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## G = C32×C5⋊D4order 360 = 23·32·5

### Direct product of C32 and C5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C32×C5⋊D4
 Chief series C1 — C5 — C10 — C30 — C3×C30 — D5×C3×C6 — C32×C5⋊D4
 Lower central C5 — C10 — C32×C5⋊D4
 Upper central C1 — C3×C6 — C62

Generators and relations for C32×C5⋊D4
G = < a,b,c,d,e | a3=b3=c5=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 240 in 96 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, D4, C32, D5, C10, C10, C12, C2×C6, C2×C6, C15, C3×C6, C3×C6, Dic5, D10, C2×C10, C3×D4, C3×D5, C30, C30, C3×C12, C62, C62, C5⋊D4, C3×C15, C3×Dic5, C6×D5, C2×C30, D4×C32, C32×D5, C3×C30, C3×C30, C3×C5⋊D4, C32×Dic5, D5×C3×C6, C6×C30, C32×C5⋊D4
Quotients: C1, C2, C3, C22, C6, D4, C32, D5, C2×C6, C3×C6, D10, C3×D4, C3×D5, C62, C5⋊D4, C6×D5, D4×C32, C32×D5, C3×C5⋊D4, D5×C3×C6, C32×C5⋊D4

Smallest permutation representation of C32×C5⋊D4
On 180 points
Generators in S180
(1 81 41)(2 82 42)(3 83 43)(4 84 44)(5 85 45)(6 86 46)(7 87 47)(8 88 48)(9 89 49)(10 90 50)(11 61 51)(12 62 52)(13 63 53)(14 64 54)(15 65 55)(16 66 56)(17 67 57)(18 68 58)(19 69 59)(20 70 60)(21 71 31)(22 72 32)(23 73 33)(24 74 34)(25 75 35)(26 76 36)(27 77 37)(28 78 38)(29 79 39)(30 80 40)(91 171 131)(92 172 132)(93 173 133)(94 174 134)(95 175 135)(96 176 136)(97 177 137)(98 178 138)(99 179 139)(100 180 140)(101 151 141)(102 152 142)(103 153 143)(104 154 144)(105 155 145)(106 156 146)(107 157 147)(108 158 148)(109 159 149)(110 160 150)(111 161 121)(112 162 122)(113 163 123)(114 164 124)(115 165 125)(116 166 126)(117 167 127)(118 168 128)(119 169 129)(120 170 130)
(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)(61 81 71)(62 82 72)(63 83 73)(64 84 74)(65 85 75)(66 86 76)(67 87 77)(68 88 78)(69 89 79)(70 90 80)(91 111 101)(92 112 102)(93 113 103)(94 114 104)(95 115 105)(96 116 106)(97 117 107)(98 118 108)(99 119 109)(100 120 110)(121 141 131)(122 142 132)(123 143 133)(124 144 134)(125 145 135)(126 146 136)(127 147 137)(128 148 138)(129 149 139)(130 150 140)(151 171 161)(152 172 162)(153 173 163)(154 174 164)(155 175 165)(156 176 166)(157 177 167)(158 178 168)(159 179 169)(160 180 170)
(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)
(1 91 6 96)(2 95 7 100)(3 94 8 99)(4 93 9 98)(5 92 10 97)(11 101 16 106)(12 105 17 110)(13 104 18 109)(14 103 19 108)(15 102 20 107)(21 111 26 116)(22 115 27 120)(23 114 28 119)(24 113 29 118)(25 112 30 117)(31 121 36 126)(32 125 37 130)(33 124 38 129)(34 123 39 128)(35 122 40 127)(41 131 46 136)(42 135 47 140)(43 134 48 139)(44 133 49 138)(45 132 50 137)(51 141 56 146)(52 145 57 150)(53 144 58 149)(54 143 59 148)(55 142 60 147)(61 151 66 156)(62 155 67 160)(63 154 68 159)(64 153 69 158)(65 152 70 157)(71 161 76 166)(72 165 77 170)(73 164 78 169)(74 163 79 168)(75 162 80 167)(81 171 86 176)(82 175 87 180)(83 174 88 179)(84 173 89 178)(85 172 90 177)
(2 5)(3 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(62 65)(63 64)(67 70)(68 69)(72 75)(73 74)(77 80)(78 79)(82 85)(83 84)(87 90)(88 89)(91 96)(92 100)(93 99)(94 98)(95 97)(101 106)(102 110)(103 109)(104 108)(105 107)(111 116)(112 120)(113 119)(114 118)(115 117)(121 126)(122 130)(123 129)(124 128)(125 127)(131 136)(132 140)(133 139)(134 138)(135 137)(141 146)(142 150)(143 149)(144 148)(145 147)(151 156)(152 160)(153 159)(154 158)(155 157)(161 166)(162 170)(163 169)(164 168)(165 167)(171 176)(172 180)(173 179)(174 178)(175 177)

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

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

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

117 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4 5A 5B 6A ··· 6H 6I ··· 6P 6Q ··· 6X 10A ··· 10F 12A ··· 12H 15A ··· 15P 30A ··· 30AV order 1 2 2 2 3 ··· 3 4 5 5 6 ··· 6 6 ··· 6 6 ··· 6 10 ··· 10 12 ··· 12 15 ··· 15 30 ··· 30 size 1 1 2 10 1 ··· 1 10 2 2 1 ··· 1 2 ··· 2 10 ··· 10 2 ··· 2 10 ··· 10 2 ··· 2 2 ··· 2

117 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D5 D10 C3×D4 C3×D5 C5⋊D4 C6×D5 C3×C5⋊D4 kernel C32×C5⋊D4 C32×Dic5 D5×C3×C6 C6×C30 C3×C5⋊D4 C3×Dic5 C6×D5 C2×C30 C3×C15 C62 C3×C6 C15 C2×C6 C32 C6 C3 # reps 1 1 1 1 8 8 8 8 1 2 2 8 16 4 16 32

Matrix representation of C32×C5⋊D4 in GL5(𝔽61)

 13 0 0 0 0 0 13 0 0 0 0 0 13 0 0 0 0 0 13 0 0 0 0 0 13
,
 47 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 44 60 0 0 0 45 60
,
 1 0 0 0 0 0 1 2 0 0 0 60 60 0 0 0 0 0 44 60 0 0 0 44 17
,
 60 0 0 0 0 0 1 0 0 0 0 60 60 0 0 0 0 0 17 1 0 0 0 17 44

G:=sub<GL(5,GF(61))| [13,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,13],[47,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,44,45,0,0,0,60,60],[1,0,0,0,0,0,1,60,0,0,0,2,60,0,0,0,0,0,44,44,0,0,0,60,17],[60,0,0,0,0,0,1,60,0,0,0,0,60,0,0,0,0,0,17,17,0,0,0,1,44] >;

C32×C5⋊D4 in GAP, Magma, Sage, TeX

C_3^2\times C_5\rtimes D_4
% in TeX

G:=Group("C3^2xC5:D4");
// GroupNames label

G:=SmallGroup(360,94);
// by ID

G=gap.SmallGroup(360,94);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-5,457,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=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,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽