Q8xC5:D5 - GroupNames

G = Q₈×C₅⋊D₅ order 400 = 2⁴·5²

Direct product of Q₈ and C₅⋊D₅

direct product, metabelian, supersoluble, monomial

Aliases: Q₈×C₅⋊D₅, C₂₀.₂₅D₁₀, C₅⋊₃(Q₈×D₅), (C₅×Q₈)⋊₂D₅, C₅²⋊₈(C₂×Q₈), (Q₈×C₅²)⋊₅C₂, C₅²⋊₄Q₈⋊₇C₂, (C₅×C₂₀).₂₅C₂², (C₅×C₁₀).₃₅C₂³, C₁₀.₃₆(C₂²×D₅), C₅²⋊₆C₄.₁₇C₂², C₄.₆(C₂×C₅⋊D₅), (C₄×C₅⋊D₅).₂C₂, C₂.₈(C₂²×C₅⋊D₅), (C₂×C₅⋊D₅).₃₀C₂², SmallGroup(400,197)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅×C₁₀ — Q₈×C₅⋊D₅

Chief series

C₁ — C₅ — C₅² — C₅×C₁₀ — C₂×C₅⋊D₅ — C₄×C₅⋊D₅ — Q₈×C₅⋊D₅

Lower central

C₅² — C₅×C₁₀ — Q₈×C₅⋊D₅

Upper central

C₁ — C₂ — Q₈

Generators and relations for Q₈×C₅⋊D₅
G = < a,b,c,d,e | a⁴=c⁵=d⁵=e²=1, b²=a², bab^-1=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c^-1, ede=d^-1 >

Subgroups: 776 in 152 conjugacy classes, 61 normal (8 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₅, C₂×C₄, Q₈, Q₈, D₅, C₁₀, C₂×Q₈, Dic₅, C₂₀, D₁₀, C₅², Dic₁₀, C₄×D₅, C₅×Q₈, C₅⋊D₅, C₅×C₁₀, Q₈×D₅, C₅²⋊₆C₄, C₅×C₂₀, C₂×C₅⋊D₅, C₅²⋊₄Q₈, C₄×C₅⋊D₅, Q₈×C₅², Q₈×C₅⋊D₅
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₅, C₂×Q₈, D₁₀, C₂²×D₅, C₅⋊D₅, Q₈×D₅, C₂×C₅⋊D₅, C₂²×C₅⋊D₅, Q₈×C₅⋊D₅

Smallest permutation representation of Q₈×C₅⋊D₅
►On 200 points

Generators in S₂₀₀

(1 79 29 54)(2 80 30 55)(3 76 26 51)(4 77 27 52)(5 78 28 53)(6 81 31 56)(7 82 32 57)(8 83 33 58)(9 84 34 59)(10 85 35 60)(11 86 36 61)(12 87 37 62)(13 88 38 63)(14 89 39 64)(15 90 40 65)(16 91 41 66)(17 92 42 67)(18 93 43 68)(19 94 44 69)(20 95 45 70)(21 96 46 71)(22 97 47 72)(23 98 48 73)(24 99 49 74)(25 100 50 75)(101 151 126 176)(102 152 127 177)(103 153 128 178)(104 154 129 179)(105 155 130 180)(106 156 131 181)(107 157 132 182)(108 158 133 183)(109 159 134 184)(110 160 135 185)(111 161 136 186)(112 162 137 187)(113 163 138 188)(114 164 139 189)(115 165 140 190)(116 166 141 191)(117 167 142 192)(118 168 143 193)(119 169 144 194)(120 170 145 195)(121 171 146 196)(122 172 147 197)(123 173 148 198)(124 174 149 199)(125 175 150 200)

(1 129 29 104)(2 130 30 105)(3 126 26 101)(4 127 27 102)(5 128 28 103)(6 131 31 106)(7 132 32 107)(8 133 33 108)(9 134 34 109)(10 135 35 110)(11 136 36 111)(12 137 37 112)(13 138 38 113)(14 139 39 114)(15 140 40 115)(16 141 41 116)(17 142 42 117)(18 143 43 118)(19 144 44 119)(20 145 45 120)(21 146 46 121)(22 147 47 122)(23 148 48 123)(24 149 49 124)(25 150 50 125)(51 176 76 151)(52 177 77 152)(53 178 78 153)(54 179 79 154)(55 180 80 155)(56 181 81 156)(57 182 82 157)(58 183 83 158)(59 184 84 159)(60 185 85 160)(61 186 86 161)(62 187 87 162)(63 188 88 163)(64 189 89 164)(65 190 90 165)(66 191 91 166)(67 192 92 167)(68 193 93 168)(69 194 94 169)(70 195 95 170)(71 196 96 171)(72 197 97 172)(73 198 98 173)(74 199 99 174)(75 200 100 175)

(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 24 19 14 9)(2 25 20 15 10)(3 21 16 11 6)(4 22 17 12 7)(5 23 18 13 8)(26 46 41 36 31)(27 47 42 37 32)(28 48 43 38 33)(29 49 44 39 34)(30 50 45 40 35)(51 71 66 61 56)(52 72 67 62 57)(53 73 68 63 58)(54 74 69 64 59)(55 75 70 65 60)(76 96 91 86 81)(77 97 92 87 82)(78 98 93 88 83)(79 99 94 89 84)(80 100 95 90 85)(101 121 116 111 106)(102 122 117 112 107)(103 123 118 113 108)(104 124 119 114 109)(105 125 120 115 110)(126 146 141 136 131)(127 147 142 137 132)(128 148 143 138 133)(129 149 144 139 134)(130 150 145 140 135)(151 171 166 161 156)(152 172 167 162 157)(153 173 168 163 158)(154 174 169 164 159)(155 175 170 165 160)(176 196 191 186 181)(177 197 192 187 182)(178 198 193 188 183)(179 199 194 189 184)(180 200 195 190 185)

(1 9)(2 8)(3 7)(4 6)(5 10)(11 22)(12 21)(13 25)(14 24)(15 23)(16 17)(18 20)(26 32)(27 31)(28 35)(29 34)(30 33)(36 47)(37 46)(38 50)(39 49)(40 48)(41 42)(43 45)(51 57)(52 56)(53 60)(54 59)(55 58)(61 72)(62 71)(63 75)(64 74)(65 73)(66 67)(68 70)(76 82)(77 81)(78 85)(79 84)(80 83)(86 97)(87 96)(88 100)(89 99)(90 98)(91 92)(93 95)(101 107)(102 106)(103 110)(104 109)(105 108)(111 122)(112 121)(113 125)(114 124)(115 123)(116 117)(118 120)(126 132)(127 131)(128 135)(129 134)(130 133)(136 147)(137 146)(138 150)(139 149)(140 148)(141 142)(143 145)(151 157)(152 156)(153 160)(154 159)(155 158)(161 172)(162 171)(163 175)(164 174)(165 173)(166 167)(168 170)(176 182)(177 181)(178 185)(179 184)(180 183)(186 197)(187 196)(188 200)(189 199)(190 198)(191 192)(193 195)

G:=sub<Sym(200)| (1,79,29,54)(2,80,30,55)(3,76,26,51)(4,77,27,52)(5,78,28,53)(6,81,31,56)(7,82,32,57)(8,83,33,58)(9,84,34,59)(10,85,35,60)(11,86,36,61)(12,87,37,62)(13,88,38,63)(14,89,39,64)(15,90,40,65)(16,91,41,66)(17,92,42,67)(18,93,43,68)(19,94,44,69)(20,95,45,70)(21,96,46,71)(22,97,47,72)(23,98,48,73)(24,99,49,74)(25,100,50,75)(101,151,126,176)(102,152,127,177)(103,153,128,178)(104,154,129,179)(105,155,130,180)(106,156,131,181)(107,157,132,182)(108,158,133,183)(109,159,134,184)(110,160,135,185)(111,161,136,186)(112,162,137,187)(113,163,138,188)(114,164,139,189)(115,165,140,190)(116,166,141,191)(117,167,142,192)(118,168,143,193)(119,169,144,194)(120,170,145,195)(121,171,146,196)(122,172,147,197)(123,173,148,198)(124,174,149,199)(125,175,150,200), (1,129,29,104)(2,130,30,105)(3,126,26,101)(4,127,27,102)(5,128,28,103)(6,131,31,106)(7,132,32,107)(8,133,33,108)(9,134,34,109)(10,135,35,110)(11,136,36,111)(12,137,37,112)(13,138,38,113)(14,139,39,114)(15,140,40,115)(16,141,41,116)(17,142,42,117)(18,143,43,118)(19,144,44,119)(20,145,45,120)(21,146,46,121)(22,147,47,122)(23,148,48,123)(24,149,49,124)(25,150,50,125)(51,176,76,151)(52,177,77,152)(53,178,78,153)(54,179,79,154)(55,180,80,155)(56,181,81,156)(57,182,82,157)(58,183,83,158)(59,184,84,159)(60,185,85,160)(61,186,86,161)(62,187,87,162)(63,188,88,163)(64,189,89,164)(65,190,90,165)(66,191,91,166)(67,192,92,167)(68,193,93,168)(69,194,94,169)(70,195,95,170)(71,196,96,171)(72,197,97,172)(73,198,98,173)(74,199,99,174)(75,200,100,175), (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), (1,24,19,14,9)(2,25,20,15,10)(3,21,16,11,6)(4,22,17,12,7)(5,23,18,13,8)(26,46,41,36,31)(27,47,42,37,32)(28,48,43,38,33)(29,49,44,39,34)(30,50,45,40,35)(51,71,66,61,56)(52,72,67,62,57)(53,73,68,63,58)(54,74,69,64,59)(55,75,70,65,60)(76,96,91,86,81)(77,97,92,87,82)(78,98,93,88,83)(79,99,94,89,84)(80,100,95,90,85)(101,121,116,111,106)(102,122,117,112,107)(103,123,118,113,108)(104,124,119,114,109)(105,125,120,115,110)(126,146,141,136,131)(127,147,142,137,132)(128,148,143,138,133)(129,149,144,139,134)(130,150,145,140,135)(151,171,166,161,156)(152,172,167,162,157)(153,173,168,163,158)(154,174,169,164,159)(155,175,170,165,160)(176,196,191,186,181)(177,197,192,187,182)(178,198,193,188,183)(179,199,194,189,184)(180,200,195,190,185), (1,9)(2,8)(3,7)(4,6)(5,10)(11,22)(12,21)(13,25)(14,24)(15,23)(16,17)(18,20)(26,32)(27,31)(28,35)(29,34)(30,33)(36,47)(37,46)(38,50)(39,49)(40,48)(41,42)(43,45)(51,57)(52,56)(53,60)(54,59)(55,58)(61,72)(62,71)(63,75)(64,74)(65,73)(66,67)(68,70)(76,82)(77,81)(78,85)(79,84)(80,83)(86,97)(87,96)(88,100)(89,99)(90,98)(91,92)(93,95)(101,107)(102,106)(103,110)(104,109)(105,108)(111,122)(112,121)(113,125)(114,124)(115,123)(116,117)(118,120)(126,132)(127,131)(128,135)(129,134)(130,133)(136,147)(137,146)(138,150)(139,149)(140,148)(141,142)(143,145)(151,157)(152,156)(153,160)(154,159)(155,158)(161,172)(162,171)(163,175)(164,174)(165,173)(166,167)(168,170)(176,182)(177,181)(178,185)(179,184)(180,183)(186,197)(187,196)(188,200)(189,199)(190,198)(191,192)(193,195)>;

G:=Group( (1,79,29,54)(2,80,30,55)(3,76,26,51)(4,77,27,52)(5,78,28,53)(6,81,31,56)(7,82,32,57)(8,83,33,58)(9,84,34,59)(10,85,35,60)(11,86,36,61)(12,87,37,62)(13,88,38,63)(14,89,39,64)(15,90,40,65)(16,91,41,66)(17,92,42,67)(18,93,43,68)(19,94,44,69)(20,95,45,70)(21,96,46,71)(22,97,47,72)(23,98,48,73)(24,99,49,74)(25,100,50,75)(101,151,126,176)(102,152,127,177)(103,153,128,178)(104,154,129,179)(105,155,130,180)(106,156,131,181)(107,157,132,182)(108,158,133,183)(109,159,134,184)(110,160,135,185)(111,161,136,186)(112,162,137,187)(113,163,138,188)(114,164,139,189)(115,165,140,190)(116,166,141,191)(117,167,142,192)(118,168,143,193)(119,169,144,194)(120,170,145,195)(121,171,146,196)(122,172,147,197)(123,173,148,198)(124,174,149,199)(125,175,150,200), (1,129,29,104)(2,130,30,105)(3,126,26,101)(4,127,27,102)(5,128,28,103)(6,131,31,106)(7,132,32,107)(8,133,33,108)(9,134,34,109)(10,135,35,110)(11,136,36,111)(12,137,37,112)(13,138,38,113)(14,139,39,114)(15,140,40,115)(16,141,41,116)(17,142,42,117)(18,143,43,118)(19,144,44,119)(20,145,45,120)(21,146,46,121)(22,147,47,122)(23,148,48,123)(24,149,49,124)(25,150,50,125)(51,176,76,151)(52,177,77,152)(53,178,78,153)(54,179,79,154)(55,180,80,155)(56,181,81,156)(57,182,82,157)(58,183,83,158)(59,184,84,159)(60,185,85,160)(61,186,86,161)(62,187,87,162)(63,188,88,163)(64,189,89,164)(65,190,90,165)(66,191,91,166)(67,192,92,167)(68,193,93,168)(69,194,94,169)(70,195,95,170)(71,196,96,171)(72,197,97,172)(73,198,98,173)(74,199,99,174)(75,200,100,175), (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), (1,24,19,14,9)(2,25,20,15,10)(3,21,16,11,6)(4,22,17,12,7)(5,23,18,13,8)(26,46,41,36,31)(27,47,42,37,32)(28,48,43,38,33)(29,49,44,39,34)(30,50,45,40,35)(51,71,66,61,56)(52,72,67,62,57)(53,73,68,63,58)(54,74,69,64,59)(55,75,70,65,60)(76,96,91,86,81)(77,97,92,87,82)(78,98,93,88,83)(79,99,94,89,84)(80,100,95,90,85)(101,121,116,111,106)(102,122,117,112,107)(103,123,118,113,108)(104,124,119,114,109)(105,125,120,115,110)(126,146,141,136,131)(127,147,142,137,132)(128,148,143,138,133)(129,149,144,139,134)(130,150,145,140,135)(151,171,166,161,156)(152,172,167,162,157)(153,173,168,163,158)(154,174,169,164,159)(155,175,170,165,160)(176,196,191,186,181)(177,197,192,187,182)(178,198,193,188,183)(179,199,194,189,184)(180,200,195,190,185), (1,9)(2,8)(3,7)(4,6)(5,10)(11,22)(12,21)(13,25)(14,24)(15,23)(16,17)(18,20)(26,32)(27,31)(28,35)(29,34)(30,33)(36,47)(37,46)(38,50)(39,49)(40,48)(41,42)(43,45)(51,57)(52,56)(53,60)(54,59)(55,58)(61,72)(62,71)(63,75)(64,74)(65,73)(66,67)(68,70)(76,82)(77,81)(78,85)(79,84)(80,83)(86,97)(87,96)(88,100)(89,99)(90,98)(91,92)(93,95)(101,107)(102,106)(103,110)(104,109)(105,108)(111,122)(112,121)(113,125)(114,124)(115,123)(116,117)(118,120)(126,132)(127,131)(128,135)(129,134)(130,133)(136,147)(137,146)(138,150)(139,149)(140,148)(141,142)(143,145)(151,157)(152,156)(153,160)(154,159)(155,158)(161,172)(162,171)(163,175)(164,174)(165,173)(166,167)(168,170)(176,182)(177,181)(178,185)(179,184)(180,183)(186,197)(187,196)(188,200)(189,199)(190,198)(191,192)(193,195) );

G=PermutationGroup([[(1,79,29,54),(2,80,30,55),(3,76,26,51),(4,77,27,52),(5,78,28,53),(6,81,31,56),(7,82,32,57),(8,83,33,58),(9,84,34,59),(10,85,35,60),(11,86,36,61),(12,87,37,62),(13,88,38,63),(14,89,39,64),(15,90,40,65),(16,91,41,66),(17,92,42,67),(18,93,43,68),(19,94,44,69),(20,95,45,70),(21,96,46,71),(22,97,47,72),(23,98,48,73),(24,99,49,74),(25,100,50,75),(101,151,126,176),(102,152,127,177),(103,153,128,178),(104,154,129,179),(105,155,130,180),(106,156,131,181),(107,157,132,182),(108,158,133,183),(109,159,134,184),(110,160,135,185),(111,161,136,186),(112,162,137,187),(113,163,138,188),(114,164,139,189),(115,165,140,190),(116,166,141,191),(117,167,142,192),(118,168,143,193),(119,169,144,194),(120,170,145,195),(121,171,146,196),(122,172,147,197),(123,173,148,198),(124,174,149,199),(125,175,150,200)], [(1,129,29,104),(2,130,30,105),(3,126,26,101),(4,127,27,102),(5,128,28,103),(6,131,31,106),(7,132,32,107),(8,133,33,108),(9,134,34,109),(10,135,35,110),(11,136,36,111),(12,137,37,112),(13,138,38,113),(14,139,39,114),(15,140,40,115),(16,141,41,116),(17,142,42,117),(18,143,43,118),(19,144,44,119),(20,145,45,120),(21,146,46,121),(22,147,47,122),(23,148,48,123),(24,149,49,124),(25,150,50,125),(51,176,76,151),(52,177,77,152),(53,178,78,153),(54,179,79,154),(55,180,80,155),(56,181,81,156),(57,182,82,157),(58,183,83,158),(59,184,84,159),(60,185,85,160),(61,186,86,161),(62,187,87,162),(63,188,88,163),(64,189,89,164),(65,190,90,165),(66,191,91,166),(67,192,92,167),(68,193,93,168),(69,194,94,169),(70,195,95,170),(71,196,96,171),(72,197,97,172),(73,198,98,173),(74,199,99,174),(75,200,100,175)], [(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)], [(1,24,19,14,9),(2,25,20,15,10),(3,21,16,11,6),(4,22,17,12,7),(5,23,18,13,8),(26,46,41,36,31),(27,47,42,37,32),(28,48,43,38,33),(29,49,44,39,34),(30,50,45,40,35),(51,71,66,61,56),(52,72,67,62,57),(53,73,68,63,58),(54,74,69,64,59),(55,75,70,65,60),(76,96,91,86,81),(77,97,92,87,82),(78,98,93,88,83),(79,99,94,89,84),(80,100,95,90,85),(101,121,116,111,106),(102,122,117,112,107),(103,123,118,113,108),(104,124,119,114,109),(105,125,120,115,110),(126,146,141,136,131),(127,147,142,137,132),(128,148,143,138,133),(129,149,144,139,134),(130,150,145,140,135),(151,171,166,161,156),(152,172,167,162,157),(153,173,168,163,158),(154,174,169,164,159),(155,175,170,165,160),(176,196,191,186,181),(177,197,192,187,182),(178,198,193,188,183),(179,199,194,189,184),(180,200,195,190,185)], [(1,9),(2,8),(3,7),(4,6),(5,10),(11,22),(12,21),(13,25),(14,24),(15,23),(16,17),(18,20),(26,32),(27,31),(28,35),(29,34),(30,33),(36,47),(37,46),(38,50),(39,49),(40,48),(41,42),(43,45),(51,57),(52,56),(53,60),(54,59),(55,58),(61,72),(62,71),(63,75),(64,74),(65,73),(66,67),(68,70),(76,82),(77,81),(78,85),(79,84),(80,83),(86,97),(87,96),(88,100),(89,99),(90,98),(91,92),(93,95),(101,107),(102,106),(103,110),(104,109),(105,108),(111,122),(112,121),(113,125),(114,124),(115,123),(116,117),(118,120),(126,132),(127,131),(128,135),(129,134),(130,133),(136,147),(137,146),(138,150),(139,149),(140,148),(141,142),(143,145),(151,157),(152,156),(153,160),(154,159),(155,158),(161,172),(162,171),(163,175),(164,174),(165,173),(166,167),(168,170),(176,182),(177,181),(178,185),(179,184),(180,183),(186,197),(187,196),(188,200),(189,199),(190,198),(191,192),(193,195)]])

70 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	5A	···	5L	10A	···	10L	20A	···	20AJ
order	1	2	2	2	4	4	4	4	4	4	5	···	5	10	···	10	20	···	20
size	1	1	25	25	2	2	2	50	50	50	2	···	2	2	···	2	4	···	4

70 irreducible representations

dim	1	1	1	1	2	2	2	4
type	+	+	+	+	-	+	+	-
image	C₁	C₂	C₂	C₂	Q₈	D₅	D₁₀	Q₈×D₅
kernel	Q₈×C₅⋊D₅	C₅²⋊₄Q₈	C₄×C₅⋊D₅	Q₈×C₅²	C₅⋊D₅	C₅×Q₈	C₂₀	C₅
# reps	1	3	3	1	2	12	36	12

Matrix representation of Q₈×C₅⋊D₅ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	40	0

40	0	0	0	0	0
0	40	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	38	21
0	0	0	0	21	3

34	1	0	0	0	0
40	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

40	7	0	0	0	0
34	7	0	0	0	0
0	0	0	1	0	0
0	0	40	34	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	40	0	0	0	0
40	0	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,38,21,0,0,0,0,21,3],[34,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,34,0,0,0,0,7,7,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Q₈×C₅⋊D₅ in GAP, Magma, Sage, TeX

Q_8\times C_5\rtimes D_5

% in TeX

G:=Group("Q8xC5:D5");

// GroupNames label

G:=SmallGroup(400,197);

// by ID

G=gap.SmallGroup(400,197);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,116,50,1924,11525]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^5=d^5=e^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;

// generators/relations

G = Q8×C5⋊D5 order 400 = 24·52

Direct product of Q8 and C5⋊D5

G = Q₈×C₅⋊D₅ order 400 = 2⁴·5²

Direct product of Q₈ and C₅⋊D₅