direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D10.D6, C23⋊3(C3⋊F5), (C22×C6)⋊4F5, (C22×C30)⋊5C4, (C6×D5).78D4, C6⋊2(C22⋊F5), C30⋊2(C22⋊C4), D5⋊(C6.D4), C10⋊(C6.D4), D10⋊8(C2×Dic3), (C23×D5).5S3, C6.44(C22×F5), C30.82(C22×C4), (C22×D5)⋊7Dic3, (C6×D5).65C23, (C22×C10)⋊7Dic3, D10.35(C3⋊D4), D10.50(C22×S3), (C22×D5).104D6, C10.13(C22×Dic3), (D5×C2×C6)⋊10C4, (C2×C6)⋊8(C2×F5), C5⋊(C2×C6.D4), (C2×C30)⋊7(C2×C4), C22⋊3(C2×C3⋊F5), C3⋊3(C2×C22⋊F5), C15⋊6(C2×C22⋊C4), (C22×C3⋊F5)⋊5C2, (C2×C3⋊F5)⋊4C22, (C6×D5)⋊31(C2×C4), D5.4(C2×C3⋊D4), (D5×C22×C6).6C2, C2.13(C22×C3⋊F5), (C3×D5).13(C2×D4), (C2×C10)⋊6(C2×Dic3), (C3×D5)⋊4(C22⋊C4), (D5×C2×C6).146C22, SmallGroup(480,1072)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1324 in 264 conjugacy classes, 81 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C22, C22 [×2], C22 [×20], C5, C6, C6 [×2], C6 [×8], C2×C4 [×8], C23, C23 [×10], D5 [×4], D5 [×2], C10, C10 [×2], C10 [×2], Dic3 [×4], C2×C6, C2×C6 [×2], C2×C6 [×20], C15, C22⋊C4 [×4], C22×C4 [×2], C24, F5 [×4], D10 [×2], D10 [×6], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×8], C22×C6, C22×C6 [×10], C3×D5 [×4], C3×D5 [×2], C30, C30 [×2], C30 [×2], C2×C22⋊C4, C2×F5 [×8], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C6.D4 [×4], C22×Dic3 [×2], C23×C6, C3⋊F5 [×4], C6×D5 [×2], C6×D5 [×6], C6×D5 [×10], C2×C30, C2×C30 [×2], C2×C30 [×2], C22⋊F5 [×4], C22×F5 [×2], C23×D5, C2×C6.D4, C2×C3⋊F5 [×4], C2×C3⋊F5 [×4], D5×C2×C6 [×2], D5×C2×C6 [×4], D5×C2×C6 [×4], C22×C30, C2×C22⋊F5, D10.D6 [×4], C22×C3⋊F5 [×2], D5×C22×C6, C2×D10.D6
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, C2×F5 [×3], C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C3⋊F5, C22⋊F5 [×2], C22×F5, C2×C6.D4, C2×C3⋊F5 [×3], C2×C22⋊F5, D10.D6 [×2], C22×C3⋊F5, C2×D10.D6
Generators and relations
G = < a,b,c,d,e | a2=b10=c2=d6=1, e2=b4c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b7, cd=dc, ece-1=b6c, ede-1=b5d-1 >
►On 120 points
Generators in S120
(1 87)(2 88)(3 89)(4 90)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 61)(18 62)(19 63)(20 64)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 71)(30 72)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 24)(22 23)(25 30)(26 29)(27 28)(31 38)(32 37)(33 36)(34 35)(39 40)(41 48)(42 47)(43 46)(44 45)(49 50)(51 52)(53 60)(54 59)(55 58)(56 57)(61 68)(62 67)(63 66)(64 65)(69 70)(71 78)(72 77)(73 76)(74 75)(79 80)(81 82)(83 90)(84 89)(85 88)(86 87)(91 98)(92 97)(93 96)(94 95)(99 100)(101 108)(102 107)(103 106)(104 105)(109 110)(111 112)(113 120)(114 119)(115 118)(116 117)
(1 16 23 117 40 105)(2 17 24 118 31 106)(3 18 25 119 32 107)(4 19 26 120 33 108)(5 20 27 111 34 109)(6 11 28 112 35 110)(7 12 29 113 36 101)(8 13 30 114 37 102)(9 14 21 115 38 103)(10 15 22 116 39 104)(41 83 66 71 53 96)(42 84 67 72 54 97)(43 85 68 73 55 98)(44 86 69 74 56 99)(45 87 70 75 57 100)(46 88 61 76 58 91)(47 89 62 77 59 92)(48 90 63 78 60 93)(49 81 64 79 51 94)(50 82 65 80 52 95)
(1 50 6 45)(2 43 5 42)(3 46 4 49)(7 48 10 47)(8 41 9 44)(11 95 16 100)(12 98 15 97)(13 91 14 94)(17 93 20 92)(18 96 19 99)(21 56 30 53)(22 59 29 60)(23 52 28 57)(24 55 27 54)(25 58 26 51)(31 68 34 67)(32 61 33 64)(35 70 40 65)(36 63 39 62)(37 66 38 69)(71 120 74 119)(72 113 73 116)(75 112 80 117)(76 115 79 114)(77 118 78 111)(81 102 88 103)(82 105 87 110)(83 108 86 107)(84 101 85 104)(89 106 90 109)
G:=sub<Sym(120)| (1,87)(2,88)(3,89)(4,90)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,71)(30,72)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,24)(22,23)(25,30)(26,29)(27,28)(31,38)(32,37)(33,36)(34,35)(39,40)(41,48)(42,47)(43,46)(44,45)(49,50)(51,52)(53,60)(54,59)(55,58)(56,57)(61,68)(62,67)(63,66)(64,65)(69,70)(71,78)(72,77)(73,76)(74,75)(79,80)(81,82)(83,90)(84,89)(85,88)(86,87)(91,98)(92,97)(93,96)(94,95)(99,100)(101,108)(102,107)(103,106)(104,105)(109,110)(111,112)(113,120)(114,119)(115,118)(116,117), (1,16,23,117,40,105)(2,17,24,118,31,106)(3,18,25,119,32,107)(4,19,26,120,33,108)(5,20,27,111,34,109)(6,11,28,112,35,110)(7,12,29,113,36,101)(8,13,30,114,37,102)(9,14,21,115,38,103)(10,15,22,116,39,104)(41,83,66,71,53,96)(42,84,67,72,54,97)(43,85,68,73,55,98)(44,86,69,74,56,99)(45,87,70,75,57,100)(46,88,61,76,58,91)(47,89,62,77,59,92)(48,90,63,78,60,93)(49,81,64,79,51,94)(50,82,65,80,52,95), (1,50,6,45)(2,43,5,42)(3,46,4,49)(7,48,10,47)(8,41,9,44)(11,95,16,100)(12,98,15,97)(13,91,14,94)(17,93,20,92)(18,96,19,99)(21,56,30,53)(22,59,29,60)(23,52,28,57)(24,55,27,54)(25,58,26,51)(31,68,34,67)(32,61,33,64)(35,70,40,65)(36,63,39,62)(37,66,38,69)(71,120,74,119)(72,113,73,116)(75,112,80,117)(76,115,79,114)(77,118,78,111)(81,102,88,103)(82,105,87,110)(83,108,86,107)(84,101,85,104)(89,106,90,109)>;
G:=Group( (1,87)(2,88)(3,89)(4,90)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,71)(30,72)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,24)(22,23)(25,30)(26,29)(27,28)(31,38)(32,37)(33,36)(34,35)(39,40)(41,48)(42,47)(43,46)(44,45)(49,50)(51,52)(53,60)(54,59)(55,58)(56,57)(61,68)(62,67)(63,66)(64,65)(69,70)(71,78)(72,77)(73,76)(74,75)(79,80)(81,82)(83,90)(84,89)(85,88)(86,87)(91,98)(92,97)(93,96)(94,95)(99,100)(101,108)(102,107)(103,106)(104,105)(109,110)(111,112)(113,120)(114,119)(115,118)(116,117), (1,16,23,117,40,105)(2,17,24,118,31,106)(3,18,25,119,32,107)(4,19,26,120,33,108)(5,20,27,111,34,109)(6,11,28,112,35,110)(7,12,29,113,36,101)(8,13,30,114,37,102)(9,14,21,115,38,103)(10,15,22,116,39,104)(41,83,66,71,53,96)(42,84,67,72,54,97)(43,85,68,73,55,98)(44,86,69,74,56,99)(45,87,70,75,57,100)(46,88,61,76,58,91)(47,89,62,77,59,92)(48,90,63,78,60,93)(49,81,64,79,51,94)(50,82,65,80,52,95), (1,50,6,45)(2,43,5,42)(3,46,4,49)(7,48,10,47)(8,41,9,44)(11,95,16,100)(12,98,15,97)(13,91,14,94)(17,93,20,92)(18,96,19,99)(21,56,30,53)(22,59,29,60)(23,52,28,57)(24,55,27,54)(25,58,26,51)(31,68,34,67)(32,61,33,64)(35,70,40,65)(36,63,39,62)(37,66,38,69)(71,120,74,119)(72,113,73,116)(75,112,80,117)(76,115,79,114)(77,118,78,111)(81,102,88,103)(82,105,87,110)(83,108,86,107)(84,101,85,104)(89,106,90,109) );
G=PermutationGroup([(1,87),(2,88),(3,89),(4,90),(5,81),(6,82),(7,83),(8,84),(9,85),(10,86),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,61),(18,62),(19,63),(20,64),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,71),(30,72),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16),(21,24),(22,23),(25,30),(26,29),(27,28),(31,38),(32,37),(33,36),(34,35),(39,40),(41,48),(42,47),(43,46),(44,45),(49,50),(51,52),(53,60),(54,59),(55,58),(56,57),(61,68),(62,67),(63,66),(64,65),(69,70),(71,78),(72,77),(73,76),(74,75),(79,80),(81,82),(83,90),(84,89),(85,88),(86,87),(91,98),(92,97),(93,96),(94,95),(99,100),(101,108),(102,107),(103,106),(104,105),(109,110),(111,112),(113,120),(114,119),(115,118),(116,117)], [(1,16,23,117,40,105),(2,17,24,118,31,106),(3,18,25,119,32,107),(4,19,26,120,33,108),(5,20,27,111,34,109),(6,11,28,112,35,110),(7,12,29,113,36,101),(8,13,30,114,37,102),(9,14,21,115,38,103),(10,15,22,116,39,104),(41,83,66,71,53,96),(42,84,67,72,54,97),(43,85,68,73,55,98),(44,86,69,74,56,99),(45,87,70,75,57,100),(46,88,61,76,58,91),(47,89,62,77,59,92),(48,90,63,78,60,93),(49,81,64,79,51,94),(50,82,65,80,52,95)], [(1,50,6,45),(2,43,5,42),(3,46,4,49),(7,48,10,47),(8,41,9,44),(11,95,16,100),(12,98,15,97),(13,91,14,94),(17,93,20,92),(18,96,19,99),(21,56,30,53),(22,59,29,60),(23,52,28,57),(24,55,27,54),(25,58,26,51),(31,68,34,67),(32,61,33,64),(35,70,40,65),(36,63,39,62),(37,66,38,69),(71,120,74,119),(72,113,73,116),(75,112,80,117),(76,115,79,114),(77,118,78,111),(81,102,88,103),(82,105,87,110),(83,108,86,107),(84,101,85,104),(89,106,90,109)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 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 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 60 | 0 |
| 0 | 0 | 0 | 1 | 0 | 60 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 60 | 0 | 1 |
| 0 | 0 | 60 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 54 | 0 | 7 |
| 0 | 0 | 0 | 10 | 54 | 7 |
| 0 | 0 | 7 | 54 | 10 | 0 |
| 0 | 0 | 7 | 0 | 54 | 17 |
| 0 | 50 | 0 | 0 | 0 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 14 | 47 | 54 |
| 0 | 0 | 0 | 14 | 54 | 7 |
| 0 | 0 | 14 | 0 | 47 | 7 |
| 0 | 0 | 14 | 7 | 0 | 54 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,1,1,1,1,0,0,60,0,0,0,0,0,0,60,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,1,1,1,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,17,0,7,7,0,0,54,10,54,0,0,0,0,54,10,54,0,0,7,7,0,17],[0,11,0,0,0,0,50,0,0,0,0,0,0,0,7,0,14,14,0,0,14,14,0,7,0,0,47,54,47,0,0,0,54,7,7,54] >;
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4H | 5 | 6A | ··· | 6G | 6H | ··· | 6O | 10A | ··· | 10G | 15A | 15B | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 2 | 30 | ··· | 30 | 4 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | Dic3 | D6 | Dic3 | C3⋊D4 | F5 | C2×F5 | C3⋊F5 | C22⋊F5 | C2×C3⋊F5 | D10.D6 |
| kernel | C2×D10.D6 | D10.D6 | C22×C3⋊F5 | D5×C22×C6 | D5×C2×C6 | C22×C30 | C23×D5 | C6×D5 | C22×D5 | C22×D5 | C22×C10 | D10 | C22×C6 | C2×C6 | C23 | C6 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 6 | 2 | 1 | 4 | 3 | 3 | 1 | 8 | 1 | 3 | 2 | 4 | 6 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_{10}.D_6 % in TeX
G:=Group("C2xD10.D6"); // GroupNames label
G:=SmallGroup(480,1072);
// by ID
G=gap.SmallGroup(480,1072);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,2693,14118,2379]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^6=1,e^2=b^4*c,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,e*b*e^-1=b^7,c*d=d*c,e*c*e^-1=b^6*c,e*d*e^-1=b^5*d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀