direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×D25, C4.6D50, C20.7D10, Dic50⋊4C2, C50.7C23, C100.6C22, D50.9C22, Dic25.4C22, C5.(Q8×D5), C25⋊2(C2×Q8), (Q8×C25)⋊2C2, (C5×Q8).3D5, (C4×D25).1C2, C2.8(C22×D25), C10.25(C22×D5), SmallGroup(400,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×D25
G = < a,b,c,d | a4=c25=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 421 in 57 conjugacy classes, 31 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, Q8, Q8, D5, C10, C2×Q8, Dic5, C20, D10, C25, Dic10, C4×D5, C5×Q8, D25, C50, Q8×D5, Dic25, C100, D50, Dic50, C4×D25, Q8×C25, Q8×D25
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, D10, C22×D5, D25, Q8×D5, D50, C22×D25, Q8×D25
►On 200 points
Generators in S200
(1 77 48 64)(2 78 49 65)(3 79 50 66)(4 80 26 67)(5 81 27 68)(6 82 28 69)(7 83 29 70)(8 84 30 71)(9 85 31 72)(10 86 32 73)(11 87 33 74)(12 88 34 75)(13 89 35 51)(14 90 36 52)(15 91 37 53)(16 92 38 54)(17 93 39 55)(18 94 40 56)(19 95 41 57)(20 96 42 58)(21 97 43 59)(22 98 44 60)(23 99 45 61)(24 100 46 62)(25 76 47 63)(101 161 131 177)(102 162 132 178)(103 163 133 179)(104 164 134 180)(105 165 135 181)(106 166 136 182)(107 167 137 183)(108 168 138 184)(109 169 139 185)(110 170 140 186)(111 171 141 187)(112 172 142 188)(113 173 143 189)(114 174 144 190)(115 175 145 191)(116 151 146 192)(117 152 147 193)(118 153 148 194)(119 154 149 195)(120 155 150 196)(121 156 126 197)(122 157 127 198)(123 158 128 199)(124 159 129 200)(125 160 130 176)
(1 142 48 112)(2 143 49 113)(3 144 50 114)(4 145 26 115)(5 146 27 116)(6 147 28 117)(7 148 29 118)(8 149 30 119)(9 150 31 120)(10 126 32 121)(11 127 33 122)(12 128 34 123)(13 129 35 124)(14 130 36 125)(15 131 37 101)(16 132 38 102)(17 133 39 103)(18 134 40 104)(19 135 41 105)(20 136 42 106)(21 137 43 107)(22 138 44 108)(23 139 45 109)(24 140 46 110)(25 141 47 111)(51 200 89 159)(52 176 90 160)(53 177 91 161)(54 178 92 162)(55 179 93 163)(56 180 94 164)(57 181 95 165)(58 182 96 166)(59 183 97 167)(60 184 98 168)(61 185 99 169)(62 186 100 170)(63 187 76 171)(64 188 77 172)(65 189 78 173)(66 190 79 174)(67 191 80 175)(68 192 81 151)(69 193 82 152)(70 194 83 153)(71 195 84 154)(72 196 85 155)(73 197 86 156)(74 198 87 157)(75 199 88 158)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)(45 50)(46 49)(47 48)(52 75)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(76 77)(78 100)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(101 122)(102 121)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(123 125)(126 132)(127 131)(128 130)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 143)(141 142)(151 167)(152 166)(153 165)(154 164)(155 163)(156 162)(157 161)(158 160)(168 175)(169 174)(170 173)(171 172)(176 199)(177 198)(178 197)(179 196)(180 195)(181 194)(182 193)(183 192)(184 191)(185 190)(186 189)(187 188)
G:=sub<Sym(200)| (1,77,48,64)(2,78,49,65)(3,79,50,66)(4,80,26,67)(5,81,27,68)(6,82,28,69)(7,83,29,70)(8,84,30,71)(9,85,31,72)(10,86,32,73)(11,87,33,74)(12,88,34,75)(13,89,35,51)(14,90,36,52)(15,91,37,53)(16,92,38,54)(17,93,39,55)(18,94,40,56)(19,95,41,57)(20,96,42,58)(21,97,43,59)(22,98,44,60)(23,99,45,61)(24,100,46,62)(25,76,47,63)(101,161,131,177)(102,162,132,178)(103,163,133,179)(104,164,134,180)(105,165,135,181)(106,166,136,182)(107,167,137,183)(108,168,138,184)(109,169,139,185)(110,170,140,186)(111,171,141,187)(112,172,142,188)(113,173,143,189)(114,174,144,190)(115,175,145,191)(116,151,146,192)(117,152,147,193)(118,153,148,194)(119,154,149,195)(120,155,150,196)(121,156,126,197)(122,157,127,198)(123,158,128,199)(124,159,129,200)(125,160,130,176), (1,142,48,112)(2,143,49,113)(3,144,50,114)(4,145,26,115)(5,146,27,116)(6,147,28,117)(7,148,29,118)(8,149,30,119)(9,150,31,120)(10,126,32,121)(11,127,33,122)(12,128,34,123)(13,129,35,124)(14,130,36,125)(15,131,37,101)(16,132,38,102)(17,133,39,103)(18,134,40,104)(19,135,41,105)(20,136,42,106)(21,137,43,107)(22,138,44,108)(23,139,45,109)(24,140,46,110)(25,141,47,111)(51,200,89,159)(52,176,90,160)(53,177,91,161)(54,178,92,162)(55,179,93,163)(56,180,94,164)(57,181,95,165)(58,182,96,166)(59,183,97,167)(60,184,98,168)(61,185,99,169)(62,186,100,170)(63,187,76,171)(64,188,77,172)(65,189,78,173)(66,190,79,174)(67,191,80,175)(68,192,81,151)(69,193,82,152)(70,194,83,153)(71,195,84,154)(72,196,85,155)(73,197,86,156)(74,198,87,157)(75,199,88,158), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(45,50)(46,49)(47,48)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(76,77)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(101,122)(102,121)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(123,125)(126,132)(127,131)(128,130)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143)(141,142)(151,167)(152,166)(153,165)(154,164)(155,163)(156,162)(157,161)(158,160)(168,175)(169,174)(170,173)(171,172)(176,199)(177,198)(178,197)(179,196)(180,195)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188)>;
G:=Group( (1,77,48,64)(2,78,49,65)(3,79,50,66)(4,80,26,67)(5,81,27,68)(6,82,28,69)(7,83,29,70)(8,84,30,71)(9,85,31,72)(10,86,32,73)(11,87,33,74)(12,88,34,75)(13,89,35,51)(14,90,36,52)(15,91,37,53)(16,92,38,54)(17,93,39,55)(18,94,40,56)(19,95,41,57)(20,96,42,58)(21,97,43,59)(22,98,44,60)(23,99,45,61)(24,100,46,62)(25,76,47,63)(101,161,131,177)(102,162,132,178)(103,163,133,179)(104,164,134,180)(105,165,135,181)(106,166,136,182)(107,167,137,183)(108,168,138,184)(109,169,139,185)(110,170,140,186)(111,171,141,187)(112,172,142,188)(113,173,143,189)(114,174,144,190)(115,175,145,191)(116,151,146,192)(117,152,147,193)(118,153,148,194)(119,154,149,195)(120,155,150,196)(121,156,126,197)(122,157,127,198)(123,158,128,199)(124,159,129,200)(125,160,130,176), (1,142,48,112)(2,143,49,113)(3,144,50,114)(4,145,26,115)(5,146,27,116)(6,147,28,117)(7,148,29,118)(8,149,30,119)(9,150,31,120)(10,126,32,121)(11,127,33,122)(12,128,34,123)(13,129,35,124)(14,130,36,125)(15,131,37,101)(16,132,38,102)(17,133,39,103)(18,134,40,104)(19,135,41,105)(20,136,42,106)(21,137,43,107)(22,138,44,108)(23,139,45,109)(24,140,46,110)(25,141,47,111)(51,200,89,159)(52,176,90,160)(53,177,91,161)(54,178,92,162)(55,179,93,163)(56,180,94,164)(57,181,95,165)(58,182,96,166)(59,183,97,167)(60,184,98,168)(61,185,99,169)(62,186,100,170)(63,187,76,171)(64,188,77,172)(65,189,78,173)(66,190,79,174)(67,191,80,175)(68,192,81,151)(69,193,82,152)(70,194,83,153)(71,195,84,154)(72,196,85,155)(73,197,86,156)(74,198,87,157)(75,199,88,158), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(45,50)(46,49)(47,48)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(76,77)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(101,122)(102,121)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(123,125)(126,132)(127,131)(128,130)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143)(141,142)(151,167)(152,166)(153,165)(154,164)(155,163)(156,162)(157,161)(158,160)(168,175)(169,174)(170,173)(171,172)(176,199)(177,198)(178,197)(179,196)(180,195)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188) );
G=PermutationGroup([[(1,77,48,64),(2,78,49,65),(3,79,50,66),(4,80,26,67),(5,81,27,68),(6,82,28,69),(7,83,29,70),(8,84,30,71),(9,85,31,72),(10,86,32,73),(11,87,33,74),(12,88,34,75),(13,89,35,51),(14,90,36,52),(15,91,37,53),(16,92,38,54),(17,93,39,55),(18,94,40,56),(19,95,41,57),(20,96,42,58),(21,97,43,59),(22,98,44,60),(23,99,45,61),(24,100,46,62),(25,76,47,63),(101,161,131,177),(102,162,132,178),(103,163,133,179),(104,164,134,180),(105,165,135,181),(106,166,136,182),(107,167,137,183),(108,168,138,184),(109,169,139,185),(110,170,140,186),(111,171,141,187),(112,172,142,188),(113,173,143,189),(114,174,144,190),(115,175,145,191),(116,151,146,192),(117,152,147,193),(118,153,148,194),(119,154,149,195),(120,155,150,196),(121,156,126,197),(122,157,127,198),(123,158,128,199),(124,159,129,200),(125,160,130,176)], [(1,142,48,112),(2,143,49,113),(3,144,50,114),(4,145,26,115),(5,146,27,116),(6,147,28,117),(7,148,29,118),(8,149,30,119),(9,150,31,120),(10,126,32,121),(11,127,33,122),(12,128,34,123),(13,129,35,124),(14,130,36,125),(15,131,37,101),(16,132,38,102),(17,133,39,103),(18,134,40,104),(19,135,41,105),(20,136,42,106),(21,137,43,107),(22,138,44,108),(23,139,45,109),(24,140,46,110),(25,141,47,111),(51,200,89,159),(52,176,90,160),(53,177,91,161),(54,178,92,162),(55,179,93,163),(56,180,94,164),(57,181,95,165),(58,182,96,166),(59,183,97,167),(60,184,98,168),(61,185,99,169),(62,186,100,170),(63,187,76,171),(64,188,77,172),(65,189,78,173),(66,190,79,174),(67,191,80,175),(68,192,81,151),(69,193,82,152),(70,194,83,153),(71,195,84,154),(72,196,85,155),(73,197,86,156),(74,198,87,157),(75,199,88,158)], [(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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36),(45,50),(46,49),(47,48),(52,75),(53,74),(54,73),(55,72),(56,71),(57,70),(58,69),(59,68),(60,67),(61,66),(62,65),(63,64),(76,77),(78,100),(79,99),(80,98),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(101,122),(102,121),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112),(123,125),(126,132),(127,131),(128,130),(133,150),(134,149),(135,148),(136,147),(137,146),(138,145),(139,144),(140,143),(141,142),(151,167),(152,166),(153,165),(154,164),(155,163),(156,162),(157,161),(158,160),(168,175),(169,174),(170,173),(171,172),(176,199),(177,198),(178,197),(179,196),(180,195),(181,194),(182,193),(183,192),(184,191),(185,190),(186,189),(187,188)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 10A | 10B | 20A | ··· | 20F | 25A | ··· | 25J | 50A | ··· | 50J | 100A | ··· | 100AD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | 10 | 20 | ··· | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 25 | 25 | 2 | 2 | 2 | 50 | 50 | 50 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | Q8 | D5 | D10 | D25 | D50 | Q8×D5 | Q8×D25 |
| kernel | Q8×D25 | Dic50 | C4×D25 | Q8×C25 | D25 | C5×Q8 | C20 | Q8 | C4 | C5 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 2 | 6 | 10 | 30 | 2 | 10 |
Matrix representation of Q8×D25 ►in GL4(𝔽101) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 8 |
| 0 | 0 | 25 | 100 |
| 100 | 0 | 0 | 0 |
| 0 | 100 | 0 | 0 |
| 0 | 0 | 87 | 53 |
| 0 | 0 | 2 | 14 |
| 51 | 97 | 0 | 0 |
| 17 | 68 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 99 | 62 | 0 | 0 |
| 70 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(101))| [1,0,0,0,0,1,0,0,0,0,1,25,0,0,8,100],[100,0,0,0,0,100,0,0,0,0,87,2,0,0,53,14],[51,17,0,0,97,68,0,0,0,0,1,0,0,0,0,1],[99,70,0,0,62,2,0,0,0,0,1,0,0,0,0,1] >;
Q8×D25 in GAP, Magma, Sage, TeX
Q_8\times D_{25} % in TeX
G:=Group("Q8xD25"); // GroupNames label
G:=SmallGroup(400,41);
// by ID
G=gap.SmallGroup(400,41);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,116,50,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^25=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations