Copied to
clipboard

G = C4×C22.F5order 320 = 26·5

Direct product of C4 and C22.F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×C22.F5, C209M4(2), Dic58M4(2), Dic5.14C42, C53(C4×M4(2)), C22.5(C4×F5), C23.39(C2×F5), (C22×C4).11F5, (C2×C10).21C42, (C22×C20).33C4, C10.18(C2×C42), (C4×Dic5).48C4, C2.6(D5⋊M4(2)), C10.24(C2×M4(2)), C10.C4217C2, C22.44(C22×F5), (C22×Dic5).30C4, Dic5.37(C22×C4), (C2×Dic5).341C23, (C4×Dic5).325C22, (C22×Dic5).269C22, C5⋊C83(C2×C4), (C4×C5⋊C8)⋊17C2, C2.18(C2×C4×F5), (C2×C5⋊C8).36C22, (C2×C4).104(C2×F5), (C2×C4×Dic5).51C2, (C2×C20).104(C2×C4), C2.2(C2×C22.F5), (C2×C22.F5).7C2, (C22×C10).57(C2×C4), (C2×C10).57(C22×C4), (C2×Dic5).128(C2×C4), SmallGroup(320,1088)

Series: Derived Chief Lower central Upper central

C1C10 — C4×C22.F5
C1C5C10Dic5C2×Dic5C2×C5⋊C8C2×C22.F5 — C4×C22.F5
C5C10 — C4×C22.F5
C1C2×C4C22×C4

Generators and relations for C4×C22.F5
 G = < a,b,c,d,e | a4=b2=c2=d5=1, e4=c, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 378 in 142 conjugacy classes, 74 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C5, C8 [×8], C2×C4 [×2], C2×C4 [×12], C23, C10 [×3], C10 [×2], C42 [×4], C2×C8 [×4], M4(2) [×8], C22×C4, C22×C4 [×2], Dic5 [×6], Dic5, C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C4×C8 [×2], C8⋊C4 [×2], C2×C42, C2×M4(2) [×2], C5⋊C8 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C4×M4(2), C4×Dic5 [×4], C2×C5⋊C8 [×4], C22.F5 [×8], C22×Dic5 [×2], C22×C20, C4×C5⋊C8 [×2], C10.C42 [×2], C2×C4×Dic5, C2×C22.F5 [×2], C4×C22.F5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×4], C22×C4 [×3], F5, C2×C42, C2×M4(2) [×2], C2×F5 [×3], C4×M4(2), C4×F5 [×2], C22.F5 [×2], C22×F5, D5⋊M4(2), C2×C4×F5, C2×C22.F5, C4×C22.F5

Smallest permutation representation of C4×C22.F5
On 160 points
Generators in S160
(1 78 88 108)(2 79 81 109)(3 80 82 110)(4 73 83 111)(5 74 84 112)(6 75 85 105)(7 76 86 106)(8 77 87 107)(9 18 132 128)(10 19 133 121)(11 20 134 122)(12 21 135 123)(13 22 136 124)(14 23 129 125)(15 24 130 126)(16 17 131 127)(25 69 150 46)(26 70 151 47)(27 71 152 48)(28 72 145 41)(29 65 146 42)(30 66 147 43)(31 67 148 44)(32 68 149 45)(33 96 140 61)(34 89 141 62)(35 90 142 63)(36 91 143 64)(37 92 144 57)(38 93 137 58)(39 94 138 59)(40 95 139 60)(49 98 115 159)(50 99 116 160)(51 100 117 153)(52 101 118 154)(53 102 119 155)(54 103 120 156)(55 104 113 157)(56 97 114 158)
(1 88)(2 85)(3 82)(4 87)(5 84)(6 81)(7 86)(8 83)(9 132)(10 129)(11 134)(12 131)(13 136)(14 133)(15 130)(16 135)(17 123)(18 128)(19 125)(20 122)(21 127)(22 124)(23 121)(24 126)(25 150)(26 147)(27 152)(28 149)(29 146)(30 151)(31 148)(32 145)(33 144)(34 141)(35 138)(36 143)(37 140)(38 137)(39 142)(40 139)(41 68)(42 65)(43 70)(44 67)(45 72)(46 69)(47 66)(48 71)(49 115)(50 120)(51 117)(52 114)(53 119)(54 116)(55 113)(56 118)(57 96)(58 93)(59 90)(60 95)(61 92)(62 89)(63 94)(64 91)(73 107)(74 112)(75 109)(76 106)(77 111)(78 108)(79 105)(80 110)(97 154)(98 159)(99 156)(100 153)(101 158)(102 155)(103 160)(104 157)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)
(1 159 36 9 65)(2 10 160 66 37)(3 67 11 38 153)(4 39 68 154 12)(5 155 40 13 69)(6 14 156 70 33)(7 71 15 34 157)(8 35 72 158 16)(17 77 90 145 56)(18 146 78 49 91)(19 50 147 92 79)(20 93 51 80 148)(21 73 94 149 52)(22 150 74 53 95)(23 54 151 96 75)(24 89 55 76 152)(25 112 119 60 124)(26 61 105 125 120)(27 126 62 113 106)(28 114 127 107 63)(29 108 115 64 128)(30 57 109 121 116)(31 122 58 117 110)(32 118 123 111 59)(41 97 131 87 142)(42 88 98 143 132)(43 144 81 133 99)(44 134 137 100 82)(45 101 135 83 138)(46 84 102 139 136)(47 140 85 129 103)(48 130 141 104 86)
(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)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G···4N4O4P4Q4R 5 8A···8P10A···10G20A···20H
order1222224444444···4444458···810···1020···20
size1111221111225···510101010410···104···44···4

56 irreducible representations

dim11111111122444444
type++++++++-
imageC1C2C2C2C2C4C4C4C4M4(2)M4(2)F5C2×F5C2×F5C22.F5C4×F5D5⋊M4(2)
kernelC4×C22.F5C4×C5⋊C8C10.C42C2×C4×Dic5C2×C22.F5C4×Dic5C22.F5C22×Dic5C22×C20Dic5C20C22×C4C2×C4C23C4C22C2
# reps122124162244121444

Matrix representation of C4×C22.F5 in GL8(𝔽41)

320000000
032000000
00100000
00010000
000040000
000004000
000000400
000000040
,
400000000
01000000
00100000
0016400000
00001000
00000100
00000010
00000001
,
400000000
040000000
004000000
000400000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00000100
0000403400
0000384034
000083277
,
040000000
320000000
0016390000
0030250000
0000171610
0000171601
000010312425
00001982425

G:=sub<GL(8,GF(41))| [32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,3,8,0,0,0,0,1,34,8,32,0,0,0,0,0,0,40,7,0,0,0,0,0,0,34,7],[0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,16,30,0,0,0,0,0,0,39,25,0,0,0,0,0,0,0,0,17,17,10,19,0,0,0,0,16,16,31,8,0,0,0,0,1,0,24,24,0,0,0,0,0,1,25,25] >;

C4×C22.F5 in GAP, Magma, Sage, TeX

C_4\times C_2^2.F_5
% in TeX

G:=Group("C4xC2^2.F5");
// GroupNames label

G:=SmallGroup(320,1088);
// by ID

G=gap.SmallGroup(320,1088);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=c^2=d^5=1,e^4=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations

׿
×
𝔽