metabelian, supersoluble, monomial
Aliases: C52⋊4Q8, C20.3D5, C5⋊2Dic10, C10.12D10, C4.(C5⋊D5), (C5×C20).1C2, C52⋊6C4.2C2, (C5×C10).11C22, C2.3(C2×C5⋊D5), SmallGroup(200,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊4Q8
G = < a,b,c,d | a5=b5=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Smallest permutation representation of C52⋊4Q8
►Regular action on 200 points
Generators in S200
(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)
(1 44 38 33 28)(2 45 39 34 29)(3 41 40 35 30)(4 42 36 31 26)(5 43 37 32 27)(6 200 25 16 11)(7 196 21 17 12)(8 197 22 18 13)(9 198 23 19 14)(10 199 24 20 15)(46 67 61 56 51)(47 68 62 57 52)(48 69 63 58 53)(49 70 64 59 54)(50 66 65 60 55)(71 92 86 81 76)(72 93 87 82 77)(73 94 88 83 78)(74 95 89 84 79)(75 91 90 85 80)(96 102 106 111 116)(97 103 107 112 117)(98 104 108 113 118)(99 105 109 114 119)(100 101 110 115 120)(121 130 131 136 141)(122 126 132 137 142)(123 127 133 138 143)(124 128 134 139 144)(125 129 135 140 145)(146 155 156 161 166)(147 151 157 162 167)(148 152 158 163 168)(149 153 159 164 169)(150 154 160 165 170)(171 180 181 186 191)(172 176 182 187 192)(173 177 183 188 193)(174 178 184 189 194)(175 179 185 190 195)
(1 98 48 73)(2 99 49 74)(3 100 50 75)(4 96 46 71)(5 97 47 72)(6 145 195 170)(7 141 191 166)(8 142 192 167)(9 143 193 168)(10 144 194 169)(11 140 190 165)(12 136 186 161)(13 137 187 162)(14 138 188 163)(15 139 189 164)(16 135 185 160)(17 131 181 156)(18 132 182 157)(19 133 183 158)(20 134 184 159)(21 130 180 155)(22 126 176 151)(23 127 177 152)(24 128 178 153)(25 129 179 154)(26 116 51 76)(27 117 52 77)(28 118 53 78)(29 119 54 79)(30 120 55 80)(31 111 56 81)(32 112 57 82)(33 113 58 83)(34 114 59 84)(35 115 60 85)(36 106 61 86)(37 107 62 87)(38 108 63 88)(39 109 64 89)(40 110 65 90)(41 101 66 91)(42 102 67 92)(43 103 68 93)(44 104 69 94)(45 105 70 95)(121 171 146 196)(122 172 147 197)(123 173 148 198)(124 174 149 199)(125 175 150 200)
(1 157 48 132)(2 156 49 131)(3 160 50 135)(4 159 46 134)(5 158 47 133)(6 115 195 85)(7 114 191 84)(8 113 192 83)(9 112 193 82)(10 111 194 81)(11 120 190 80)(12 119 186 79)(13 118 187 78)(14 117 188 77)(15 116 189 76)(16 100 185 75)(17 99 181 74)(18 98 182 73)(19 97 183 72)(20 96 184 71)(21 105 180 95)(22 104 176 94)(23 103 177 93)(24 102 178 92)(25 101 179 91)(26 164 51 139)(27 163 52 138)(28 162 53 137)(29 161 54 136)(30 165 55 140)(31 169 56 144)(32 168 57 143)(33 167 58 142)(34 166 59 141)(35 170 60 145)(36 149 61 124)(37 148 62 123)(38 147 63 122)(39 146 64 121)(40 150 65 125)(41 154 66 129)(42 153 67 128)(43 152 68 127)(44 151 69 126)(45 155 70 130)(86 199 106 174)(87 198 107 173)(88 197 108 172)(89 196 109 171)(90 200 110 175)
G:=sub<Sym(200)| (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), (1,44,38,33,28)(2,45,39,34,29)(3,41,40,35,30)(4,42,36,31,26)(5,43,37,32,27)(6,200,25,16,11)(7,196,21,17,12)(8,197,22,18,13)(9,198,23,19,14)(10,199,24,20,15)(46,67,61,56,51)(47,68,62,57,52)(48,69,63,58,53)(49,70,64,59,54)(50,66,65,60,55)(71,92,86,81,76)(72,93,87,82,77)(73,94,88,83,78)(74,95,89,84,79)(75,91,90,85,80)(96,102,106,111,116)(97,103,107,112,117)(98,104,108,113,118)(99,105,109,114,119)(100,101,110,115,120)(121,130,131,136,141)(122,126,132,137,142)(123,127,133,138,143)(124,128,134,139,144)(125,129,135,140,145)(146,155,156,161,166)(147,151,157,162,167)(148,152,158,163,168)(149,153,159,164,169)(150,154,160,165,170)(171,180,181,186,191)(172,176,182,187,192)(173,177,183,188,193)(174,178,184,189,194)(175,179,185,190,195), (1,98,48,73)(2,99,49,74)(3,100,50,75)(4,96,46,71)(5,97,47,72)(6,145,195,170)(7,141,191,166)(8,142,192,167)(9,143,193,168)(10,144,194,169)(11,140,190,165)(12,136,186,161)(13,137,187,162)(14,138,188,163)(15,139,189,164)(16,135,185,160)(17,131,181,156)(18,132,182,157)(19,133,183,158)(20,134,184,159)(21,130,180,155)(22,126,176,151)(23,127,177,152)(24,128,178,153)(25,129,179,154)(26,116,51,76)(27,117,52,77)(28,118,53,78)(29,119,54,79)(30,120,55,80)(31,111,56,81)(32,112,57,82)(33,113,58,83)(34,114,59,84)(35,115,60,85)(36,106,61,86)(37,107,62,87)(38,108,63,88)(39,109,64,89)(40,110,65,90)(41,101,66,91)(42,102,67,92)(43,103,68,93)(44,104,69,94)(45,105,70,95)(121,171,146,196)(122,172,147,197)(123,173,148,198)(124,174,149,199)(125,175,150,200), (1,157,48,132)(2,156,49,131)(3,160,50,135)(4,159,46,134)(5,158,47,133)(6,115,195,85)(7,114,191,84)(8,113,192,83)(9,112,193,82)(10,111,194,81)(11,120,190,80)(12,119,186,79)(13,118,187,78)(14,117,188,77)(15,116,189,76)(16,100,185,75)(17,99,181,74)(18,98,182,73)(19,97,183,72)(20,96,184,71)(21,105,180,95)(22,104,176,94)(23,103,177,93)(24,102,178,92)(25,101,179,91)(26,164,51,139)(27,163,52,138)(28,162,53,137)(29,161,54,136)(30,165,55,140)(31,169,56,144)(32,168,57,143)(33,167,58,142)(34,166,59,141)(35,170,60,145)(36,149,61,124)(37,148,62,123)(38,147,63,122)(39,146,64,121)(40,150,65,125)(41,154,66,129)(42,153,67,128)(43,152,68,127)(44,151,69,126)(45,155,70,130)(86,199,106,174)(87,198,107,173)(88,197,108,172)(89,196,109,171)(90,200,110,175)>;
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), (1,44,38,33,28)(2,45,39,34,29)(3,41,40,35,30)(4,42,36,31,26)(5,43,37,32,27)(6,200,25,16,11)(7,196,21,17,12)(8,197,22,18,13)(9,198,23,19,14)(10,199,24,20,15)(46,67,61,56,51)(47,68,62,57,52)(48,69,63,58,53)(49,70,64,59,54)(50,66,65,60,55)(71,92,86,81,76)(72,93,87,82,77)(73,94,88,83,78)(74,95,89,84,79)(75,91,90,85,80)(96,102,106,111,116)(97,103,107,112,117)(98,104,108,113,118)(99,105,109,114,119)(100,101,110,115,120)(121,130,131,136,141)(122,126,132,137,142)(123,127,133,138,143)(124,128,134,139,144)(125,129,135,140,145)(146,155,156,161,166)(147,151,157,162,167)(148,152,158,163,168)(149,153,159,164,169)(150,154,160,165,170)(171,180,181,186,191)(172,176,182,187,192)(173,177,183,188,193)(174,178,184,189,194)(175,179,185,190,195), (1,98,48,73)(2,99,49,74)(3,100,50,75)(4,96,46,71)(5,97,47,72)(6,145,195,170)(7,141,191,166)(8,142,192,167)(9,143,193,168)(10,144,194,169)(11,140,190,165)(12,136,186,161)(13,137,187,162)(14,138,188,163)(15,139,189,164)(16,135,185,160)(17,131,181,156)(18,132,182,157)(19,133,183,158)(20,134,184,159)(21,130,180,155)(22,126,176,151)(23,127,177,152)(24,128,178,153)(25,129,179,154)(26,116,51,76)(27,117,52,77)(28,118,53,78)(29,119,54,79)(30,120,55,80)(31,111,56,81)(32,112,57,82)(33,113,58,83)(34,114,59,84)(35,115,60,85)(36,106,61,86)(37,107,62,87)(38,108,63,88)(39,109,64,89)(40,110,65,90)(41,101,66,91)(42,102,67,92)(43,103,68,93)(44,104,69,94)(45,105,70,95)(121,171,146,196)(122,172,147,197)(123,173,148,198)(124,174,149,199)(125,175,150,200), (1,157,48,132)(2,156,49,131)(3,160,50,135)(4,159,46,134)(5,158,47,133)(6,115,195,85)(7,114,191,84)(8,113,192,83)(9,112,193,82)(10,111,194,81)(11,120,190,80)(12,119,186,79)(13,118,187,78)(14,117,188,77)(15,116,189,76)(16,100,185,75)(17,99,181,74)(18,98,182,73)(19,97,183,72)(20,96,184,71)(21,105,180,95)(22,104,176,94)(23,103,177,93)(24,102,178,92)(25,101,179,91)(26,164,51,139)(27,163,52,138)(28,162,53,137)(29,161,54,136)(30,165,55,140)(31,169,56,144)(32,168,57,143)(33,167,58,142)(34,166,59,141)(35,170,60,145)(36,149,61,124)(37,148,62,123)(38,147,63,122)(39,146,64,121)(40,150,65,125)(41,154,66,129)(42,153,67,128)(43,152,68,127)(44,151,69,126)(45,155,70,130)(86,199,106,174)(87,198,107,173)(88,197,108,172)(89,196,109,171)(90,200,110,175) );
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)], [(1,44,38,33,28),(2,45,39,34,29),(3,41,40,35,30),(4,42,36,31,26),(5,43,37,32,27),(6,200,25,16,11),(7,196,21,17,12),(8,197,22,18,13),(9,198,23,19,14),(10,199,24,20,15),(46,67,61,56,51),(47,68,62,57,52),(48,69,63,58,53),(49,70,64,59,54),(50,66,65,60,55),(71,92,86,81,76),(72,93,87,82,77),(73,94,88,83,78),(74,95,89,84,79),(75,91,90,85,80),(96,102,106,111,116),(97,103,107,112,117),(98,104,108,113,118),(99,105,109,114,119),(100,101,110,115,120),(121,130,131,136,141),(122,126,132,137,142),(123,127,133,138,143),(124,128,134,139,144),(125,129,135,140,145),(146,155,156,161,166),(147,151,157,162,167),(148,152,158,163,168),(149,153,159,164,169),(150,154,160,165,170),(171,180,181,186,191),(172,176,182,187,192),(173,177,183,188,193),(174,178,184,189,194),(175,179,185,190,195)], [(1,98,48,73),(2,99,49,74),(3,100,50,75),(4,96,46,71),(5,97,47,72),(6,145,195,170),(7,141,191,166),(8,142,192,167),(9,143,193,168),(10,144,194,169),(11,140,190,165),(12,136,186,161),(13,137,187,162),(14,138,188,163),(15,139,189,164),(16,135,185,160),(17,131,181,156),(18,132,182,157),(19,133,183,158),(20,134,184,159),(21,130,180,155),(22,126,176,151),(23,127,177,152),(24,128,178,153),(25,129,179,154),(26,116,51,76),(27,117,52,77),(28,118,53,78),(29,119,54,79),(30,120,55,80),(31,111,56,81),(32,112,57,82),(33,113,58,83),(34,114,59,84),(35,115,60,85),(36,106,61,86),(37,107,62,87),(38,108,63,88),(39,109,64,89),(40,110,65,90),(41,101,66,91),(42,102,67,92),(43,103,68,93),(44,104,69,94),(45,105,70,95),(121,171,146,196),(122,172,147,197),(123,173,148,198),(124,174,149,199),(125,175,150,200)], [(1,157,48,132),(2,156,49,131),(3,160,50,135),(4,159,46,134),(5,158,47,133),(6,115,195,85),(7,114,191,84),(8,113,192,83),(9,112,193,82),(10,111,194,81),(11,120,190,80),(12,119,186,79),(13,118,187,78),(14,117,188,77),(15,116,189,76),(16,100,185,75),(17,99,181,74),(18,98,182,73),(19,97,183,72),(20,96,184,71),(21,105,180,95),(22,104,176,94),(23,103,177,93),(24,102,178,92),(25,101,179,91),(26,164,51,139),(27,163,52,138),(28,162,53,137),(29,161,54,136),(30,165,55,140),(31,169,56,144),(32,168,57,143),(33,167,58,142),(34,166,59,141),(35,170,60,145),(36,149,61,124),(37,148,62,123),(38,147,63,122),(39,146,64,121),(40,150,65,125),(41,154,66,129),(42,153,67,128),(43,152,68,127),(44,151,69,126),(45,155,70,130),(86,199,106,174),(87,198,107,173),(88,197,108,172),(89,196,109,171),(90,200,110,175)]])
C52⋊4Q8 is a maximal subgroup of
C52⋊3SD16 C52⋊3Q16 C40⋊2D5 C40.D5 C52⋊8SD16 C52⋊7Q16 D5×Dic10 D20⋊5D5 C20.50D10 C20.D10 Q8×C5⋊D5
C52⋊4Q8 is a maximal quotient of
C102.22C22 C20⋊3Dic5
53 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | ··· | 5L | 10A | ··· | 10L | 20A | ··· | 20X |
| order | 1 | 2 | 4 | 4 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 50 | 50 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
53 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D5 | D10 | Dic10 |
| kernel | C52⋊4Q8 | C52⋊6C4 | C5×C20 | C52 | C20 | C10 | C5 |
| # reps | 1 | 2 | 1 | 1 | 12 | 12 | 24 |
Matrix representation of C52⋊4Q8 ►in GL4(𝔽41) generated by
| 34 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 40 | 34 | 0 | 0 |
| 0 | 0 | 7 | 1 |
| 0 | 0 | 33 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 30 | 28 |
| 0 | 0 | 22 | 11 |
| 1 | 7 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 14 | 10 |
| 0 | 0 | 9 | 27 |
G:=sub<GL(4,GF(41))| [34,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,1,34,0,0,0,0,7,33,0,0,1,40],[40,0,0,0,0,40,0,0,0,0,30,22,0,0,28,11],[1,0,0,0,7,40,0,0,0,0,14,9,0,0,10,27] >;
C52⋊4Q8 in GAP, Magma, Sage, TeX
C_5^2\rtimes_4Q_8
% in TeX
G:=Group("C5^2:4Q8"); // GroupNames label
G:=SmallGroup(200,32);
// by ID
G=gap.SmallGroup(200,32);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,20,61,26,643,4004]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
Export