direct product, metacyclic, nilpotent (class 2), monomial
Aliases: Q8×C52, C20.5C10, C2.2C102, C4.(C5×C10), (C5×C20).5C2, C10.9(C2×C10), (C5×C10).17C22, SmallGroup(200,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C52
G = < a,b,c,d | a5=b5=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of Q8×C52
►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 67 56 35 8)(2 68 57 31 9)(3 69 58 32 10)(4 70 59 33 6)(5 66 60 34 7)(11 194 176 189 22)(12 195 177 190 23)(13 191 178 186 24)(14 192 179 187 25)(15 193 180 188 21)(16 41 30 71 84)(17 42 26 72 85)(18 43 27 73 81)(19 44 28 74 82)(20 45 29 75 83)(36 49 165 154 198)(37 50 161 155 199)(38 46 162 151 200)(39 47 163 152 196)(40 48 164 153 197)(51 100 122 135 62)(52 96 123 131 63)(53 97 124 132 64)(54 98 125 133 65)(55 99 121 134 61)(76 89 116 110 94)(77 90 117 106 95)(78 86 118 107 91)(79 87 119 108 92)(80 88 120 109 93)(101 150 172 185 112)(102 146 173 181 113)(103 147 174 182 114)(104 148 175 183 115)(105 149 171 184 111)(126 139 166 160 144)(127 140 167 156 145)(128 136 168 157 141)(129 137 169 158 142)(130 138 170 159 143)
(1 132 82 90)(2 133 83 86)(3 134 84 87)(4 135 85 88)(5 131 81 89)(6 122 72 80)(7 123 73 76)(8 124 74 77)(9 125 75 78)(10 121 71 79)(11 102 152 160)(12 103 153 156)(13 104 154 157)(14 105 155 158)(15 101 151 159)(16 119 69 61)(17 120 70 62)(18 116 66 63)(19 117 67 64)(20 118 68 65)(21 112 162 170)(22 113 163 166)(23 114 164 167)(24 115 165 168)(25 111 161 169)(26 93 33 100)(27 94 34 96)(28 95 35 97)(29 91 31 98)(30 92 32 99)(36 128 178 175)(37 129 179 171)(38 130 180 172)(39 126 176 173)(40 127 177 174)(41 108 58 55)(42 109 59 51)(43 110 60 52)(44 106 56 53)(45 107 57 54)(46 138 188 185)(47 139 189 181)(48 140 190 182)(49 136 186 183)(50 137 187 184)(141 191 148 198)(142 192 149 199)(143 193 150 200)(144 194 146 196)(145 195 147 197)
(1 182 82 140)(2 183 83 136)(3 184 84 137)(4 185 85 138)(5 181 81 139)(6 172 72 130)(7 173 73 126)(8 174 74 127)(9 175 75 128)(10 171 71 129)(11 110 152 52)(12 106 153 53)(13 107 154 54)(14 108 155 55)(15 109 151 51)(16 169 69 111)(17 170 70 112)(18 166 66 113)(19 167 67 114)(20 168 68 115)(21 120 162 62)(22 116 163 63)(23 117 164 64)(24 118 165 65)(25 119 161 61)(26 143 33 150)(27 144 34 146)(28 145 35 147)(29 141 31 148)(30 142 32 149)(36 125 178 78)(37 121 179 79)(38 122 180 80)(39 123 176 76)(40 124 177 77)(41 158 58 105)(42 159 59 101)(43 160 60 102)(44 156 56 103)(45 157 57 104)(46 135 188 88)(47 131 189 89)(48 132 190 90)(49 133 186 86)(50 134 187 87)(91 198 98 191)(92 199 99 192)(93 200 100 193)(94 196 96 194)(95 197 97 195)
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), (1,67,56,35,8)(2,68,57,31,9)(3,69,58,32,10)(4,70,59,33,6)(5,66,60,34,7)(11,194,176,189,22)(12,195,177,190,23)(13,191,178,186,24)(14,192,179,187,25)(15,193,180,188,21)(16,41,30,71,84)(17,42,26,72,85)(18,43,27,73,81)(19,44,28,74,82)(20,45,29,75,83)(36,49,165,154,198)(37,50,161,155,199)(38,46,162,151,200)(39,47,163,152,196)(40,48,164,153,197)(51,100,122,135,62)(52,96,123,131,63)(53,97,124,132,64)(54,98,125,133,65)(55,99,121,134,61)(76,89,116,110,94)(77,90,117,106,95)(78,86,118,107,91)(79,87,119,108,92)(80,88,120,109,93)(101,150,172,185,112)(102,146,173,181,113)(103,147,174,182,114)(104,148,175,183,115)(105,149,171,184,111)(126,139,166,160,144)(127,140,167,156,145)(128,136,168,157,141)(129,137,169,158,142)(130,138,170,159,143), (1,132,82,90)(2,133,83,86)(3,134,84,87)(4,135,85,88)(5,131,81,89)(6,122,72,80)(7,123,73,76)(8,124,74,77)(9,125,75,78)(10,121,71,79)(11,102,152,160)(12,103,153,156)(13,104,154,157)(14,105,155,158)(15,101,151,159)(16,119,69,61)(17,120,70,62)(18,116,66,63)(19,117,67,64)(20,118,68,65)(21,112,162,170)(22,113,163,166)(23,114,164,167)(24,115,165,168)(25,111,161,169)(26,93,33,100)(27,94,34,96)(28,95,35,97)(29,91,31,98)(30,92,32,99)(36,128,178,175)(37,129,179,171)(38,130,180,172)(39,126,176,173)(40,127,177,174)(41,108,58,55)(42,109,59,51)(43,110,60,52)(44,106,56,53)(45,107,57,54)(46,138,188,185)(47,139,189,181)(48,140,190,182)(49,136,186,183)(50,137,187,184)(141,191,148,198)(142,192,149,199)(143,193,150,200)(144,194,146,196)(145,195,147,197), (1,182,82,140)(2,183,83,136)(3,184,84,137)(4,185,85,138)(5,181,81,139)(6,172,72,130)(7,173,73,126)(8,174,74,127)(9,175,75,128)(10,171,71,129)(11,110,152,52)(12,106,153,53)(13,107,154,54)(14,108,155,55)(15,109,151,51)(16,169,69,111)(17,170,70,112)(18,166,66,113)(19,167,67,114)(20,168,68,115)(21,120,162,62)(22,116,163,63)(23,117,164,64)(24,118,165,65)(25,119,161,61)(26,143,33,150)(27,144,34,146)(28,145,35,147)(29,141,31,148)(30,142,32,149)(36,125,178,78)(37,121,179,79)(38,122,180,80)(39,123,176,76)(40,124,177,77)(41,158,58,105)(42,159,59,101)(43,160,60,102)(44,156,56,103)(45,157,57,104)(46,135,188,88)(47,131,189,89)(48,132,190,90)(49,133,186,86)(50,134,187,87)(91,198,98,191)(92,199,99,192)(93,200,100,193)(94,196,96,194)(95,197,97,195)>;
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), (1,67,56,35,8)(2,68,57,31,9)(3,69,58,32,10)(4,70,59,33,6)(5,66,60,34,7)(11,194,176,189,22)(12,195,177,190,23)(13,191,178,186,24)(14,192,179,187,25)(15,193,180,188,21)(16,41,30,71,84)(17,42,26,72,85)(18,43,27,73,81)(19,44,28,74,82)(20,45,29,75,83)(36,49,165,154,198)(37,50,161,155,199)(38,46,162,151,200)(39,47,163,152,196)(40,48,164,153,197)(51,100,122,135,62)(52,96,123,131,63)(53,97,124,132,64)(54,98,125,133,65)(55,99,121,134,61)(76,89,116,110,94)(77,90,117,106,95)(78,86,118,107,91)(79,87,119,108,92)(80,88,120,109,93)(101,150,172,185,112)(102,146,173,181,113)(103,147,174,182,114)(104,148,175,183,115)(105,149,171,184,111)(126,139,166,160,144)(127,140,167,156,145)(128,136,168,157,141)(129,137,169,158,142)(130,138,170,159,143), (1,132,82,90)(2,133,83,86)(3,134,84,87)(4,135,85,88)(5,131,81,89)(6,122,72,80)(7,123,73,76)(8,124,74,77)(9,125,75,78)(10,121,71,79)(11,102,152,160)(12,103,153,156)(13,104,154,157)(14,105,155,158)(15,101,151,159)(16,119,69,61)(17,120,70,62)(18,116,66,63)(19,117,67,64)(20,118,68,65)(21,112,162,170)(22,113,163,166)(23,114,164,167)(24,115,165,168)(25,111,161,169)(26,93,33,100)(27,94,34,96)(28,95,35,97)(29,91,31,98)(30,92,32,99)(36,128,178,175)(37,129,179,171)(38,130,180,172)(39,126,176,173)(40,127,177,174)(41,108,58,55)(42,109,59,51)(43,110,60,52)(44,106,56,53)(45,107,57,54)(46,138,188,185)(47,139,189,181)(48,140,190,182)(49,136,186,183)(50,137,187,184)(141,191,148,198)(142,192,149,199)(143,193,150,200)(144,194,146,196)(145,195,147,197), (1,182,82,140)(2,183,83,136)(3,184,84,137)(4,185,85,138)(5,181,81,139)(6,172,72,130)(7,173,73,126)(8,174,74,127)(9,175,75,128)(10,171,71,129)(11,110,152,52)(12,106,153,53)(13,107,154,54)(14,108,155,55)(15,109,151,51)(16,169,69,111)(17,170,70,112)(18,166,66,113)(19,167,67,114)(20,168,68,115)(21,120,162,62)(22,116,163,63)(23,117,164,64)(24,118,165,65)(25,119,161,61)(26,143,33,150)(27,144,34,146)(28,145,35,147)(29,141,31,148)(30,142,32,149)(36,125,178,78)(37,121,179,79)(38,122,180,80)(39,123,176,76)(40,124,177,77)(41,158,58,105)(42,159,59,101)(43,160,60,102)(44,156,56,103)(45,157,57,104)(46,135,188,88)(47,131,189,89)(48,132,190,90)(49,133,186,86)(50,134,187,87)(91,198,98,191)(92,199,99,192)(93,200,100,193)(94,196,96,194)(95,197,97,195) );
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)], [(1,67,56,35,8),(2,68,57,31,9),(3,69,58,32,10),(4,70,59,33,6),(5,66,60,34,7),(11,194,176,189,22),(12,195,177,190,23),(13,191,178,186,24),(14,192,179,187,25),(15,193,180,188,21),(16,41,30,71,84),(17,42,26,72,85),(18,43,27,73,81),(19,44,28,74,82),(20,45,29,75,83),(36,49,165,154,198),(37,50,161,155,199),(38,46,162,151,200),(39,47,163,152,196),(40,48,164,153,197),(51,100,122,135,62),(52,96,123,131,63),(53,97,124,132,64),(54,98,125,133,65),(55,99,121,134,61),(76,89,116,110,94),(77,90,117,106,95),(78,86,118,107,91),(79,87,119,108,92),(80,88,120,109,93),(101,150,172,185,112),(102,146,173,181,113),(103,147,174,182,114),(104,148,175,183,115),(105,149,171,184,111),(126,139,166,160,144),(127,140,167,156,145),(128,136,168,157,141),(129,137,169,158,142),(130,138,170,159,143)], [(1,132,82,90),(2,133,83,86),(3,134,84,87),(4,135,85,88),(5,131,81,89),(6,122,72,80),(7,123,73,76),(8,124,74,77),(9,125,75,78),(10,121,71,79),(11,102,152,160),(12,103,153,156),(13,104,154,157),(14,105,155,158),(15,101,151,159),(16,119,69,61),(17,120,70,62),(18,116,66,63),(19,117,67,64),(20,118,68,65),(21,112,162,170),(22,113,163,166),(23,114,164,167),(24,115,165,168),(25,111,161,169),(26,93,33,100),(27,94,34,96),(28,95,35,97),(29,91,31,98),(30,92,32,99),(36,128,178,175),(37,129,179,171),(38,130,180,172),(39,126,176,173),(40,127,177,174),(41,108,58,55),(42,109,59,51),(43,110,60,52),(44,106,56,53),(45,107,57,54),(46,138,188,185),(47,139,189,181),(48,140,190,182),(49,136,186,183),(50,137,187,184),(141,191,148,198),(142,192,149,199),(143,193,150,200),(144,194,146,196),(145,195,147,197)], [(1,182,82,140),(2,183,83,136),(3,184,84,137),(4,185,85,138),(5,181,81,139),(6,172,72,130),(7,173,73,126),(8,174,74,127),(9,175,75,128),(10,171,71,129),(11,110,152,52),(12,106,153,53),(13,107,154,54),(14,108,155,55),(15,109,151,51),(16,169,69,111),(17,170,70,112),(18,166,66,113),(19,167,67,114),(20,168,68,115),(21,120,162,62),(22,116,163,63),(23,117,164,64),(24,118,165,65),(25,119,161,61),(26,143,33,150),(27,144,34,146),(28,145,35,147),(29,141,31,148),(30,142,32,149),(36,125,178,78),(37,121,179,79),(38,122,180,80),(39,123,176,76),(40,124,177,77),(41,158,58,105),(42,159,59,101),(43,160,60,102),(44,156,56,103),(45,157,57,104),(46,135,188,88),(47,131,189,89),(48,132,190,90),(49,133,186,86),(50,134,187,87),(91,198,98,191),(92,199,99,192),(93,200,100,193),(94,196,96,194),(95,197,97,195)]])
Q8×C52 is a maximal subgroup of
C52⋊10SD16 C52⋊7Q16 C20.26D10
125 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | ··· | 5X | 10A | ··· | 10X | 20A | ··· | 20BT |
| order | 1 | 2 | 4 | 4 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C5 | C10 | Q8 | C5×Q8 |
| kernel | Q8×C52 | C5×C20 | C5×Q8 | C20 | C52 | C5 |
| # reps | 1 | 3 | 24 | 72 | 1 | 24 |
Matrix representation of Q8×C52 ►in GL3(𝔽41) generated by
| 10 | 0 | 0 |
| 0 | 37 | 0 |
| 0 | 0 | 37 |
| 18 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 40 | 39 |
| 0 | 1 | 1 |
| 40 | 0 | 0 |
| 0 | 23 | 6 |
| 0 | 21 | 18 |
G:=sub<GL(3,GF(41))| [10,0,0,0,37,0,0,0,37],[18,0,0,0,1,0,0,0,1],[1,0,0,0,40,1,0,39,1],[40,0,0,0,23,21,0,6,18] >;
Q8×C52 in GAP, Magma, Sage, TeX
Q_8\times C_5^2
% in TeX
G:=Group("Q8xC5^2"); // GroupNames label
G:=SmallGroup(200,39);
// by ID
G=gap.SmallGroup(200,39);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-2,500,1021,506]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export