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## G = C2×C52⋊7D4order 400 = 24·52

### Direct product of C2 and C52⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C2×C52⋊7D4
 Chief series C1 — C5 — C52 — C5×C10 — C2×C5⋊D5 — C22×C5⋊D5 — C2×C52⋊7D4
 Lower central C52 — C5×C10 — C2×C52⋊7D4
 Upper central C1 — C22 — C23

Generators and relations for C2×C527D4
G = < a,b,c,d,e | a2=b5=c5=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1192 in 216 conjugacy classes, 75 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, D4, C23, C23, D5, C10, C10, C2×D4, Dic5, D10, C2×C10, C2×C10, C52, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C5⋊D5, C5×C10, C5×C10, C5×C10, C2×C5⋊D4, C526C4, C2×C5⋊D5, C2×C5⋊D5, C102, C102, C102, C2×C526C4, C527D4, C22×C5⋊D5, C2×C102, C2×C527D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C5⋊D4, C22×D5, C5⋊D5, C2×C5⋊D4, C2×C5⋊D5, C527D4, C22×C5⋊D5, C2×C527D4

Smallest permutation representation of C2×C527D4
On 200 points
Generators in S200
(1 98)(2 99)(3 100)(4 96)(5 97)(6 145)(7 141)(8 142)(9 143)(10 144)(11 140)(12 136)(13 137)(14 138)(15 139)(16 135)(17 131)(18 132)(19 133)(20 134)(21 130)(22 126)(23 127)(24 128)(25 129)(26 116)(27 117)(28 118)(29 119)(30 120)(31 111)(32 112)(33 113)(34 114)(35 115)(36 106)(37 107)(38 108)(39 109)(40 110)(41 101)(42 102)(43 103)(44 104)(45 105)(46 71)(47 72)(48 73)(49 74)(50 75)(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)(121 196)(122 197)(123 198)(124 199)(125 200)(146 171)(147 172)(148 173)(149 174)(150 175)(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)
(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 44 38 33 28)(2 45 39 34 29)(3 41 40 35 30)(4 42 36 31 26)(5 43 37 32 27)(6 200 25 16 11)(7 196 21 17 12)(8 197 22 18 13)(9 198 23 19 14)(10 199 24 20 15)(46 67 61 56 51)(47 68 62 57 52)(48 69 63 58 53)(49 70 64 59 54)(50 66 65 60 55)(71 92 86 81 76)(72 93 87 82 77)(73 94 88 83 78)(74 95 89 84 79)(75 91 90 85 80)(96 102 106 111 116)(97 103 107 112 117)(98 104 108 113 118)(99 105 109 114 119)(100 101 110 115 120)(121 130 131 136 141)(122 126 132 137 142)(123 127 133 138 143)(124 128 134 139 144)(125 129 135 140 145)(146 155 156 161 166)(147 151 157 162 167)(148 152 158 163 168)(149 153 159 164 169)(150 154 160 165 170)(171 180 181 186 191)(172 176 182 187 192)(173 177 183 188 193)(174 178 184 189 194)(175 179 185 190 195)
(1 18 48 182)(2 17 49 181)(3 16 50 185)(4 20 46 184)(5 19 47 183)(6 60 195 35)(7 59 191 34)(8 58 192 33)(9 57 193 32)(10 56 194 31)(11 55 190 30)(12 54 186 29)(13 53 187 28)(14 52 188 27)(15 51 189 26)(21 70 180 45)(22 69 176 44)(23 68 177 43)(24 67 178 42)(25 66 179 41)(36 199 61 174)(37 198 62 173)(38 197 63 172)(39 196 64 171)(40 200 65 175)(71 159 96 134)(72 158 97 133)(73 157 98 132)(74 156 99 131)(75 160 100 135)(76 164 116 139)(77 163 117 138)(78 162 118 137)(79 161 119 136)(80 165 120 140)(81 169 111 144)(82 168 112 143)(83 167 113 142)(84 166 114 141)(85 170 115 145)(86 149 106 124)(87 148 107 123)(88 147 108 122)(89 146 109 121)(90 150 110 125)(91 154 101 129)(92 153 102 128)(93 152 103 127)(94 151 104 126)(95 155 105 130)
(1 132)(2 131)(3 135)(4 134)(5 133)(6 115)(7 114)(8 113)(9 112)(10 111)(11 120)(12 119)(13 118)(14 117)(15 116)(16 100)(17 99)(18 98)(19 97)(20 96)(21 105)(22 104)(23 103)(24 102)(25 101)(26 139)(27 138)(28 137)(29 136)(30 140)(31 144)(32 143)(33 142)(34 141)(35 145)(36 124)(37 123)(38 122)(39 121)(40 125)(41 129)(42 128)(43 127)(44 126)(45 130)(46 159)(47 158)(48 157)(49 156)(50 160)(51 164)(52 163)(53 162)(54 161)(55 165)(56 169)(57 168)(58 167)(59 166)(60 170)(61 149)(62 148)(63 147)(64 146)(65 150)(66 154)(67 153)(68 152)(69 151)(70 155)(71 184)(72 183)(73 182)(74 181)(75 185)(76 189)(77 188)(78 187)(79 186)(80 190)(81 194)(82 193)(83 192)(84 191)(85 195)(86 174)(87 173)(88 172)(89 171)(90 175)(91 179)(92 178)(93 177)(94 176)(95 180)(106 199)(107 198)(108 197)(109 196)(110 200)

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

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

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

106 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A ··· 5L 10A ··· 10CF order 1 2 2 2 2 2 2 2 4 4 5 ··· 5 10 ··· 10 size 1 1 1 1 2 2 50 50 50 50 2 ··· 2 2 ··· 2

106 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 D4 D5 D10 C5⋊D4 kernel C2×C52⋊7D4 C2×C52⋊6C4 C52⋊7D4 C22×C5⋊D5 C2×C102 C5×C10 C22×C10 C2×C10 C10 # reps 1 1 4 1 1 2 12 36 48

Matrix representation of C2×C527D4 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 16 0 0 0 0 18 0 0 0 0 37 0 0 0 0 10
,
 10 0 0 0 0 37 0 0 0 0 37 0 0 0 0 10
,
 0 1 0 0 40 0 0 0 0 0 0 40 0 0 1 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,18,0,0,0,0,37,0,0,0,0,10],[10,0,0,0,0,37,0,0,0,0,37,0,0,0,0,10],[0,40,0,0,1,0,0,0,0,0,0,1,0,0,40,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C2×C527D4 in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes_7D_4
% in TeX

G:=Group("C2xC5^2:7D4");
// GroupNames label

G:=SmallGroup(400,200);
// by ID

G=gap.SmallGroup(400,200);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,218,1924,11525]);
// Polycyclic

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

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