direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×D4×C10, C60⋊8C23, C30.88C24, C6⋊2(D4×C10), (C6×D4)⋊5C10, C30⋊15(C2×D4), (C2×C20)⋊29D6, C12⋊(C22×C10), (D4×C30)⋊19C2, D12⋊7(C2×C10), C23⋊4(S3×C10), C20⋊7(C22×S3), (C2×C30)⋊8C23, C15⋊16(C22×D4), (C10×D12)⋊27C2, (C2×D12)⋊11C10, (S3×C23)⋊4C10, (C2×C60)⋊27C22, D6⋊2(C22×C10), (C22×C10)⋊13D6, C6.5(C23×C10), (S3×C20)⋊22C22, (S3×C10)⋊11C23, (D4×C15)⋊36C22, (C5×D12)⋊37C22, C10.73(S3×C23), (C5×Dic3)⋊9C23, (C22×C30)⋊16C22, Dic3⋊1(C22×C10), (C10×Dic3)⋊35C22, C3⋊2(D4×C2×C10), C4⋊1(S3×C2×C10), (S3×C2×C4)⋊3C10, (S3×C2×C20)⋊13C2, (C2×C4)⋊6(S3×C10), (C2×C6)⋊(C22×C10), C22⋊2(S3×C2×C10), (C4×S3)⋊3(C2×C10), (C2×C12)⋊2(C2×C10), (C3×D4)⋊5(C2×C10), (C2×C3⋊D4)⋊9C10, C3⋊D4⋊1(C2×C10), C2.6(S3×C22×C10), (C10×C3⋊D4)⋊24C2, (S3×C2×C10)⋊22C22, (S3×C22×C10)⋊10C2, (C22×C6)⋊4(C2×C10), (C2×C10)⋊8(C22×S3), (C22×S3)⋊6(C2×C10), (C2×Dic3)⋊8(C2×C10), (C5×C3⋊D4)⋊17C22, SmallGroup(480,1154)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1124 in 472 conjugacy classes, 194 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×4], C22 [×34], C5, S3 [×4], S3 [×4], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×5], D4 [×4], D4 [×12], C23 [×2], C23 [×19], C10, C10 [×2], C10 [×12], Dic3 [×2], C12 [×2], D6 [×10], D6 [×20], C2×C6, C2×C6 [×4], C2×C6 [×4], C15, C22×C4, C2×D4, C2×D4 [×11], C24 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C2×C10 [×34], C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×S3 [×10], C22×S3 [×8], C22×C6 [×2], C5×S3 [×4], C5×S3 [×4], C30, C30 [×2], C30 [×4], C22×D4, C2×C20, C2×C20 [×5], C5×D4 [×4], C5×D4 [×12], C22×C10 [×2], C22×C10 [×19], S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4 [×2], C6×D4, S3×C23 [×2], C5×Dic3 [×2], C60 [×2], S3×C10 [×10], S3×C10 [×20], C2×C30, C2×C30 [×4], C2×C30 [×4], C22×C20, D4×C10, D4×C10 [×11], C23×C10 [×2], C2×S3×D4, S3×C20 [×4], C5×D12 [×4], C10×Dic3, C5×C3⋊D4 [×8], C2×C60, D4×C15 [×4], S3×C2×C10, S3×C2×C10 [×10], S3×C2×C10 [×8], C22×C30 [×2], D4×C2×C10, S3×C2×C20, C10×D12, C5×S3×D4 [×8], C10×C3⋊D4 [×2], D4×C30, S3×C22×C10 [×2], S3×D4×C10
Quotients:
C1, C2 [×15], C22 [×35], C5, S3, D4 [×4], C23 [×15], C10 [×15], D6 [×7], C2×D4 [×6], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, C22×D4, C5×D4 [×4], C22×C10 [×15], S3×D4 [×2], S3×C23, S3×C10 [×7], D4×C10 [×6], C23×C10, C2×S3×D4, S3×C2×C10 [×7], D4×C2×C10, C5×S3×D4 [×2], S3×C22×C10, S3×D4×C10
Generators and relations
G = < a,b,c,d,e | a10=b3=c2=d4=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
(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 37)(2 11 38)(3 12 39)(4 13 40)(5 14 31)(6 15 32)(7 16 33)(8 17 34)(9 18 35)(10 19 36)(21 104 96)(22 105 97)(23 106 98)(24 107 99)(25 108 100)(26 109 91)(27 110 92)(28 101 93)(29 102 94)(30 103 95)(41 78 66)(42 79 67)(43 80 68)(44 71 69)(45 72 70)(46 73 61)(47 74 62)(48 75 63)(49 76 64)(50 77 65)(51 88 117)(52 89 118)(53 90 119)(54 81 120)(55 82 111)(56 83 112)(57 84 113)(58 85 114)(59 86 115)(60 87 116)
(1 112)(2 113)(3 114)(4 115)(5 116)(6 117)(7 118)(8 119)(9 120)(10 111)(11 84)(12 85)(13 86)(14 87)(15 88)(16 89)(17 90)(18 81)(19 82)(20 83)(21 76)(22 77)(23 78)(24 79)(25 80)(26 71)(27 72)(28 73)(29 74)(30 75)(31 60)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(41 106)(42 107)(43 108)(44 109)(45 110)(46 101)(47 102)(48 103)(49 104)(50 105)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)(69 91)(70 92)
(1 69 117 96)(2 70 118 97)(3 61 119 98)(4 62 120 99)(5 63 111 100)(6 64 112 91)(7 65 113 92)(8 66 114 93)(9 67 115 94)(10 68 116 95)(11 45 52 22)(12 46 53 23)(13 47 54 24)(14 48 55 25)(15 49 56 26)(16 50 57 27)(17 41 58 28)(18 42 59 29)(19 43 60 30)(20 44 51 21)(31 75 82 108)(32 76 83 109)(33 77 84 110)(34 78 85 101)(35 79 86 102)(36 80 87 103)(37 71 88 104)(38 72 89 105)(39 73 90 106)(40 74 81 107)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 91)(7 92)(8 93)(9 94)(10 95)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 21)(31 108)(32 109)(33 110)(34 101)(35 102)(36 103)(37 104)(38 105)(39 106)(40 107)(41 58)(42 59)(43 60)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(50 57)(61 119)(62 120)(63 111)(64 112)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 88)(72 89)(73 90)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)
G:=sub<Sym(120)| (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), (1,20,37)(2,11,38)(3,12,39)(4,13,40)(5,14,31)(6,15,32)(7,16,33)(8,17,34)(9,18,35)(10,19,36)(21,104,96)(22,105,97)(23,106,98)(24,107,99)(25,108,100)(26,109,91)(27,110,92)(28,101,93)(29,102,94)(30,103,95)(41,78,66)(42,79,67)(43,80,68)(44,71,69)(45,72,70)(46,73,61)(47,74,62)(48,75,63)(49,76,64)(50,77,65)(51,88,117)(52,89,118)(53,90,119)(54,81,120)(55,82,111)(56,83,112)(57,84,113)(58,85,114)(59,86,115)(60,87,116), (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,111)(11,84)(12,85)(13,86)(14,87)(15,88)(16,89)(17,90)(18,81)(19,82)(20,83)(21,76)(22,77)(23,78)(24,79)(25,80)(26,71)(27,72)(28,73)(29,74)(30,75)(31,60)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,106)(42,107)(43,108)(44,109)(45,110)(46,101)(47,102)(48,103)(49,104)(50,105)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,91)(70,92), (1,69,117,96)(2,70,118,97)(3,61,119,98)(4,62,120,99)(5,63,111,100)(6,64,112,91)(7,65,113,92)(8,66,114,93)(9,67,115,94)(10,68,116,95)(11,45,52,22)(12,46,53,23)(13,47,54,24)(14,48,55,25)(15,49,56,26)(16,50,57,27)(17,41,58,28)(18,42,59,29)(19,43,60,30)(20,44,51,21)(31,75,82,108)(32,76,83,109)(33,77,84,110)(34,78,85,101)(35,79,86,102)(36,80,87,103)(37,71,88,104)(38,72,89,105)(39,73,90,106)(40,74,81,107), (1,96)(2,97)(3,98)(4,99)(5,100)(6,91)(7,92)(8,93)(9,94)(10,95)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,21)(31,108)(32,109)(33,110)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,58)(42,59)(43,60)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(50,57)(61,119)(62,120)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,88)(72,89)(73,90)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)>;
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), (1,20,37)(2,11,38)(3,12,39)(4,13,40)(5,14,31)(6,15,32)(7,16,33)(8,17,34)(9,18,35)(10,19,36)(21,104,96)(22,105,97)(23,106,98)(24,107,99)(25,108,100)(26,109,91)(27,110,92)(28,101,93)(29,102,94)(30,103,95)(41,78,66)(42,79,67)(43,80,68)(44,71,69)(45,72,70)(46,73,61)(47,74,62)(48,75,63)(49,76,64)(50,77,65)(51,88,117)(52,89,118)(53,90,119)(54,81,120)(55,82,111)(56,83,112)(57,84,113)(58,85,114)(59,86,115)(60,87,116), (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,111)(11,84)(12,85)(13,86)(14,87)(15,88)(16,89)(17,90)(18,81)(19,82)(20,83)(21,76)(22,77)(23,78)(24,79)(25,80)(26,71)(27,72)(28,73)(29,74)(30,75)(31,60)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,106)(42,107)(43,108)(44,109)(45,110)(46,101)(47,102)(48,103)(49,104)(50,105)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,91)(70,92), (1,69,117,96)(2,70,118,97)(3,61,119,98)(4,62,120,99)(5,63,111,100)(6,64,112,91)(7,65,113,92)(8,66,114,93)(9,67,115,94)(10,68,116,95)(11,45,52,22)(12,46,53,23)(13,47,54,24)(14,48,55,25)(15,49,56,26)(16,50,57,27)(17,41,58,28)(18,42,59,29)(19,43,60,30)(20,44,51,21)(31,75,82,108)(32,76,83,109)(33,77,84,110)(34,78,85,101)(35,79,86,102)(36,80,87,103)(37,71,88,104)(38,72,89,105)(39,73,90,106)(40,74,81,107), (1,96)(2,97)(3,98)(4,99)(5,100)(6,91)(7,92)(8,93)(9,94)(10,95)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,21)(31,108)(32,109)(33,110)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,58)(42,59)(43,60)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(50,57)(61,119)(62,120)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,88)(72,89)(73,90)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87) );
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)], [(1,20,37),(2,11,38),(3,12,39),(4,13,40),(5,14,31),(6,15,32),(7,16,33),(8,17,34),(9,18,35),(10,19,36),(21,104,96),(22,105,97),(23,106,98),(24,107,99),(25,108,100),(26,109,91),(27,110,92),(28,101,93),(29,102,94),(30,103,95),(41,78,66),(42,79,67),(43,80,68),(44,71,69),(45,72,70),(46,73,61),(47,74,62),(48,75,63),(49,76,64),(50,77,65),(51,88,117),(52,89,118),(53,90,119),(54,81,120),(55,82,111),(56,83,112),(57,84,113),(58,85,114),(59,86,115),(60,87,116)], [(1,112),(2,113),(3,114),(4,115),(5,116),(6,117),(7,118),(8,119),(9,120),(10,111),(11,84),(12,85),(13,86),(14,87),(15,88),(16,89),(17,90),(18,81),(19,82),(20,83),(21,76),(22,77),(23,78),(24,79),(25,80),(26,71),(27,72),(28,73),(29,74),(30,75),(31,60),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59),(41,106),(42,107),(43,108),(44,109),(45,110),(46,101),(47,102),(48,103),(49,104),(50,105),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,99),(68,100),(69,91),(70,92)], [(1,69,117,96),(2,70,118,97),(3,61,119,98),(4,62,120,99),(5,63,111,100),(6,64,112,91),(7,65,113,92),(8,66,114,93),(9,67,115,94),(10,68,116,95),(11,45,52,22),(12,46,53,23),(13,47,54,24),(14,48,55,25),(15,49,56,26),(16,50,57,27),(17,41,58,28),(18,42,59,29),(19,43,60,30),(20,44,51,21),(31,75,82,108),(32,76,83,109),(33,77,84,110),(34,78,85,101),(35,79,86,102),(36,80,87,103),(37,71,88,104),(38,72,89,105),(39,73,90,106),(40,74,81,107)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,91),(7,92),(8,93),(9,94),(10,95),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,21),(31,108),(32,109),(33,110),(34,101),(35,102),(36,103),(37,104),(38,105),(39,106),(40,107),(41,58),(42,59),(43,60),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(50,57),(61,119),(62,120),(63,111),(64,112),(65,113),(66,114),(67,115),(68,116),(69,117),(70,118),(71,88),(72,89),(73,90),(74,81),(75,82),(76,83),(77,84),(78,85),(79,86),(80,87)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 41 | 0 | 0 | 0 |
| 0 | 41 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 59 |
| 0 | 0 | 1 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 59 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(61))| [41,0,0,0,0,41,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,60,1,0,0,59,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,59,1] >;
150 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 | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10L | 10M | ··· | 10AB | 10AC | ··· | 10AR | 10AS | ··· | 10BH | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30L | 30M | ··· | 30AB | 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 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D6 | D6 | D6 | C5×S3 | C5×D4 | S3×C10 | S3×C10 | S3×C10 | S3×D4 | C5×S3×D4 |
| kernel | S3×D4×C10 | S3×C2×C20 | C10×D12 | C5×S3×D4 | C10×C3⋊D4 | D4×C30 | S3×C22×C10 | C2×S3×D4 | S3×C2×C4 | C2×D12 | S3×D4 | C2×C3⋊D4 | C6×D4 | S3×C23 | D4×C10 | S3×C10 | C2×C20 | C5×D4 | C22×C10 | C2×D4 | D6 | C2×C4 | D4 | C23 | C10 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 4 | 4 | 4 | 32 | 8 | 4 | 8 | 1 | 4 | 1 | 4 | 2 | 4 | 16 | 4 | 16 | 8 | 2 | 8 |
In GAP, Magma, Sage, TeX
S_3\times D_4\times C_{10} % in TeX
G:=Group("S3xD4xC10"); // GroupNames label
G:=SmallGroup(480,1154);
// by ID
G=gap.SmallGroup(480,1154);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,633,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^3=c^2=d^4=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