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## G = C4○D4×C52order 400 = 24·52

### Direct product of C52 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C52
 Chief series C1 — C2 — C10 — C5×C10 — C102 — D4×C52 — C4○D4×C52
 Lower central C1 — C2 — C4○D4×C52
 Upper central C1 — C5×C20 — C4○D4×C52

Generators and relations for C4○D4×C52
G = < a,b,c,d,e | a5=b5=c4=e2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 184 in 160 conjugacy classes, 136 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, D4, Q8, C10, C10, C4○D4, C20, C2×C10, C52, C2×C20, C5×D4, C5×Q8, C5×C10, C5×C10, C5×C4○D4, C5×C20, C5×C20, C102, C10×C20, D4×C52, Q8×C52, C4○D4×C52
Quotients: C1, C2, C22, C5, C23, C10, C4○D4, C2×C10, C52, C22×C10, C5×C10, C5×C4○D4, C102, C2×C102, C4○D4×C52

Smallest permutation representation of C4○D4×C52
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 60 75 44 20)(2 56 71 45 16)(3 57 72 41 17)(4 58 73 42 18)(5 59 74 43 19)(6 33 27 66 81)(7 34 28 67 82)(8 35 29 68 83)(9 31 30 69 84)(10 32 26 70 85)(11 36 195 165 190)(12 37 191 161 186)(13 38 192 162 187)(14 39 193 163 188)(15 40 194 164 189)(21 46 155 180 200)(22 47 151 176 196)(23 48 152 177 197)(24 49 153 178 198)(25 50 154 179 199)(51 76 96 116 131)(52 77 97 117 132)(53 78 98 118 133)(54 79 99 119 134)(55 80 100 120 135)(61 86 106 121 91)(62 87 107 122 92)(63 88 108 123 93)(64 89 109 124 94)(65 90 110 125 95)(101 126 146 166 181)(102 127 147 167 182)(103 128 148 168 183)(104 129 149 169 184)(105 130 150 170 185)(111 136 156 171 141)(112 137 157 172 142)(113 138 158 173 143)(114 139 159 174 144)(115 140 160 175 145)
(1 132 82 90)(2 133 83 86)(3 134 84 87)(4 135 85 88)(5 131 81 89)(6 109 59 51)(7 110 60 52)(8 106 56 53)(9 107 57 54)(10 108 58 55)(11 160 152 102)(12 156 153 103)(13 157 154 104)(14 158 155 105)(15 159 151 101)(16 118 68 61)(17 119 69 62)(18 120 70 63)(19 116 66 64)(20 117 67 65)(21 170 163 113)(22 166 164 114)(23 167 165 115)(24 168 161 111)(25 169 162 112)(26 93 42 100)(27 94 43 96)(28 95 44 97)(29 91 45 98)(30 92 41 99)(31 122 72 79)(32 123 73 80)(33 124 74 76)(34 125 75 77)(35 121 71 78)(36 175 177 127)(37 171 178 128)(38 172 179 129)(39 173 180 130)(40 174 176 126)(46 185 188 138)(47 181 189 139)(48 182 190 140)(49 183 186 136)(50 184 187 137)(141 198 148 191)(142 199 149 192)(143 200 150 193)(144 196 146 194)(145 197 147 195)
(1 182 82 140)(2 183 83 136)(3 184 84 137)(4 185 85 138)(5 181 81 139)(6 159 59 101)(7 160 60 102)(8 156 56 103)(9 157 57 104)(10 158 58 105)(11 110 152 52)(12 106 153 53)(13 107 154 54)(14 108 155 55)(15 109 151 51)(16 168 68 111)(17 169 69 112)(18 170 70 113)(19 166 66 114)(20 167 67 115)(21 120 163 63)(22 116 164 64)(23 117 165 65)(24 118 161 61)(25 119 162 62)(26 143 42 150)(27 144 43 146)(28 145 44 147)(29 141 45 148)(30 142 41 149)(31 172 72 129)(32 173 73 130)(33 174 74 126)(34 175 75 127)(35 171 71 128)(36 125 177 77)(37 121 178 78)(38 122 179 79)(39 123 180 80)(40 124 176 76)(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)
(1 140)(2 136)(3 137)(4 138)(5 139)(6 101)(7 102)(8 103)(9 104)(10 105)(11 110)(12 106)(13 107)(14 108)(15 109)(16 111)(17 112)(18 113)(19 114)(20 115)(21 120)(22 116)(23 117)(24 118)(25 119)(26 150)(27 146)(28 147)(29 148)(30 149)(31 129)(32 130)(33 126)(34 127)(35 128)(36 125)(37 121)(38 122)(39 123)(40 124)(41 142)(42 143)(43 144)(44 145)(45 141)(46 135)(47 131)(48 132)(49 133)(50 134)(51 151)(52 152)(53 153)(54 154)(55 155)(56 156)(57 157)(58 158)(59 159)(60 160)(61 161)(62 162)(63 163)(64 164)(65 165)(66 166)(67 167)(68 168)(69 169)(70 170)(71 171)(72 172)(73 173)(74 174)(75 175)(76 176)(77 177)(78 178)(79 179)(80 180)(81 181)(82 182)(83 183)(84 184)(85 185)(86 186)(87 187)(88 188)(89 189)(90 190)(91 191)(92 192)(93 193)(94 194)(95 195)(96 196)(97 197)(98 198)(99 199)(100 200)

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

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

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

250 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A ··· 5X 10A ··· 10X 10Y ··· 10CR 20A ··· 20AV 20AW ··· 20DP order 1 2 2 2 2 4 4 4 4 4 5 ··· 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 2 2 2 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

250 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C5 C10 C10 C10 C4○D4 C5×C4○D4 kernel C4○D4×C52 C10×C20 D4×C52 Q8×C52 C5×C4○D4 C2×C20 C5×D4 C5×Q8 C52 C5 # reps 1 3 3 1 24 72 72 24 2 48

Matrix representation of C4○D4×C52 in GL3(𝔽41) generated by

 16 0 0 0 18 0 0 0 18
,
 16 0 0 0 10 0 0 0 10
,
 1 0 0 0 32 0 0 0 32
,
 1 0 0 0 0 1 0 40 0
,
 40 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(41))| [16,0,0,0,18,0,0,0,18],[16,0,0,0,10,0,0,0,10],[1,0,0,0,32,0,0,0,32],[1,0,0,0,0,40,0,1,0],[40,0,0,0,0,1,0,1,0] >;

C4○D4×C52 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_5^2
% in TeX

G:=Group("C4oD4xC5^2");
// GroupNames label

G:=SmallGroup(400,204);
// by ID

G=gap.SmallGroup(400,204);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-5,-2,2425,914]);
// Polycyclic

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

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