direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×S3×D20, C60⋊3C23, D30⋊2C23, D60⋊32C22, C30.16C24, C10⋊1(S3×D4), C6⋊1(C2×D20), C30⋊2(C2×D4), (C6×D20)⋊8C2, (C2×C12)⋊4D10, (C2×C20)⋊26D6, C15⋊2(C22×D4), C3⋊1(C22×D20), (S3×C10)⋊12D4, (C2×D60)⋊24C2, (C4×S3)⋊16D10, C20⋊5(C22×S3), (C6×D5)⋊1C23, (C22×D5)⋊9D6, C12⋊2(C22×D5), (C2×C60)⋊12C22, D10⋊1(C22×S3), C6.16(C23×D5), (S3×C20)⋊18C22, (C2×Dic3)⋊21D10, (C3×D20)⋊23C22, C3⋊D20⋊10C22, C10.16(S3×C23), Dic3⋊4(C22×D5), (C5×Dic3)⋊5C23, D6.32(C22×D5), (S3×C10).29C23, (C2×C30).235C23, (C22×S3).91D10, (C22×D15)⋊9C22, (C10×Dic3)⋊26C22, C5⋊1(C2×S3×D4), C4⋊3(C2×S3×D5), (S3×C2×C4)⋊5D5, (S3×C2×C20)⋊6C2, (C2×C4)⋊8(S3×D5), (C5×S3)⋊1(C2×D4), (D5×C2×C6)⋊4C22, (C22×S3×D5)⋊6C2, (C2×S3×D5)⋊10C22, (C2×C3⋊D20)⋊19C2, C2.19(C22×S3×D5), C22.104(C2×S3×D5), (S3×C2×C10).102C22, (C2×C6).245(C22×D5), (C2×C10).245(C22×S3), SmallGroup(480,1088)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 3004 in 472 conjugacy classes, 132 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×38], C5, S3 [×4], S3 [×4], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×5], D4 [×16], C23 [×21], D5 [×8], C10, C10 [×2], C10 [×4], Dic3 [×2], C12 [×2], D6 [×6], D6 [×24], C2×C6, C2×C6 [×8], C15, C22×C4, C2×D4 [×12], C24 [×2], C20 [×2], C20 [×2], D10 [×4], D10 [×28], C2×C10, C2×C10 [×6], C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×S3 [×18], C22×C6 [×2], C5×S3 [×4], C3×D5 [×4], D15 [×4], C30, C30 [×2], C22×D4, D20 [×4], D20 [×12], C2×C20, C2×C20 [×5], C22×D5 [×2], C22×D5 [×18], C22×C10, S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4 [×2], C6×D4, S3×C23 [×2], C5×Dic3 [×2], C60 [×2], S3×D5 [×16], C6×D5 [×4], C6×D5 [×4], S3×C10 [×6], D30 [×4], D30 [×4], C2×C30, C2×D20, C2×D20 [×11], C22×C20, C23×D5 [×2], C2×S3×D4, C3⋊D20 [×8], C3×D20 [×4], S3×C20 [×4], C10×Dic3, D60 [×4], C2×C60, C2×S3×D5 [×8], C2×S3×D5 [×8], D5×C2×C6 [×2], S3×C2×C10, C22×D15 [×2], C22×D20, S3×D20 [×8], C2×C3⋊D20 [×2], C6×D20, S3×C2×C20, C2×D60, C22×S3×D5 [×2], C2×S3×D20
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], C22×S3 [×7], C22×D4, D20 [×4], C22×D5 [×7], S3×D4 [×2], S3×C23, S3×D5, C2×D20 [×6], C23×D5, C2×S3×D4, C2×S3×D5 [×3], C22×D20, S3×D20 [×2], C22×S3×D5, C2×S3×D20
Generators and relations
G = < a,b,c,d,e | a2=b3=c2=d20=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
►On 120 points
Generators in S120
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 61)(19 62)(20 63)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 111)(42 112)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)
(1 22 101)(2 23 102)(3 24 103)(4 25 104)(5 26 105)(6 27 106)(7 28 107)(8 29 108)(9 30 109)(10 31 110)(11 32 111)(12 33 112)(13 34 113)(14 35 114)(15 36 115)(16 37 116)(17 38 117)(18 39 118)(19 40 119)(20 21 120)(41 74 86)(42 75 87)(43 76 88)(44 77 89)(45 78 90)(46 79 91)(47 80 92)(48 61 93)(49 62 94)(50 63 95)(51 64 96)(52 65 97)(53 66 98)(54 67 99)(55 68 100)(56 69 81)(57 70 82)(58 71 83)(59 72 84)(60 73 85)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 120)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 96)(42 97)(43 98)(44 99)(45 100)(46 81)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(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)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 22)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 32)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 66)(62 65)(63 64)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 90)(82 89)(83 88)(84 87)(85 86)(91 100)(92 99)(93 98)(94 97)(95 96)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)
G:=sub<Sym(120)| (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,61)(19,62)(20,63)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,111)(42,112)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110), (1,22,101)(2,23,102)(3,24,103)(4,25,104)(5,26,105)(6,27,106)(7,28,107)(8,29,108)(9,30,109)(10,31,110)(11,32,111)(12,33,112)(13,34,113)(14,35,114)(15,36,115)(16,37,116)(17,38,117)(18,39,118)(19,40,119)(20,21,120)(41,74,86)(42,75,87)(43,76,88)(44,77,89)(45,78,90)(46,79,91)(47,80,92)(48,61,93)(49,62,94)(50,63,95)(51,64,96)(52,65,97)(53,66,98)(54,67,99)(55,68,100)(56,69,81)(57,70,82)(58,71,83)(59,72,84)(60,73,85), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,96)(42,97)(43,98)(44,99)(45,100)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,22)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)>;
G:=Group( (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,61)(19,62)(20,63)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,111)(42,112)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110), (1,22,101)(2,23,102)(3,24,103)(4,25,104)(5,26,105)(6,27,106)(7,28,107)(8,29,108)(9,30,109)(10,31,110)(11,32,111)(12,33,112)(13,34,113)(14,35,114)(15,36,115)(16,37,116)(17,38,117)(18,39,118)(19,40,119)(20,21,120)(41,74,86)(42,75,87)(43,76,88)(44,77,89)(45,78,90)(46,79,91)(47,80,92)(48,61,93)(49,62,94)(50,63,95)(51,64,96)(52,65,97)(53,66,98)(54,67,99)(55,68,100)(56,69,81)(57,70,82)(58,71,83)(59,72,84)(60,73,85), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,96)(42,97)(43,98)(44,99)(45,100)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,22)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111) );
G=PermutationGroup([(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,61),(19,62),(20,63),(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(27,81),(28,82),(29,83),(30,84),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,91),(38,92),(39,93),(40,94),(41,111),(42,112),(43,113),(44,114),(45,115),(46,116),(47,117),(48,118),(49,119),(50,120),(51,101),(52,102),(53,103),(54,104),(55,105),(56,106),(57,107),(58,108),(59,109),(60,110)], [(1,22,101),(2,23,102),(3,24,103),(4,25,104),(5,26,105),(6,27,106),(7,28,107),(8,29,108),(9,30,109),(10,31,110),(11,32,111),(12,33,112),(13,34,113),(14,35,114),(15,36,115),(16,37,116),(17,38,117),(18,39,118),(19,40,119),(20,21,120),(41,74,86),(42,75,87),(43,76,88),(44,77,89),(45,78,90),(46,79,91),(47,80,92),(48,61,93),(49,62,94),(50,63,95),(51,64,96),(52,65,97),(53,66,98),(54,67,99),(55,68,100),(56,69,81),(57,70,82),(58,71,83),(59,72,84),(60,73,85)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,110),(22,111),(23,112),(24,113),(25,114),(26,115),(27,116),(28,117),(29,118),(30,119),(31,120),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(41,96),(42,97),(43,98),(44,99),(45,100),(46,81),(47,82),(48,83),(49,84),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91),(57,92),(58,93),(59,94),(60,95),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,22),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(31,32),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,66),(62,65),(63,64),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(81,90),(82,89),(83,88),(84,87),(85,86),(91,100),(92,99),(93,98),(94,97),(95,96),(101,120),(102,119),(103,118),(104,117),(105,116),(106,115),(107,114),(108,113),(109,112),(110,111)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 34 | 32 | 0 | 0 | 0 | 0 |
| 2 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 29 | 0 | 0 |
| 0 | 0 | 59 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 25 | 32 | 0 | 0 | 0 | 0 |
| 11 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 29 | 0 | 0 |
| 0 | 0 | 50 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[34,2,0,0,0,0,32,36,0,0,0,0,0,0,27,59,0,0,0,0,29,25,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[25,11,0,0,0,0,32,36,0,0,0,0,0,0,36,50,0,0,0,0,29,25,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | D10 | D20 | S3×D4 | S3×D5 | C2×S3×D5 | C2×S3×D5 | S3×D20 |
| kernel | C2×S3×D20 | S3×D20 | C2×C3⋊D20 | C6×D20 | S3×C2×C20 | C2×D60 | C22×S3×D5 | C2×D20 | S3×C10 | S3×C2×C4 | D20 | C2×C20 | C22×D5 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | D6 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 4 | 1 | 2 | 8 | 2 | 2 | 2 | 16 | 2 | 2 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times S_3\times D_{20} % in TeX
G:=Group("C2xS3xD20"); // GroupNames label
G:=SmallGroup(480,1088);
// by ID
G=gap.SmallGroup(480,1088);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^20=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀