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G = C23×C5⋊D5  order 400 = 24·52

Direct product of C23 and C5⋊D5

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C23×C5⋊D5, C52⋊3C24, C102⋊10C22, (C2×C10)⋊8D10, C5⋊2(C23×D5), (C2×C102)⋊5C2, (C5×C10)⋊3C23, (C22×C10)⋊3D5, C10⋊2(C22×D5), SmallGroup(400,220)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C52 — C23×C5⋊D5
Chief series
C1 — C5 — C52 — C5⋊D5 — C2×C5⋊D5 — C22×C5⋊D5 — C23×C5⋊D5
Lower central
C52 — C23×C5⋊D5
Upper central
C1 — C23

Generators and relations for C23×C5⋊D5
 G = < a,b,c,d,e,f | a2=b2=c2=d5=e5=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 2984 in 536 conjugacy classes, 179 normal (5 characteristic)
C1, C2, C2, C22, C22, C5, C23, C23, D5, C10, C24, D10, C2×C10, C52, C22×D5, C22×C10, C5⋊D5, C5×C10, C23×D5, C2×C5⋊D5, C102, C22×C5⋊D5, C2×C102, C23×C5⋊D5
Quotients: C1, C2, C22, C23, D5, C24, D10, C22×D5, C5⋊D5, C23×D5, C2×C5⋊D5, C22×C5⋊D5, C23×C5⋊D5

Smallest permutation representation of C23×C5⋊D5
►On 200 points
Generators in S200
(1 179)(2 180)(3 176)(4 177)(5 178)(6 181)(7 182)(8 183)(9 184)(10 185)(11 186)(12 187)(13 188)(14 189)(15 190)(16 191)(17 192)(18 193)(19 194)(20 195)(21 196)(22 197)(23 198)(24 199)(25 200)(26 151)(27 152)(28 153)(29 154)(30 155)(31 156)(32 157)(33 158)(34 159)(35 160)(36 161)(37 162)(38 163)(39 164)(40 165)(41 166)(42 167)(43 168)(44 169)(45 170)(46 171)(47 172)(48 173)(49 174)(50 175)(51 126)(52 127)(53 128)(54 129)(55 130)(56 131)(57 132)(58 133)(59 134)(60 135)(61 136)(62 137)(63 138)(64 139)(65 140)(66 141)(67 142)(68 143)(69 144)(70 145)(71 146)(72 147)(73 148)(74 149)(75 150)(76 101)(77 102)(78 103)(79 104)(80 105)(81 106)(82 107)(83 108)(84 109)(85 110)(86 111)(87 112)(88 113)(89 114)(90 115)(91 116)(92 117)(93 118)(94 119)(95 120)(96 121)(97 122)(98 123)(99 124)(100 125)

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

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

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

112 conjugacy classes

class 1 2A···2G2H···2O5A···5L10A···10CF
order12···22···25···510···10
size11···125···252···22···2

112 irreducible representations

dim11122
type+++++
imageC1C2C2D5D10
kernelC23×C5⋊D5C22×C5⋊D5C2×C102C22×C10C2×C10
# reps11411284

Matrix representation of C23×C5⋊D5 ►in GL6(𝔽11)

100000
010000
0010000
0001000
000010
000001
,
100000
0100000
0010000
0001000
000010
000001
,
1000000
010000
0010000
0001000
000010
000001
,
100000
010000
001000
000100
0000101
000064
,
100000
010000
000100
0010300
000073
000070
,
1000000
010000
0001000
0010000
0000110
0000010

G:=sub<GL(6,GF(11))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,6,0,0,0,0,1,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,10,0,0,0,0,1,3,0,0,0,0,0,0,7,7,0,0,0,0,3,0],[10,0,0,0,0,0,0,1,0,0,0,0,0,0,0,10,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,10,10] >;
 

C23×C5⋊D5 in GAP, Magma, Sage, TeX 

C_2^3\times C_5\rtimes D_5
 
% in TeX
 
G:=Group("C2^3xC5:D5");
 
// GroupNames label
 
G:=SmallGroup(400,220);
 
// by ID
 
G=gap.SmallGroup(400,220);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,1924,11525]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^5=e^5=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

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