direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic3⋊F5, D10.4Dic6, C30⋊(C4⋊C4), C6⋊2(C4⋊F5), (C2×F5).8D6, (C6×D5).4Q8, D5⋊(Dic3⋊C4), C10⋊(Dic3⋊C4), (D5×Dic3)⋊6C4, (C2×Dic3)⋊4F5, Dic3⋊4(C2×F5), (C6×D5).31D4, Dic15⋊6(C2×C4), (C10×Dic3)⋊6C4, (C2×Dic15)⋊7C4, D10.27(C4×S3), D5.2(C2×Dic6), (C22×F5).2S3, C6.20(C22×F5), C22.19(S3×F5), C30.20(C22×C4), (C6×D5).32C23, (C22×D5).76D6, (C6×F5).12C22, D10.15(C3⋊D4), D10.35(C22×S3), (D5×Dic3).18C22, C3⋊3(C2×C4⋊F5), C15⋊2(C2×C4⋊C4), C5⋊(C2×Dic3⋊C4), (C3×D5)⋊(C4⋊C4), (C2×C6×F5).3C2, C2.22(C2×S3×F5), C10.20(S3×C2×C4), (C3×D5).5(C2×D4), D5.1(C2×C3⋊D4), (C2×C6).20(C2×F5), (C3×D5).4(C2×Q8), (C2×C30).15(C2×C4), (C2×C10).17(C4×S3), (C22×C3⋊F5).3C2, (C5×Dic3)⋊6(C2×C4), (C6×D5).25(C2×C4), (C2×D5×Dic3).13C2, (D5×C2×C6).69C22, (C2×C3⋊F5).12C22, SmallGroup(480,1001)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 884 in 184 conjugacy classes, 70 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×8], C22, C22 [×6], C5, C6, C6 [×2], C6 [×4], C2×C4 [×14], C23, D5 [×2], D5 [×2], C10, C10 [×2], Dic3 [×2], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×6], C15, C4⋊C4 [×4], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×4], C2×C10, C2×Dic3, C2×Dic3 [×9], C2×C12 [×4], C22×C6, C3×D5 [×2], C3×D5 [×2], C30, C30 [×2], C2×C4⋊C4, C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×2], C2×F5 [×6], C22×D5, Dic3⋊C4 [×4], C22×Dic3 [×2], C22×C12, C5×Dic3 [×2], Dic15 [×2], C3×F5 [×2], C3⋊F5 [×2], C6×D5 [×2], C6×D5 [×4], C2×C30, C4⋊F5 [×4], C2×C4×D5, C22×F5, C22×F5, C2×Dic3⋊C4, D5×Dic3 [×4], C10×Dic3, C2×Dic15, C6×F5 [×2], C6×F5 [×2], C2×C3⋊F5 [×2], C2×C3⋊F5 [×2], D5×C2×C6, C2×C4⋊F5, Dic3⋊F5 [×4], C2×D5×Dic3, C2×C6×F5, C22×C3⋊F5, C2×Dic3⋊F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4, C2×F5 [×3], Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, C2×C3⋊D4, C4⋊F5 [×2], C22×F5, C2×Dic3⋊C4, S3×F5, C2×C4⋊F5, Dic3⋊F5 [×2], C2×S3×F5, C2×Dic3⋊F5
Generators and relations
G = < a,b,c,d,e | a2=b6=d5=e4=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d3 >
►On 120 points
Generators in S120
(1 59)(2 60)(3 55)(4 56)(5 57)(6 58)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 105)(14 106)(15 107)(16 108)(17 103)(18 104)(19 101)(20 102)(21 97)(22 98)(23 99)(24 100)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 69)(32 70)(33 71)(34 72)(35 67)(36 68)(37 65)(38 66)(39 61)(40 62)(41 63)(42 64)(43 76)(44 77)(45 78)(46 73)(47 74)(48 75)(49 82)(50 83)(51 84)(52 79)(53 80)(54 81)(85 118)(86 119)(87 120)(88 115)(89 116)(90 117)
(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)
(1 115 4 118)(2 120 5 117)(3 119 6 116)(7 54 10 51)(8 53 11 50)(9 52 12 49)(13 48 16 45)(14 47 17 44)(15 46 18 43)(19 31 22 34)(20 36 23 33)(21 35 24 32)(25 41 28 38)(26 40 29 37)(27 39 30 42)(55 86 58 89)(56 85 59 88)(57 90 60 87)(61 96 64 93)(62 95 65 92)(63 94 66 91)(67 100 70 97)(68 99 71 102)(69 98 72 101)(73 104 76 107)(74 103 77 106)(75 108 78 105)(79 114 82 111)(80 113 83 110)(81 112 84 109)
(1 36 44 53 38)(2 31 45 54 39)(3 32 46 49 40)(4 33 47 50 41)(5 34 48 51 42)(6 35 43 52 37)(7 27 117 19 16)(8 28 118 20 17)(9 29 119 21 18)(10 30 120 22 13)(11 25 115 23 14)(12 26 116 24 15)(55 70 73 82 62)(56 71 74 83 63)(57 72 75 84 64)(58 67 76 79 65)(59 68 77 80 66)(60 69 78 81 61)(85 102 103 110 94)(86 97 104 111 95)(87 98 105 112 96)(88 99 106 113 91)(89 100 107 114 92)(90 101 108 109 93)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 55)(7 101 16 93)(8 102 17 94)(9 97 18 95)(10 98 13 96)(11 99 14 91)(12 100 15 92)(19 108 27 109)(20 103 28 110)(21 104 29 111)(22 105 30 112)(23 106 25 113)(24 107 26 114)(31 75 39 84)(32 76 40 79)(33 77 41 80)(34 78 42 81)(35 73 37 82)(36 74 38 83)(43 62 52 70)(44 63 53 71)(45 64 54 72)(46 65 49 67)(47 66 50 68)(48 61 51 69)(85 118)(86 119)(87 120)(88 115)(89 116)(90 117)
G:=sub<Sym(120)| (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,105)(14,106)(15,107)(16,108)(17,103)(18,104)(19,101)(20,102)(21,97)(22,98)(23,99)(24,100)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,69)(32,70)(33,71)(34,72)(35,67)(36,68)(37,65)(38,66)(39,61)(40,62)(41,63)(42,64)(43,76)(44,77)(45,78)(46,73)(47,74)(48,75)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(85,118)(86,119)(87,120)(88,115)(89,116)(90,117), (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), (1,115,4,118)(2,120,5,117)(3,119,6,116)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,48,16,45)(14,47,17,44)(15,46,18,43)(19,31,22,34)(20,36,23,33)(21,35,24,32)(25,41,28,38)(26,40,29,37)(27,39,30,42)(55,86,58,89)(56,85,59,88)(57,90,60,87)(61,96,64,93)(62,95,65,92)(63,94,66,91)(67,100,70,97)(68,99,71,102)(69,98,72,101)(73,104,76,107)(74,103,77,106)(75,108,78,105)(79,114,82,111)(80,113,83,110)(81,112,84,109), (1,36,44,53,38)(2,31,45,54,39)(3,32,46,49,40)(4,33,47,50,41)(5,34,48,51,42)(6,35,43,52,37)(7,27,117,19,16)(8,28,118,20,17)(9,29,119,21,18)(10,30,120,22,13)(11,25,115,23,14)(12,26,116,24,15)(55,70,73,82,62)(56,71,74,83,63)(57,72,75,84,64)(58,67,76,79,65)(59,68,77,80,66)(60,69,78,81,61)(85,102,103,110,94)(86,97,104,111,95)(87,98,105,112,96)(88,99,106,113,91)(89,100,107,114,92)(90,101,108,109,93), (1,56)(2,57)(3,58)(4,59)(5,60)(6,55)(7,101,16,93)(8,102,17,94)(9,97,18,95)(10,98,13,96)(11,99,14,91)(12,100,15,92)(19,108,27,109)(20,103,28,110)(21,104,29,111)(22,105,30,112)(23,106,25,113)(24,107,26,114)(31,75,39,84)(32,76,40,79)(33,77,41,80)(34,78,42,81)(35,73,37,82)(36,74,38,83)(43,62,52,70)(44,63,53,71)(45,64,54,72)(46,65,49,67)(47,66,50,68)(48,61,51,69)(85,118)(86,119)(87,120)(88,115)(89,116)(90,117)>;
G:=Group( (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,105)(14,106)(15,107)(16,108)(17,103)(18,104)(19,101)(20,102)(21,97)(22,98)(23,99)(24,100)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,69)(32,70)(33,71)(34,72)(35,67)(36,68)(37,65)(38,66)(39,61)(40,62)(41,63)(42,64)(43,76)(44,77)(45,78)(46,73)(47,74)(48,75)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(85,118)(86,119)(87,120)(88,115)(89,116)(90,117), (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), (1,115,4,118)(2,120,5,117)(3,119,6,116)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,48,16,45)(14,47,17,44)(15,46,18,43)(19,31,22,34)(20,36,23,33)(21,35,24,32)(25,41,28,38)(26,40,29,37)(27,39,30,42)(55,86,58,89)(56,85,59,88)(57,90,60,87)(61,96,64,93)(62,95,65,92)(63,94,66,91)(67,100,70,97)(68,99,71,102)(69,98,72,101)(73,104,76,107)(74,103,77,106)(75,108,78,105)(79,114,82,111)(80,113,83,110)(81,112,84,109), (1,36,44,53,38)(2,31,45,54,39)(3,32,46,49,40)(4,33,47,50,41)(5,34,48,51,42)(6,35,43,52,37)(7,27,117,19,16)(8,28,118,20,17)(9,29,119,21,18)(10,30,120,22,13)(11,25,115,23,14)(12,26,116,24,15)(55,70,73,82,62)(56,71,74,83,63)(57,72,75,84,64)(58,67,76,79,65)(59,68,77,80,66)(60,69,78,81,61)(85,102,103,110,94)(86,97,104,111,95)(87,98,105,112,96)(88,99,106,113,91)(89,100,107,114,92)(90,101,108,109,93), (1,56)(2,57)(3,58)(4,59)(5,60)(6,55)(7,101,16,93)(8,102,17,94)(9,97,18,95)(10,98,13,96)(11,99,14,91)(12,100,15,92)(19,108,27,109)(20,103,28,110)(21,104,29,111)(22,105,30,112)(23,106,25,113)(24,107,26,114)(31,75,39,84)(32,76,40,79)(33,77,41,80)(34,78,42,81)(35,73,37,82)(36,74,38,83)(43,62,52,70)(44,63,53,71)(45,64,54,72)(46,65,49,67)(47,66,50,68)(48,61,51,69)(85,118)(86,119)(87,120)(88,115)(89,116)(90,117) );
G=PermutationGroup([(1,59),(2,60),(3,55),(4,56),(5,57),(6,58),(7,109),(8,110),(9,111),(10,112),(11,113),(12,114),(13,105),(14,106),(15,107),(16,108),(17,103),(18,104),(19,101),(20,102),(21,97),(22,98),(23,99),(24,100),(25,91),(26,92),(27,93),(28,94),(29,95),(30,96),(31,69),(32,70),(33,71),(34,72),(35,67),(36,68),(37,65),(38,66),(39,61),(40,62),(41,63),(42,64),(43,76),(44,77),(45,78),(46,73),(47,74),(48,75),(49,82),(50,83),(51,84),(52,79),(53,80),(54,81),(85,118),(86,119),(87,120),(88,115),(89,116),(90,117)], [(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)], [(1,115,4,118),(2,120,5,117),(3,119,6,116),(7,54,10,51),(8,53,11,50),(9,52,12,49),(13,48,16,45),(14,47,17,44),(15,46,18,43),(19,31,22,34),(20,36,23,33),(21,35,24,32),(25,41,28,38),(26,40,29,37),(27,39,30,42),(55,86,58,89),(56,85,59,88),(57,90,60,87),(61,96,64,93),(62,95,65,92),(63,94,66,91),(67,100,70,97),(68,99,71,102),(69,98,72,101),(73,104,76,107),(74,103,77,106),(75,108,78,105),(79,114,82,111),(80,113,83,110),(81,112,84,109)], [(1,36,44,53,38),(2,31,45,54,39),(3,32,46,49,40),(4,33,47,50,41),(5,34,48,51,42),(6,35,43,52,37),(7,27,117,19,16),(8,28,118,20,17),(9,29,119,21,18),(10,30,120,22,13),(11,25,115,23,14),(12,26,116,24,15),(55,70,73,82,62),(56,71,74,83,63),(57,72,75,84,64),(58,67,76,79,65),(59,68,77,80,66),(60,69,78,81,61),(85,102,103,110,94),(86,97,104,111,95),(87,98,105,112,96),(88,99,106,113,91),(89,100,107,114,92),(90,101,108,109,93)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,55),(7,101,16,93),(8,102,17,94),(9,97,18,95),(10,98,13,96),(11,99,14,91),(12,100,15,92),(19,108,27,109),(20,103,28,110),(21,104,29,111),(22,105,30,112),(23,106,25,113),(24,107,26,114),(31,75,39,84),(32,76,40,79),(33,77,41,80),(34,78,42,81),(35,73,37,82),(36,74,38,83),(43,62,52,70),(44,63,53,71),(45,64,54,72),(46,65,49,67),(47,66,50,68),(48,61,51,69),(85,118),(86,119),(87,120),(88,115),(89,116),(90,117)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 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 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 48 | 0 | 0 | 0 | 0 | 0 |
| 44 | 14 | 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 | 1 |
| 57 | 53 | 0 | 0 | 0 | 0 |
| 25 | 4 | 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 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 60 | 60 | 60 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 50 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[48,44,0,0,0,0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[57,25,0,0,0,0,53,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,1,0,0,60,0,0,0,1,0,60,0,0,0,0,1,60],[11,50,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,1,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,60,0,1] >;
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | 10B | 10C | 12A | ··· | 12H | 15 | 20A | 20B | 20C | 20D | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | ··· | 30 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 10 | ··· | 10 | 8 | 12 | 12 | 12 | 12 | 8 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | + | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | Q8 | D6 | D6 | Dic6 | C4×S3 | C3⋊D4 | C4×S3 | F5 | C2×F5 | C2×F5 | C4⋊F5 | S3×F5 | Dic3⋊F5 | C2×S3×F5 |
| kernel | C2×Dic3⋊F5 | Dic3⋊F5 | C2×D5×Dic3 | C2×C6×F5 | C22×C3⋊F5 | D5×Dic3 | C10×Dic3 | C2×Dic15 | C22×F5 | C6×D5 | C6×D5 | C2×F5 | C22×D5 | D10 | D10 | D10 | C2×C10 | C2×Dic3 | Dic3 | C2×C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 2 | 1 | 2 | 1 | 4 | 1 | 2 | 1 |
In GAP, Magma, Sage, TeX
C_2\times Dic_3\rtimes F_5
% in TeX
G:=Group("C2xDic3:F5"); // GroupNames label
G:=SmallGroup(480,1001);
// by ID
G=gap.SmallGroup(480,1001);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,120,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=d^5=e^4=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=d^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀