Copied to
clipboard

G = C10.152- 1+4order 320 = 26·5

15th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.152- 1+4, C4⋊C4.188D10, C22⋊Q8.9D5, (Q8×Dic5)⋊12C2, (C2×C20).50C23, (C2×Q8).123D10, C22⋊C4.13D10, C4.Dic1023C2, Dic5⋊Q813C2, Dic53Q824C2, C20.209(C4○D4), C4.72(D42D5), (C2×C10).169C24, (C22×C4).233D10, Dic5.Q817C2, C20.48D4.16C2, C4⋊Dic5.311C22, (Q8×C10).104C22, (C2×Dic5).84C23, C23.D10.2C2, C23.116(C22×D5), C22.190(C23×D5), (C22×C20).249C22, (C22×C10).197C23, C53(C22.35C24), (C4×Dic5).111C22, C10.D4.24C22, C23.D5.114C22, C2.34(D4.10D10), C2.16(Q8.10D10), (C2×Dic10).165C22, C23.21D10.24C2, C10.89(C2×C4○D4), (C5×C22⋊Q8).9C2, C2.45(C2×D42D5), (C5×C4⋊C4).155C22, (C2×C4).182(C22×D5), (C5×C22⋊C4).24C22, SmallGroup(320,1297)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.152- 1+4
C1C5C10C2×C10C2×Dic5C4×Dic5Dic53Q8 — C10.152- 1+4
C5C2×C10 — C10.152- 1+4
C1C22C22⋊Q8

Generators and relations for C10.152- 1+4
 G = < a,b,c,d,e | a10=b4=1, c2=a5, d2=e2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=b-1, dbd-1=a5b, be=eb, dcd-1=a5c, ce=ec, ede-1=b2d >

Subgroups: 526 in 192 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×13], C22, C22 [×3], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×4], C23, C10 [×3], C10, C42 [×6], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×17], C22×C4, C2×Q8, C2×Q8, Dic5 [×8], C20 [×2], C20 [×5], C2×C10, C2×C10 [×3], C42⋊C2, C4×Q8 [×2], C22⋊Q8, C22⋊Q8, C42.C2 [×5], C422C2 [×4], C4⋊Q8, Dic10 [×2], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×C10, C22.35C24, C4×Dic5 [×2], C4×Dic5 [×4], C10.D4 [×10], C4⋊Dic5 [×3], C4⋊Dic5 [×4], C23.D5 [×2], C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C22×C20, Q8×C10, C23.D10 [×4], Dic53Q8, Dic5.Q8 [×2], C4.Dic10, C4.Dic10 [×2], C20.48D4, C23.21D10, Dic5⋊Q8, Q8×Dic5, C5×C22⋊Q8, C10.152- 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2- 1+4 [×2], C22×D5 [×7], C22.35C24, D42D5 [×2], C23×D5, C2×D42D5, Q8.10D10, D4.10D10, C10.152- 1+4

Smallest permutation representation of C10.152- 1+4
On 160 points
Generators in S160
(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)
(1 58 13 48)(2 59 14 49)(3 60 15 50)(4 51 16 41)(5 52 17 42)(6 53 18 43)(7 54 19 44)(8 55 20 45)(9 56 11 46)(10 57 12 47)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 126 91 136)(82 127 92 137)(83 128 93 138)(84 129 94 139)(85 130 95 140)(86 121 96 131)(87 122 97 132)(88 123 98 133)(89 124 99 134)(90 125 100 135)(101 141 111 151)(102 142 112 152)(103 143 113 153)(104 144 114 154)(105 145 115 155)(106 146 116 156)(107 147 117 157)(108 148 118 158)(109 149 119 159)(110 150 120 160)
(1 132 6 137)(2 131 7 136)(3 140 8 135)(4 139 9 134)(5 138 10 133)(11 124 16 129)(12 123 17 128)(13 122 18 127)(14 121 19 126)(15 130 20 125)(21 159 26 154)(22 158 27 153)(23 157 28 152)(24 156 29 151)(25 155 30 160)(31 149 36 144)(32 148 37 143)(33 147 38 142)(34 146 39 141)(35 145 40 150)(41 84 46 89)(42 83 47 88)(43 82 48 87)(44 81 49 86)(45 90 50 85)(51 94 56 99)(52 93 57 98)(53 92 58 97)(54 91 59 96)(55 100 60 95)(61 109 66 104)(62 108 67 103)(63 107 68 102)(64 106 69 101)(65 105 70 110)(71 119 76 114)(72 118 77 113)(73 117 78 112)(74 116 79 111)(75 115 80 120)
(1 107 13 117)(2 106 14 116)(3 105 15 115)(4 104 16 114)(5 103 17 113)(6 102 18 112)(7 101 19 111)(8 110 20 120)(9 109 11 119)(10 108 12 118)(21 94 31 84)(22 93 32 83)(23 92 33 82)(24 91 34 81)(25 100 35 90)(26 99 36 89)(27 98 37 88)(28 97 38 87)(29 96 39 86)(30 95 40 85)(41 159 51 149)(42 158 52 148)(43 157 53 147)(44 156 54 146)(45 155 55 145)(46 154 56 144)(47 153 57 143)(48 152 58 142)(49 151 59 141)(50 160 60 150)(61 124 71 134)(62 123 72 133)(63 122 73 132)(64 121 74 131)(65 130 75 140)(66 129 76 139)(67 128 77 138)(68 127 78 137)(69 126 79 136)(70 125 80 135)
(1 53 13 43)(2 54 14 44)(3 55 15 45)(4 56 16 46)(5 57 17 47)(6 58 18 48)(7 59 19 49)(8 60 20 50)(9 51 11 41)(10 52 12 42)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 131 91 121)(82 132 92 122)(83 133 93 123)(84 134 94 124)(85 135 95 125)(86 136 96 126)(87 137 97 127)(88 138 98 128)(89 139 99 129)(90 140 100 130)(101 151 111 141)(102 152 112 142)(103 153 113 143)(104 154 114 144)(105 155 115 145)(106 156 116 146)(107 157 117 147)(108 158 118 148)(109 159 119 149)(110 160 120 150)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D4A4B4C···4G4H4I4J4K4L···4Q5A5B10A···10F10G10H10I10J20A···20H20I···20P
order12222444···444444···45510···101010101020···2020···20
size11114224···41010101020···20222···244444···48···8

50 irreducible representations

dim11111111112222224444
type+++++++++++++++---
imageC1C2C2C2C2C2C2C2C2C2D5C4○D4D10D10D10D102- 1+4D42D5Q8.10D10D4.10D10
kernelC10.152- 1+4C23.D10Dic53Q8Dic5.Q8C4.Dic10C20.48D4C23.21D10Dic5⋊Q8Q8×Dic5C5×C22⋊Q8C22⋊Q8C20C22⋊C4C4⋊C4C22×C4C2×Q8C10C4C2C2
# reps14123111112446222444

Matrix representation of C10.152- 1+4 in GL8(𝔽41)

400000000
040000000
0026350000
002780000
00001000
00000100
00000010
00000001
,
400000000
401000000
00100000
00010000
000071400
0000143400
0000003427
000000277
,
320000000
329000000
002230000
007390000
0000003427
000000277
000071400
0000143400
,
3218000000
329000000
0039180000
003420000
000000040
00000010
000004000
00001000
,
10000000
01000000
004000000
000400000
000071400
0000143400
000000714
0000001434

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,26,27,0,0,0,0,0,0,35,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,14,0,0,0,0,0,0,14,34,0,0,0,0,0,0,0,0,34,27,0,0,0,0,0,0,27,7],[32,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,2,7,0,0,0,0,0,0,23,39,0,0,0,0,0,0,0,0,0,0,7,14,0,0,0,0,0,0,14,34,0,0,0,0,34,27,0,0,0,0,0,0,27,7,0,0],[32,32,0,0,0,0,0,0,18,9,0,0,0,0,0,0,0,0,39,34,0,0,0,0,0,0,18,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,7,14,0,0,0,0,0,0,14,34,0,0,0,0,0,0,0,0,7,14,0,0,0,0,0,0,14,34] >;

C10.152- 1+4 in GAP, Magma, Sage, TeX

C_{10}._{15}2_-^{1+4}
% in TeX

G:=Group("C10.15ES-(2,2)");
// GroupNames label

G:=SmallGroup(320,1297);
// by ID

G=gap.SmallGroup(320,1297);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,219,268,675,297,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=b^4=1,c^2=a^5,d^2=e^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,d*b*d^-1=a^5*b,b*e=e*b,d*c*d^-1=a^5*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations

׿
×
𝔽