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## G = C22×C3⋊D15order 360 = 23·32·5

### Direct product of C22 and C3⋊D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C22×C3⋊D15
 Chief series C1 — C5 — C15 — C3×C15 — C3⋊D15 — C2×C3⋊D15 — C22×C3⋊D15
 Lower central C3×C15 — C22×C3⋊D15
 Upper central C1 — C22

Generators and relations for C22×C3⋊D15
G = < a,b,c,d,e | a2=b2=c3=d15=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1512 in 192 conjugacy classes, 71 normal (9 characteristic)
C1, C2, C2, C3, C22, C22, C5, S3, C6, C23, C32, D5, C10, D6, C2×C6, C15, C3⋊S3, C3×C6, D10, C2×C10, C22×S3, D15, C30, C2×C3⋊S3, C62, C22×D5, C3×C15, D30, C2×C30, C22×C3⋊S3, C3⋊D15, C3×C30, C22×D15, C2×C3⋊D15, C6×C30, C22×C3⋊D15
Quotients: C1, C2, C22, S3, C23, D5, D6, C3⋊S3, D10, C22×S3, D15, C2×C3⋊S3, C22×D5, D30, C22×C3⋊S3, C3⋊D15, C22×D15, C2×C3⋊D15, C22×C3⋊D15

Smallest permutation representation of C22×C3⋊D15
On 180 points
Generators in S180
(1 102)(2 103)(3 104)(4 105)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 108)(17 109)(18 110)(19 111)(20 112)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 106)(30 107)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 127)(38 128)(39 129)(40 130)(41 131)(42 132)(43 133)(44 134)(45 135)(46 138)(47 139)(48 140)(49 141)(50 142)(51 143)(52 144)(53 145)(54 146)(55 147)(56 148)(57 149)(58 150)(59 136)(60 137)(61 151)(62 152)(63 153)(64 154)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 161)(72 162)(73 163)(74 164)(75 165)(76 166)(77 167)(78 168)(79 169)(80 170)(81 171)(82 172)(83 173)(84 174)(85 175)(86 176)(87 177)(88 178)(89 179)(90 180)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 72)(17 73)(18 74)(19 75)(20 61)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 80)(32 81)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 76)(43 77)(44 78)(45 79)(91 136)(92 137)(93 138)(94 139)(95 140)(96 141)(97 142)(98 143)(99 144)(100 145)(101 146)(102 147)(103 148)(104 149)(105 150)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(112 151)(113 152)(114 153)(115 154)(116 155)(117 156)(118 157)(119 158)(120 159)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(127 176)(128 177)(129 178)(130 179)(131 180)(132 166)(133 167)(134 168)(135 169)
(1 35 19)(2 36 20)(3 37 21)(4 38 22)(5 39 23)(6 40 24)(7 41 25)(8 42 26)(9 43 27)(10 44 28)(11 45 29)(12 31 30)(13 32 16)(14 33 17)(15 34 18)(46 90 66)(47 76 67)(48 77 68)(49 78 69)(50 79 70)(51 80 71)(52 81 72)(53 82 73)(54 83 74)(55 84 75)(56 85 61)(57 86 62)(58 87 63)(59 88 64)(60 89 65)(91 129 115)(92 130 116)(93 131 117)(94 132 118)(95 133 119)(96 134 120)(97 135 106)(98 121 107)(99 122 108)(100 123 109)(101 124 110)(102 125 111)(103 126 112)(104 127 113)(105 128 114)(136 178 154)(137 179 155)(138 180 156)(139 166 157)(140 167 158)(141 168 159)(142 169 160)(143 170 161)(144 171 162)(145 172 163)(146 173 164)(147 174 165)(148 175 151)(149 176 152)(150 177 153)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 45)(24 44)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)(46 48)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 90)(69 89)(70 88)(71 87)(72 86)(73 85)(74 84)(75 83)(91 97)(92 96)(93 95)(98 105)(99 104)(100 103)(101 102)(106 129)(107 128)(108 127)(109 126)(110 125)(111 124)(112 123)(113 122)(114 121)(115 135)(116 134)(117 133)(118 132)(119 131)(120 130)(136 142)(137 141)(138 140)(143 150)(144 149)(145 148)(146 147)(151 172)(152 171)(153 170)(154 169)(155 168)(156 167)(157 166)(158 180)(159 179)(160 178)(161 177)(162 176)(163 175)(164 174)(165 173)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 5A 5B 6A ··· 6L 10A ··· 10F 15A ··· 15P 30A ··· 30AV order 1 2 2 2 2 2 2 2 3 3 3 3 5 5 6 ··· 6 10 ··· 10 15 ··· 15 30 ··· 30 size 1 1 1 1 45 45 45 45 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 D5 D6 D10 D15 D30 kernel C22×C3⋊D15 C2×C3⋊D15 C6×C30 C2×C30 C62 C30 C3×C6 C2×C6 C6 # reps 1 6 1 4 2 12 6 16 48

Matrix representation of C22×C3⋊D15 in GL4(𝔽31) generated by

 30 0 0 0 0 30 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 30 0 0 0 0 30
,
 16 5 0 0 26 14 0 0 0 0 19 3 0 0 28 11
,
 23 20 0 0 11 15 0 0 0 0 13 13 0 0 18 30
,
 12 12 0 0 1 19 0 0 0 0 13 13 0 0 30 18
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,30,0,0,0,0,30],[16,26,0,0,5,14,0,0,0,0,19,28,0,0,3,11],[23,11,0,0,20,15,0,0,0,0,13,18,0,0,13,30],[12,1,0,0,12,19,0,0,0,0,13,30,0,0,13,18] >;

C22×C3⋊D15 in GAP, Magma, Sage, TeX

C_2^2\times C_3\rtimes D_{15}
% in TeX

G:=Group("C2^2xC3:D15");
// GroupNames label

G:=SmallGroup(360,161);
// by ID

G=gap.SmallGroup(360,161);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,387,1444,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^15=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,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽