metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2D25, C4.7D50, D100⋊4C2, C20.8D10, C50.8C23, C100.7C22, D50.3C22, Dic25.9C22, (C4×D25)⋊3C2, C25⋊3(C4○D4), (Q8×C25)⋊3C2, (C5×Q8).4D5, C5.(Q8⋊2D5), C2.9(C22×D25), C10.26(C22×D5), SmallGroup(400,42)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2D25
G = < a,b,c,d | a4=c25=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >
Subgroups: 545 in 60 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, D4, Q8, D5, C10, C4○D4, Dic5, C20, D10, C25, C4×D5, D20, C5×Q8, D25, C50, Q8⋊2D5, Dic25, C100, D50, C4×D25, D100, Q8×C25, Q8⋊2D25
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C22×D5, D25, Q8⋊2D5, D50, C22×D25, Q8⋊2D25
►On 200 points
Generators in S200
(1 78 45 69)(2 79 46 70)(3 80 47 71)(4 81 48 72)(5 82 49 73)(6 83 50 74)(7 84 26 75)(8 85 27 51)(9 86 28 52)(10 87 29 53)(11 88 30 54)(12 89 31 55)(13 90 32 56)(14 91 33 57)(15 92 34 58)(16 93 35 59)(17 94 36 60)(18 95 37 61)(19 96 38 62)(20 97 39 63)(21 98 40 64)(22 99 41 65)(23 100 42 66)(24 76 43 67)(25 77 44 68)(101 172 141 179)(102 173 142 180)(103 174 143 181)(104 175 144 182)(105 151 145 183)(106 152 146 184)(107 153 147 185)(108 154 148 186)(109 155 149 187)(110 156 150 188)(111 157 126 189)(112 158 127 190)(113 159 128 191)(114 160 129 192)(115 161 130 193)(116 162 131 194)(117 163 132 195)(118 164 133 196)(119 165 134 197)(120 166 135 198)(121 167 136 199)(122 168 137 200)(123 169 138 176)(124 170 139 177)(125 171 140 178)
(1 134 45 119)(2 135 46 120)(3 136 47 121)(4 137 48 122)(5 138 49 123)(6 139 50 124)(7 140 26 125)(8 141 27 101)(9 142 28 102)(10 143 29 103)(11 144 30 104)(12 145 31 105)(13 146 32 106)(14 147 33 107)(15 148 34 108)(16 149 35 109)(17 150 36 110)(18 126 37 111)(19 127 38 112)(20 128 39 113)(21 129 40 114)(22 130 41 115)(23 131 42 116)(24 132 43 117)(25 133 44 118)(51 179 85 172)(52 180 86 173)(53 181 87 174)(54 182 88 175)(55 183 89 151)(56 184 90 152)(57 185 91 153)(58 186 92 154)(59 187 93 155)(60 188 94 156)(61 189 95 157)(62 190 96 158)(63 191 97 159)(64 192 98 160)(65 193 99 161)(66 194 100 162)(67 195 76 163)(68 196 77 164)(69 197 78 165)(70 198 79 166)(71 199 80 167)(72 200 81 168)(73 176 82 169)(74 177 83 170)(75 178 84 171)
(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 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 75)(20 74)(21 73)(22 72)(23 71)(24 70)(25 69)(26 96)(27 95)(28 94)(29 93)(30 92)(31 91)(32 90)(33 89)(34 88)(35 87)(36 86)(37 85)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 100)(48 99)(49 98)(50 97)(101 157)(102 156)(103 155)(104 154)(105 153)(106 152)(107 151)(108 175)(109 174)(110 173)(111 172)(112 171)(113 170)(114 169)(115 168)(116 167)(117 166)(118 165)(119 164)(120 163)(121 162)(122 161)(123 160)(124 159)(125 158)(126 179)(127 178)(128 177)(129 176)(130 200)(131 199)(132 198)(133 197)(134 196)(135 195)(136 194)(137 193)(138 192)(139 191)(140 190)(141 189)(142 188)(143 187)(144 186)(145 185)(146 184)(147 183)(148 182)(149 181)(150 180)
G:=sub<Sym(200)| (1,78,45,69)(2,79,46,70)(3,80,47,71)(4,81,48,72)(5,82,49,73)(6,83,50,74)(7,84,26,75)(8,85,27,51)(9,86,28,52)(10,87,29,53)(11,88,30,54)(12,89,31,55)(13,90,32,56)(14,91,33,57)(15,92,34,58)(16,93,35,59)(17,94,36,60)(18,95,37,61)(19,96,38,62)(20,97,39,63)(21,98,40,64)(22,99,41,65)(23,100,42,66)(24,76,43,67)(25,77,44,68)(101,172,141,179)(102,173,142,180)(103,174,143,181)(104,175,144,182)(105,151,145,183)(106,152,146,184)(107,153,147,185)(108,154,148,186)(109,155,149,187)(110,156,150,188)(111,157,126,189)(112,158,127,190)(113,159,128,191)(114,160,129,192)(115,161,130,193)(116,162,131,194)(117,163,132,195)(118,164,133,196)(119,165,134,197)(120,166,135,198)(121,167,136,199)(122,168,137,200)(123,169,138,176)(124,170,139,177)(125,171,140,178), (1,134,45,119)(2,135,46,120)(3,136,47,121)(4,137,48,122)(5,138,49,123)(6,139,50,124)(7,140,26,125)(8,141,27,101)(9,142,28,102)(10,143,29,103)(11,144,30,104)(12,145,31,105)(13,146,32,106)(14,147,33,107)(15,148,34,108)(16,149,35,109)(17,150,36,110)(18,126,37,111)(19,127,38,112)(20,128,39,113)(21,129,40,114)(22,130,41,115)(23,131,42,116)(24,132,43,117)(25,133,44,118)(51,179,85,172)(52,180,86,173)(53,181,87,174)(54,182,88,175)(55,183,89,151)(56,184,90,152)(57,185,91,153)(58,186,92,154)(59,187,93,155)(60,188,94,156)(61,189,95,157)(62,190,96,158)(63,191,97,159)(64,192,98,160)(65,193,99,161)(66,194,100,162)(67,195,76,163)(68,196,77,164)(69,197,78,165)(70,198,79,166)(71,199,80,167)(72,200,81,168)(73,176,82,169)(74,177,83,170)(75,178,84,171), (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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,75)(20,74)(21,73)(22,72)(23,71)(24,70)(25,69)(26,96)(27,95)(28,94)(29,93)(30,92)(31,91)(32,90)(33,89)(34,88)(35,87)(36,86)(37,85)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,100)(48,99)(49,98)(50,97)(101,157)(102,156)(103,155)(104,154)(105,153)(106,152)(107,151)(108,175)(109,174)(110,173)(111,172)(112,171)(113,170)(114,169)(115,168)(116,167)(117,166)(118,165)(119,164)(120,163)(121,162)(122,161)(123,160)(124,159)(125,158)(126,179)(127,178)(128,177)(129,176)(130,200)(131,199)(132,198)(133,197)(134,196)(135,195)(136,194)(137,193)(138,192)(139,191)(140,190)(141,189)(142,188)(143,187)(144,186)(145,185)(146,184)(147,183)(148,182)(149,181)(150,180)>;
G:=Group( (1,78,45,69)(2,79,46,70)(3,80,47,71)(4,81,48,72)(5,82,49,73)(6,83,50,74)(7,84,26,75)(8,85,27,51)(9,86,28,52)(10,87,29,53)(11,88,30,54)(12,89,31,55)(13,90,32,56)(14,91,33,57)(15,92,34,58)(16,93,35,59)(17,94,36,60)(18,95,37,61)(19,96,38,62)(20,97,39,63)(21,98,40,64)(22,99,41,65)(23,100,42,66)(24,76,43,67)(25,77,44,68)(101,172,141,179)(102,173,142,180)(103,174,143,181)(104,175,144,182)(105,151,145,183)(106,152,146,184)(107,153,147,185)(108,154,148,186)(109,155,149,187)(110,156,150,188)(111,157,126,189)(112,158,127,190)(113,159,128,191)(114,160,129,192)(115,161,130,193)(116,162,131,194)(117,163,132,195)(118,164,133,196)(119,165,134,197)(120,166,135,198)(121,167,136,199)(122,168,137,200)(123,169,138,176)(124,170,139,177)(125,171,140,178), (1,134,45,119)(2,135,46,120)(3,136,47,121)(4,137,48,122)(5,138,49,123)(6,139,50,124)(7,140,26,125)(8,141,27,101)(9,142,28,102)(10,143,29,103)(11,144,30,104)(12,145,31,105)(13,146,32,106)(14,147,33,107)(15,148,34,108)(16,149,35,109)(17,150,36,110)(18,126,37,111)(19,127,38,112)(20,128,39,113)(21,129,40,114)(22,130,41,115)(23,131,42,116)(24,132,43,117)(25,133,44,118)(51,179,85,172)(52,180,86,173)(53,181,87,174)(54,182,88,175)(55,183,89,151)(56,184,90,152)(57,185,91,153)(58,186,92,154)(59,187,93,155)(60,188,94,156)(61,189,95,157)(62,190,96,158)(63,191,97,159)(64,192,98,160)(65,193,99,161)(66,194,100,162)(67,195,76,163)(68,196,77,164)(69,197,78,165)(70,198,79,166)(71,199,80,167)(72,200,81,168)(73,176,82,169)(74,177,83,170)(75,178,84,171), (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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,75)(20,74)(21,73)(22,72)(23,71)(24,70)(25,69)(26,96)(27,95)(28,94)(29,93)(30,92)(31,91)(32,90)(33,89)(34,88)(35,87)(36,86)(37,85)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,100)(48,99)(49,98)(50,97)(101,157)(102,156)(103,155)(104,154)(105,153)(106,152)(107,151)(108,175)(109,174)(110,173)(111,172)(112,171)(113,170)(114,169)(115,168)(116,167)(117,166)(118,165)(119,164)(120,163)(121,162)(122,161)(123,160)(124,159)(125,158)(126,179)(127,178)(128,177)(129,176)(130,200)(131,199)(132,198)(133,197)(134,196)(135,195)(136,194)(137,193)(138,192)(139,191)(140,190)(141,189)(142,188)(143,187)(144,186)(145,185)(146,184)(147,183)(148,182)(149,181)(150,180) );
G=PermutationGroup([[(1,78,45,69),(2,79,46,70),(3,80,47,71),(4,81,48,72),(5,82,49,73),(6,83,50,74),(7,84,26,75),(8,85,27,51),(9,86,28,52),(10,87,29,53),(11,88,30,54),(12,89,31,55),(13,90,32,56),(14,91,33,57),(15,92,34,58),(16,93,35,59),(17,94,36,60),(18,95,37,61),(19,96,38,62),(20,97,39,63),(21,98,40,64),(22,99,41,65),(23,100,42,66),(24,76,43,67),(25,77,44,68),(101,172,141,179),(102,173,142,180),(103,174,143,181),(104,175,144,182),(105,151,145,183),(106,152,146,184),(107,153,147,185),(108,154,148,186),(109,155,149,187),(110,156,150,188),(111,157,126,189),(112,158,127,190),(113,159,128,191),(114,160,129,192),(115,161,130,193),(116,162,131,194),(117,163,132,195),(118,164,133,196),(119,165,134,197),(120,166,135,198),(121,167,136,199),(122,168,137,200),(123,169,138,176),(124,170,139,177),(125,171,140,178)], [(1,134,45,119),(2,135,46,120),(3,136,47,121),(4,137,48,122),(5,138,49,123),(6,139,50,124),(7,140,26,125),(8,141,27,101),(9,142,28,102),(10,143,29,103),(11,144,30,104),(12,145,31,105),(13,146,32,106),(14,147,33,107),(15,148,34,108),(16,149,35,109),(17,150,36,110),(18,126,37,111),(19,127,38,112),(20,128,39,113),(21,129,40,114),(22,130,41,115),(23,131,42,116),(24,132,43,117),(25,133,44,118),(51,179,85,172),(52,180,86,173),(53,181,87,174),(54,182,88,175),(55,183,89,151),(56,184,90,152),(57,185,91,153),(58,186,92,154),(59,187,93,155),(60,188,94,156),(61,189,95,157),(62,190,96,158),(63,191,97,159),(64,192,98,160),(65,193,99,161),(66,194,100,162),(67,195,76,163),(68,196,77,164),(69,197,78,165),(70,198,79,166),(71,199,80,167),(72,200,81,168),(73,176,82,169),(74,177,83,170),(75,178,84,171)], [(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)], [(1,68),(2,67),(3,66),(4,65),(5,64),(6,63),(7,62),(8,61),(9,60),(10,59),(11,58),(12,57),(13,56),(14,55),(15,54),(16,53),(17,52),(18,51),(19,75),(20,74),(21,73),(22,72),(23,71),(24,70),(25,69),(26,96),(27,95),(28,94),(29,93),(30,92),(31,91),(32,90),(33,89),(34,88),(35,87),(36,86),(37,85),(38,84),(39,83),(40,82),(41,81),(42,80),(43,79),(44,78),(45,77),(46,76),(47,100),(48,99),(49,98),(50,97),(101,157),(102,156),(103,155),(104,154),(105,153),(106,152),(107,151),(108,175),(109,174),(110,173),(111,172),(112,171),(113,170),(114,169),(115,168),(116,167),(117,166),(118,165),(119,164),(120,163),(121,162),(122,161),(123,160),(124,159),(125,158),(126,179),(127,178),(128,177),(129,176),(130,200),(131,199),(132,198),(133,197),(134,196),(135,195),(136,194),(137,193),(138,192),(139,191),(140,190),(141,189),(142,188),(143,187),(144,186),(145,185),(146,184),(147,183),(148,182),(149,181),(150,180)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 10A | 10B | 20A | ··· | 20F | 25A | ··· | 25J | 50A | ··· | 50J | 100A | ··· | 100AD |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | 10 | 20 | ··· | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 50 | 50 | 50 | 2 | 2 | 2 | 25 | 25 | 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 | D5 | C4○D4 | D10 | D25 | D50 | Q8⋊2D5 | Q8⋊2D25 |
| kernel | Q8⋊2D25 | C4×D25 | D100 | Q8×C25 | C5×Q8 | C25 | C20 | Q8 | C4 | C5 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 2 | 6 | 10 | 30 | 2 | 10 |
Matrix representation of Q8⋊2D25 ►in GL4(𝔽101) generated by
| 100 | 0 | 0 | 0 |
| 0 | 100 | 0 | 0 |
| 0 | 0 | 1 | 81 |
| 0 | 0 | 91 | 100 |
| 100 | 0 | 0 | 0 |
| 0 | 100 | 0 | 0 |
| 0 | 0 | 10 | 2 |
| 0 | 0 | 0 | 91 |
| 4 | 55 | 0 | 0 |
| 46 | 52 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 61 | 72 | 0 | 0 |
| 83 | 40 | 0 | 0 |
| 0 | 0 | 1 | 81 |
| 0 | 0 | 0 | 100 |
G:=sub<GL(4,GF(101))| [100,0,0,0,0,100,0,0,0,0,1,91,0,0,81,100],[100,0,0,0,0,100,0,0,0,0,10,0,0,0,2,91],[4,46,0,0,55,52,0,0,0,0,1,0,0,0,0,1],[61,83,0,0,72,40,0,0,0,0,1,0,0,0,81,100] >;
Q8⋊2D25 in GAP, Magma, Sage, TeX
Q_8\rtimes_2D_{25} % in TeX
G:=Group("Q8:2D25"); // GroupNames label
G:=SmallGroup(400,42);
// by ID
G=gap.SmallGroup(400,42);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,116,50,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^25=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations