 0 |
| 33 | 14 | 1 | 1 | 0 | 0 | 0 | 0 |
| 7 | 16 | 35 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 22 | 27 | 22 |
| 0 | 0 | 0 | 0 | 5 | 0 | 19 | 1 |
| 0 | 0 | 0 | 0 | 19 | 1 | 0 | 5 |
| 0 | 0 | 0 | 0 | 40 | 4 | 40 | 18 |
G:=sub<GL(8,GF(41))| [0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,1,40],[6,8,33,7,0,0,0,0,15,8,14,16,0,0,0,0,2,27,1,35,0,0,0,0,34,40,1,26,0,0,0,0,0,0,0,0,23,5,19,40,0,0,0,0,22,0,1,4,0,0,0,0,27,19,0,40,0,0,0,0,22,1,5,18] >;
C52⋊4C8 in GAP, Magma, Sage, TeX
C_5^2\rtimes_4C_8
% in TeX
G:=Group("C5^2:4C8"); // GroupNames label
G:=SmallGroup(200,20);
// by ID
G=gap.SmallGroup(200,20);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,10,26,323,328,2004,2009]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^8=1,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C52⋊4C8 in TeX
Character table of C52⋊4C8 in TeX