direct product, metabelian, nilpotent (class 2), monomial
Aliases: C22⋊C4×C52, C102⋊9C4, C22.2C102, C102.36C22, C22⋊(C5×C20), (C2×C10)⋊3C20, (C10×C20)⋊3C2, (C2×C20)⋊2C10, C23.(C5×C10), C2.1(C10×C20), C10.18(C5×D4), (C5×C10).39D4, C2.1(D4×C52), C10.20(C2×C20), (C2×C102).1C2, (C22×C10).3C10, (C2×C4)⋊1(C5×C10), (C5×C10).68(C2×C4), (C2×C10).18(C2×C10), SmallGroup(400,109)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×C52
G = < a,b,c,d,e | a5=b5=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >
Subgroups: 184 in 136 conjugacy classes, 88 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C23, C10, C10, C22⋊C4, C20, C2×C10, C2×C10, C52, C2×C20, C22×C10, C5×C10, C5×C10, C5×C10, C5×C22⋊C4, C5×C20, C102, C102, C102, C10×C20, C2×C102, C22⋊C4×C52
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, C20, C2×C10, C52, C2×C20, C5×D4, C5×C10, C5×C22⋊C4, C5×C20, C102, C10×C20, D4×C52, C22⋊C4×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 40 93 108 60)(2 36 94 109 56)(3 37 95 110 57)(4 38 91 106 58)(5 39 92 107 59)(6 83 70 52 158)(7 84 66 53 159)(8 85 67 54 160)(9 81 68 55 156)(10 82 69 51 157)(11 88 27 133 64)(12 89 28 134 65)(13 90 29 135 61)(14 86 30 131 62)(15 87 26 132 63)(16 79 165 33 43)(17 80 161 34 44)(18 76 162 35 45)(19 77 163 31 41)(20 78 164 32 42)(21 182 169 122 152)(22 183 170 123 153)(23 184 166 124 154)(24 185 167 125 155)(25 181 168 121 151)(46 145 118 195 177)(47 141 119 191 178)(48 142 120 192 179)(49 143 116 193 180)(50 144 117 194 176)(71 113 190 172 130)(72 114 186 173 126)(73 115 187 174 127)(74 111 188 175 128)(75 112 189 171 129)(96 138 148 101 198)(97 139 149 102 199)(98 140 150 103 200)(99 136 146 104 196)(100 137 147 105 197)
(21 117)(22 118)(23 119)(24 120)(25 116)(46 123)(47 124)(48 125)(49 121)(50 122)(71 147)(72 148)(73 149)(74 150)(75 146)(96 173)(97 174)(98 175)(99 171)(100 172)(101 114)(102 115)(103 111)(104 112)(105 113)(126 138)(127 139)(128 140)(129 136)(130 137)(141 154)(142 155)(143 151)(144 152)(145 153)(166 178)(167 179)(168 180)(169 176)(170 177)(181 193)(182 194)(183 195)(184 191)(185 192)(186 198)(187 199)(188 200)(189 196)(190 197)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 18)(7 19)(8 20)(9 16)(10 17)(11 109)(12 110)(13 106)(14 107)(15 108)(21 117)(22 118)(23 119)(24 120)(25 116)(31 53)(32 54)(33 55)(34 51)(35 52)(36 133)(37 134)(38 135)(39 131)(40 132)(41 159)(42 160)(43 156)(44 157)(45 158)(46 123)(47 124)(48 125)(49 121)(50 122)(56 88)(57 89)(58 90)(59 86)(60 87)(61 91)(62 92)(63 93)(64 94)(65 95)(66 163)(67 164)(68 165)(69 161)(70 162)(71 147)(72 148)(73 149)(74 150)(75 146)(76 83)(77 84)(78 85)(79 81)(80 82)(96 173)(97 174)(98 175)(99 171)(100 172)(101 114)(102 115)(103 111)(104 112)(105 113)(126 138)(127 139)(128 140)(129 136)(130 137)(141 154)(142 155)(143 151)(144 152)(145 153)(166 178)(167 179)(168 180)(169 176)(170 177)(181 193)(182 194)(183 195)(184 191)(185 192)(186 198)(187 199)(188 200)(189 196)(190 197)
(1 173 68 167)(2 174 69 168)(3 175 70 169)(4 171 66 170)(5 172 67 166)(6 21 110 111)(7 22 106 112)(8 23 107 113)(9 24 108 114)(10 25 109 115)(11 102 17 116)(12 103 18 117)(13 104 19 118)(14 105 20 119)(15 101 16 120)(26 96 165 179)(27 97 161 180)(28 98 162 176)(29 99 163 177)(30 100 164 178)(31 46 135 136)(32 47 131 137)(33 48 132 138)(34 49 133 139)(35 50 134 140)(36 127 51 121)(37 128 52 122)(38 129 53 123)(39 130 54 124)(40 126 55 125)(41 145 61 146)(42 141 62 147)(43 142 63 148)(44 143 64 149)(45 144 65 150)(56 187 82 181)(57 188 83 182)(58 189 84 183)(59 190 85 184)(60 186 81 185)(71 160 154 92)(72 156 155 93)(73 157 151 94)(74 158 152 95)(75 159 153 91)(76 194 89 200)(77 195 90 196)(78 191 86 197)(79 192 87 198)(80 193 88 199)
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), (1,40,93,108,60)(2,36,94,109,56)(3,37,95,110,57)(4,38,91,106,58)(5,39,92,107,59)(6,83,70,52,158)(7,84,66,53,159)(8,85,67,54,160)(9,81,68,55,156)(10,82,69,51,157)(11,88,27,133,64)(12,89,28,134,65)(13,90,29,135,61)(14,86,30,131,62)(15,87,26,132,63)(16,79,165,33,43)(17,80,161,34,44)(18,76,162,35,45)(19,77,163,31,41)(20,78,164,32,42)(21,182,169,122,152)(22,183,170,123,153)(23,184,166,124,154)(24,185,167,125,155)(25,181,168,121,151)(46,145,118,195,177)(47,141,119,191,178)(48,142,120,192,179)(49,143,116,193,180)(50,144,117,194,176)(71,113,190,172,130)(72,114,186,173,126)(73,115,187,174,127)(74,111,188,175,128)(75,112,189,171,129)(96,138,148,101,198)(97,139,149,102,199)(98,140,150,103,200)(99,136,146,104,196)(100,137,147,105,197), (21,117)(22,118)(23,119)(24,120)(25,116)(46,123)(47,124)(48,125)(49,121)(50,122)(71,147)(72,148)(73,149)(74,150)(75,146)(96,173)(97,174)(98,175)(99,171)(100,172)(101,114)(102,115)(103,111)(104,112)(105,113)(126,138)(127,139)(128,140)(129,136)(130,137)(141,154)(142,155)(143,151)(144,152)(145,153)(166,178)(167,179)(168,180)(169,176)(170,177)(181,193)(182,194)(183,195)(184,191)(185,192)(186,198)(187,199)(188,200)(189,196)(190,197), (1,26)(2,27)(3,28)(4,29)(5,30)(6,18)(7,19)(8,20)(9,16)(10,17)(11,109)(12,110)(13,106)(14,107)(15,108)(21,117)(22,118)(23,119)(24,120)(25,116)(31,53)(32,54)(33,55)(34,51)(35,52)(36,133)(37,134)(38,135)(39,131)(40,132)(41,159)(42,160)(43,156)(44,157)(45,158)(46,123)(47,124)(48,125)(49,121)(50,122)(56,88)(57,89)(58,90)(59,86)(60,87)(61,91)(62,92)(63,93)(64,94)(65,95)(66,163)(67,164)(68,165)(69,161)(70,162)(71,147)(72,148)(73,149)(74,150)(75,146)(76,83)(77,84)(78,85)(79,81)(80,82)(96,173)(97,174)(98,175)(99,171)(100,172)(101,114)(102,115)(103,111)(104,112)(105,113)(126,138)(127,139)(128,140)(129,136)(130,137)(141,154)(142,155)(143,151)(144,152)(145,153)(166,178)(167,179)(168,180)(169,176)(170,177)(181,193)(182,194)(183,195)(184,191)(185,192)(186,198)(187,199)(188,200)(189,196)(190,197), (1,173,68,167)(2,174,69,168)(3,175,70,169)(4,171,66,170)(5,172,67,166)(6,21,110,111)(7,22,106,112)(8,23,107,113)(9,24,108,114)(10,25,109,115)(11,102,17,116)(12,103,18,117)(13,104,19,118)(14,105,20,119)(15,101,16,120)(26,96,165,179)(27,97,161,180)(28,98,162,176)(29,99,163,177)(30,100,164,178)(31,46,135,136)(32,47,131,137)(33,48,132,138)(34,49,133,139)(35,50,134,140)(36,127,51,121)(37,128,52,122)(38,129,53,123)(39,130,54,124)(40,126,55,125)(41,145,61,146)(42,141,62,147)(43,142,63,148)(44,143,64,149)(45,144,65,150)(56,187,82,181)(57,188,83,182)(58,189,84,183)(59,190,85,184)(60,186,81,185)(71,160,154,92)(72,156,155,93)(73,157,151,94)(74,158,152,95)(75,159,153,91)(76,194,89,200)(77,195,90,196)(78,191,86,197)(79,192,87,198)(80,193,88,199)>;
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), (1,40,93,108,60)(2,36,94,109,56)(3,37,95,110,57)(4,38,91,106,58)(5,39,92,107,59)(6,83,70,52,158)(7,84,66,53,159)(8,85,67,54,160)(9,81,68,55,156)(10,82,69,51,157)(11,88,27,133,64)(12,89,28,134,65)(13,90,29,135,61)(14,86,30,131,62)(15,87,26,132,63)(16,79,165,33,43)(17,80,161,34,44)(18,76,162,35,45)(19,77,163,31,41)(20,78,164,32,42)(21,182,169,122,152)(22,183,170,123,153)(23,184,166,124,154)(24,185,167,125,155)(25,181,168,121,151)(46,145,118,195,177)(47,141,119,191,178)(48,142,120,192,179)(49,143,116,193,180)(50,144,117,194,176)(71,113,190,172,130)(72,114,186,173,126)(73,115,187,174,127)(74,111,188,175,128)(75,112,189,171,129)(96,138,148,101,198)(97,139,149,102,199)(98,140,150,103,200)(99,136,146,104,196)(100,137,147,105,197), (21,117)(22,118)(23,119)(24,120)(25,116)(46,123)(47,124)(48,125)(49,121)(50,122)(71,147)(72,148)(73,149)(74,150)(75,146)(96,173)(97,174)(98,175)(99,171)(100,172)(101,114)(102,115)(103,111)(104,112)(105,113)(126,138)(127,139)(128,140)(129,136)(130,137)(141,154)(142,155)(143,151)(144,152)(145,153)(166,178)(167,179)(168,180)(169,176)(170,177)(181,193)(182,194)(183,195)(184,191)(185,192)(186,198)(187,199)(188,200)(189,196)(190,197), (1,26)(2,27)(3,28)(4,29)(5,30)(6,18)(7,19)(8,20)(9,16)(10,17)(11,109)(12,110)(13,106)(14,107)(15,108)(21,117)(22,118)(23,119)(24,120)(25,116)(31,53)(32,54)(33,55)(34,51)(35,52)(36,133)(37,134)(38,135)(39,131)(40,132)(41,159)(42,160)(43,156)(44,157)(45,158)(46,123)(47,124)(48,125)(49,121)(50,122)(56,88)(57,89)(58,90)(59,86)(60,87)(61,91)(62,92)(63,93)(64,94)(65,95)(66,163)(67,164)(68,165)(69,161)(70,162)(71,147)(72,148)(73,149)(74,150)(75,146)(76,83)(77,84)(78,85)(79,81)(80,82)(96,173)(97,174)(98,175)(99,171)(100,172)(101,114)(102,115)(103,111)(104,112)(105,113)(126,138)(127,139)(128,140)(129,136)(130,137)(141,154)(142,155)(143,151)(144,152)(145,153)(166,178)(167,179)(168,180)(169,176)(170,177)(181,193)(182,194)(183,195)(184,191)(185,192)(186,198)(187,199)(188,200)(189,196)(190,197), (1,173,68,167)(2,174,69,168)(3,175,70,169)(4,171,66,170)(5,172,67,166)(6,21,110,111)(7,22,106,112)(8,23,107,113)(9,24,108,114)(10,25,109,115)(11,102,17,116)(12,103,18,117)(13,104,19,118)(14,105,20,119)(15,101,16,120)(26,96,165,179)(27,97,161,180)(28,98,162,176)(29,99,163,177)(30,100,164,178)(31,46,135,136)(32,47,131,137)(33,48,132,138)(34,49,133,139)(35,50,134,140)(36,127,51,121)(37,128,52,122)(38,129,53,123)(39,130,54,124)(40,126,55,125)(41,145,61,146)(42,141,62,147)(43,142,63,148)(44,143,64,149)(45,144,65,150)(56,187,82,181)(57,188,83,182)(58,189,84,183)(59,190,85,184)(60,186,81,185)(71,160,154,92)(72,156,155,93)(73,157,151,94)(74,158,152,95)(75,159,153,91)(76,194,89,200)(77,195,90,196)(78,191,86,197)(79,192,87,198)(80,193,88,199) );
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)], [(1,40,93,108,60),(2,36,94,109,56),(3,37,95,110,57),(4,38,91,106,58),(5,39,92,107,59),(6,83,70,52,158),(7,84,66,53,159),(8,85,67,54,160),(9,81,68,55,156),(10,82,69,51,157),(11,88,27,133,64),(12,89,28,134,65),(13,90,29,135,61),(14,86,30,131,62),(15,87,26,132,63),(16,79,165,33,43),(17,80,161,34,44),(18,76,162,35,45),(19,77,163,31,41),(20,78,164,32,42),(21,182,169,122,152),(22,183,170,123,153),(23,184,166,124,154),(24,185,167,125,155),(25,181,168,121,151),(46,145,118,195,177),(47,141,119,191,178),(48,142,120,192,179),(49,143,116,193,180),(50,144,117,194,176),(71,113,190,172,130),(72,114,186,173,126),(73,115,187,174,127),(74,111,188,175,128),(75,112,189,171,129),(96,138,148,101,198),(97,139,149,102,199),(98,140,150,103,200),(99,136,146,104,196),(100,137,147,105,197)], [(21,117),(22,118),(23,119),(24,120),(25,116),(46,123),(47,124),(48,125),(49,121),(50,122),(71,147),(72,148),(73,149),(74,150),(75,146),(96,173),(97,174),(98,175),(99,171),(100,172),(101,114),(102,115),(103,111),(104,112),(105,113),(126,138),(127,139),(128,140),(129,136),(130,137),(141,154),(142,155),(143,151),(144,152),(145,153),(166,178),(167,179),(168,180),(169,176),(170,177),(181,193),(182,194),(183,195),(184,191),(185,192),(186,198),(187,199),(188,200),(189,196),(190,197)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,18),(7,19),(8,20),(9,16),(10,17),(11,109),(12,110),(13,106),(14,107),(15,108),(21,117),(22,118),(23,119),(24,120),(25,116),(31,53),(32,54),(33,55),(34,51),(35,52),(36,133),(37,134),(38,135),(39,131),(40,132),(41,159),(42,160),(43,156),(44,157),(45,158),(46,123),(47,124),(48,125),(49,121),(50,122),(56,88),(57,89),(58,90),(59,86),(60,87),(61,91),(62,92),(63,93),(64,94),(65,95),(66,163),(67,164),(68,165),(69,161),(70,162),(71,147),(72,148),(73,149),(74,150),(75,146),(76,83),(77,84),(78,85),(79,81),(80,82),(96,173),(97,174),(98,175),(99,171),(100,172),(101,114),(102,115),(103,111),(104,112),(105,113),(126,138),(127,139),(128,140),(129,136),(130,137),(141,154),(142,155),(143,151),(144,152),(145,153),(166,178),(167,179),(168,180),(169,176),(170,177),(181,193),(182,194),(183,195),(184,191),(185,192),(186,198),(187,199),(188,200),(189,196),(190,197)], [(1,173,68,167),(2,174,69,168),(3,175,70,169),(4,171,66,170),(5,172,67,166),(6,21,110,111),(7,22,106,112),(8,23,107,113),(9,24,108,114),(10,25,109,115),(11,102,17,116),(12,103,18,117),(13,104,19,118),(14,105,20,119),(15,101,16,120),(26,96,165,179),(27,97,161,180),(28,98,162,176),(29,99,163,177),(30,100,164,178),(31,46,135,136),(32,47,131,137),(33,48,132,138),(34,49,133,139),(35,50,134,140),(36,127,51,121),(37,128,52,122),(38,129,53,123),(39,130,54,124),(40,126,55,125),(41,145,61,146),(42,141,62,147),(43,142,63,148),(44,143,64,149),(45,144,65,150),(56,187,82,181),(57,188,83,182),(58,189,84,183),(59,190,85,184),(60,186,81,185),(71,160,154,92),(72,156,155,93),(73,157,151,94),(74,158,152,95),(75,159,153,91),(76,194,89,200),(77,195,90,196),(78,191,86,197),(79,192,87,198),(80,193,88,199)]])
250 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | ··· | 5X | 10A | ··· | 10BT | 10BU | ··· | 10DP | 20A | ··· | 20CR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
250 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | D4 | C5×D4 |
| kernel | C22⋊C4×C52 | C10×C20 | C2×C102 | C102 | C5×C22⋊C4 | C2×C20 | C22×C10 | C2×C10 | C5×C10 | C10 |
| # reps | 1 | 2 | 1 | 4 | 24 | 48 | 24 | 96 | 2 | 48 |
Matrix representation of C22⋊C4×C52 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 39 | 40 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,1,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,32,0,0,0,0,1,39,0,0,0,40] >;
C22⋊C4×C52 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times C_5^2
% in TeX
G:=Group("C2^2:C4xC5^2"); // GroupNames label
G:=SmallGroup(400,109);
// by ID
G=gap.SmallGroup(400,109);
# by ID
G:=PCGroup([6,-2,-2,-5,-5,-2,-2,1200,1225]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^5=c^2=d^2=e^4=1,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,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations