direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×C22⋊F5, D5.3(C6×D4), (C22×C6)⋊3F5, C23⋊3(C3×F5), C22⋊3(C6×F5), D10⋊8(C2×C12), (C22×C30)⋊8C4, (C6×D5).82D4, (C22×F5)⋊3C6, (C6×F5)⋊4C22, C30⋊3(C22⋊C4), (C22×C10)⋊6C12, D10.20(C3×D4), (C22×D5)⋊7C12, (C23×D5).5C6, C6.57(C22×F5), C30.95(C22×C4), (C6×D5).72C23, C10.13(C22×C12), D10.13(C22×C6), C5⋊(C6×C22⋊C4), (C2×F5)⋊(C2×C6), (C2×C6×F5)⋊5C2, C10⋊(C3×C22⋊C4), (D5×C2×C6)⋊12C4, D5⋊(C3×C22⋊C4), (C2×C6)⋊7(C2×F5), (C2×C30)⋊9(C2×C4), C2.13(C2×C6×F5), C15⋊7(C2×C22⋊C4), (C2×C10)⋊5(C2×C12), (C6×D5)⋊33(C2×C4), (D5×C22×C6).8C2, (C3×D5).14(C2×D4), (C3×D5)⋊5(C22⋊C4), (D5×C2×C6).152C22, (C22×D5).41(C2×C6), SmallGroup(480,1059)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1000 in 264 conjugacy classes, 92 normal (28 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], C12 [×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×C12 [×8], C22×C6, C22×C6 [×10], C3×D5 [×4], C3×D5 [×2], C30, C30 [×2], C30 [×2], C2×C22⋊C4, C2×F5 [×4], C2×F5 [×4], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C3×C22⋊C4 [×4], C22×C12 [×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, C6×C22⋊C4, C6×F5 [×4], C6×F5 [×4], D5×C2×C6 [×2], D5×C2×C6 [×4], D5×C2×C6 [×4], C22×C30, C2×C22⋊F5, C3×C22⋊F5 [×4], C2×C6×F5 [×2], D5×C22×C6, C6×C22⋊F5
Quotients:
C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×4], C23, C12 [×4], C2×C6 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C12 [×6], C3×D4 [×4], C22×C6, C2×C22⋊C4, C2×F5 [×3], C3×C22⋊C4 [×4], C22×C12, C6×D4 [×2], C3×F5, C22⋊F5 [×2], C22×F5, C6×C22⋊C4, C6×F5 [×3], C2×C22⋊F5, C3×C22⋊F5 [×2], C2×C6×F5, C6×C22⋊F5
Generators and relations
G = < a,b,c,d,e | a6=b2=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >
►On 120 points
Generators in S120
(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 54)(2 49)(3 50)(4 51)(5 52)(6 53)(7 59)(8 60)(9 55)(10 56)(11 57)(12 58)(13 96)(14 91)(15 92)(16 93)(17 94)(18 95)(19 41)(20 42)(21 37)(22 38)(23 39)(24 40)(25 32)(26 33)(27 34)(28 35)(29 36)(30 31)(43 88)(44 89)(45 90)(46 85)(47 86)(48 87)(61 113)(62 114)(63 109)(64 110)(65 111)(66 112)(67 119)(68 120)(69 115)(70 116)(71 117)(72 118)(73 106)(74 107)(75 108)(76 103)(77 104)(78 105)(79 101)(80 102)(81 97)(82 98)(83 99)(84 100)
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 90)(8 85)(9 86)(10 87)(11 88)(12 89)(13 111)(14 112)(15 113)(16 114)(17 109)(18 110)(19 104)(20 105)(21 106)(22 107)(23 108)(24 103)(25 117)(26 118)(27 119)(28 120)(29 115)(30 116)(31 70)(32 71)(33 72)(34 67)(35 68)(36 69)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 57)(44 58)(45 59)(46 60)(47 55)(48 56)(49 82)(50 83)(51 84)(52 79)(53 80)(54 81)(61 92)(62 93)(63 94)(64 95)(65 96)(66 91)
(1 29 41 17 59)(2 30 42 18 60)(3 25 37 13 55)(4 26 38 14 56)(5 27 39 15 57)(6 28 40 16 58)(7 54 36 19 94)(8 49 31 20 95)(9 50 32 21 96)(10 51 33 22 91)(11 52 34 23 92)(12 53 35 24 93)(43 101 119 75 113)(44 102 120 76 114)(45 97 115 77 109)(46 98 116 78 110)(47 99 117 73 111)(48 100 118 74 112)(61 88 79 67 108)(62 89 80 68 103)(63 90 81 69 104)(64 85 82 70 105)(65 86 83 71 106)(66 87 84 72 107)
(1 100)(2 101)(3 102)(4 97)(5 98)(6 99)(7 91 36 22)(8 92 31 23)(9 93 32 24)(10 94 33 19)(11 95 34 20)(12 96 35 21)(13 120 37 44)(14 115 38 45)(15 116 39 46)(16 117 40 47)(17 118 41 48)(18 119 42 43)(25 76 55 114)(26 77 56 109)(27 78 57 110)(28 73 58 111)(29 74 59 112)(30 75 60 113)(49 52)(50 53)(51 54)(61 70 108 85)(62 71 103 86)(63 72 104 87)(64 67 105 88)(65 68 106 89)(66 69 107 90)(79 82)(80 83)(81 84)
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), (1,54)(2,49)(3,50)(4,51)(5,52)(6,53)(7,59)(8,60)(9,55)(10,56)(11,57)(12,58)(13,96)(14,91)(15,92)(16,93)(17,94)(18,95)(19,41)(20,42)(21,37)(22,38)(23,39)(24,40)(25,32)(26,33)(27,34)(28,35)(29,36)(30,31)(43,88)(44,89)(45,90)(46,85)(47,86)(48,87)(61,113)(62,114)(63,109)(64,110)(65,111)(66,112)(67,119)(68,120)(69,115)(70,116)(71,117)(72,118)(73,106)(74,107)(75,108)(76,103)(77,104)(78,105)(79,101)(80,102)(81,97)(82,98)(83,99)(84,100), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,90)(8,85)(9,86)(10,87)(11,88)(12,89)(13,111)(14,112)(15,113)(16,114)(17,109)(18,110)(19,104)(20,105)(21,106)(22,107)(23,108)(24,103)(25,117)(26,118)(27,119)(28,120)(29,115)(30,116)(31,70)(32,71)(33,72)(34,67)(35,68)(36,69)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,57)(44,58)(45,59)(46,60)(47,55)(48,56)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(61,92)(62,93)(63,94)(64,95)(65,96)(66,91), (1,29,41,17,59)(2,30,42,18,60)(3,25,37,13,55)(4,26,38,14,56)(5,27,39,15,57)(6,28,40,16,58)(7,54,36,19,94)(8,49,31,20,95)(9,50,32,21,96)(10,51,33,22,91)(11,52,34,23,92)(12,53,35,24,93)(43,101,119,75,113)(44,102,120,76,114)(45,97,115,77,109)(46,98,116,78,110)(47,99,117,73,111)(48,100,118,74,112)(61,88,79,67,108)(62,89,80,68,103)(63,90,81,69,104)(64,85,82,70,105)(65,86,83,71,106)(66,87,84,72,107), (1,100)(2,101)(3,102)(4,97)(5,98)(6,99)(7,91,36,22)(8,92,31,23)(9,93,32,24)(10,94,33,19)(11,95,34,20)(12,96,35,21)(13,120,37,44)(14,115,38,45)(15,116,39,46)(16,117,40,47)(17,118,41,48)(18,119,42,43)(25,76,55,114)(26,77,56,109)(27,78,57,110)(28,73,58,111)(29,74,59,112)(30,75,60,113)(49,52)(50,53)(51,54)(61,70,108,85)(62,71,103,86)(63,72,104,87)(64,67,105,88)(65,68,106,89)(66,69,107,90)(79,82)(80,83)(81,84)>;
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), (1,54)(2,49)(3,50)(4,51)(5,52)(6,53)(7,59)(8,60)(9,55)(10,56)(11,57)(12,58)(13,96)(14,91)(15,92)(16,93)(17,94)(18,95)(19,41)(20,42)(21,37)(22,38)(23,39)(24,40)(25,32)(26,33)(27,34)(28,35)(29,36)(30,31)(43,88)(44,89)(45,90)(46,85)(47,86)(48,87)(61,113)(62,114)(63,109)(64,110)(65,111)(66,112)(67,119)(68,120)(69,115)(70,116)(71,117)(72,118)(73,106)(74,107)(75,108)(76,103)(77,104)(78,105)(79,101)(80,102)(81,97)(82,98)(83,99)(84,100), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,90)(8,85)(9,86)(10,87)(11,88)(12,89)(13,111)(14,112)(15,113)(16,114)(17,109)(18,110)(19,104)(20,105)(21,106)(22,107)(23,108)(24,103)(25,117)(26,118)(27,119)(28,120)(29,115)(30,116)(31,70)(32,71)(33,72)(34,67)(35,68)(36,69)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,57)(44,58)(45,59)(46,60)(47,55)(48,56)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(61,92)(62,93)(63,94)(64,95)(65,96)(66,91), (1,29,41,17,59)(2,30,42,18,60)(3,25,37,13,55)(4,26,38,14,56)(5,27,39,15,57)(6,28,40,16,58)(7,54,36,19,94)(8,49,31,20,95)(9,50,32,21,96)(10,51,33,22,91)(11,52,34,23,92)(12,53,35,24,93)(43,101,119,75,113)(44,102,120,76,114)(45,97,115,77,109)(46,98,116,78,110)(47,99,117,73,111)(48,100,118,74,112)(61,88,79,67,108)(62,89,80,68,103)(63,90,81,69,104)(64,85,82,70,105)(65,86,83,71,106)(66,87,84,72,107), (1,100)(2,101)(3,102)(4,97)(5,98)(6,99)(7,91,36,22)(8,92,31,23)(9,93,32,24)(10,94,33,19)(11,95,34,20)(12,96,35,21)(13,120,37,44)(14,115,38,45)(15,116,39,46)(16,117,40,47)(17,118,41,48)(18,119,42,43)(25,76,55,114)(26,77,56,109)(27,78,57,110)(28,73,58,111)(29,74,59,112)(30,75,60,113)(49,52)(50,53)(51,54)(61,70,108,85)(62,71,103,86)(63,72,104,87)(64,67,105,88)(65,68,106,89)(66,69,107,90)(79,82)(80,83)(81,84) );
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)], [(1,54),(2,49),(3,50),(4,51),(5,52),(6,53),(7,59),(8,60),(9,55),(10,56),(11,57),(12,58),(13,96),(14,91),(15,92),(16,93),(17,94),(18,95),(19,41),(20,42),(21,37),(22,38),(23,39),(24,40),(25,32),(26,33),(27,34),(28,35),(29,36),(30,31),(43,88),(44,89),(45,90),(46,85),(47,86),(48,87),(61,113),(62,114),(63,109),(64,110),(65,111),(66,112),(67,119),(68,120),(69,115),(70,116),(71,117),(72,118),(73,106),(74,107),(75,108),(76,103),(77,104),(78,105),(79,101),(80,102),(81,97),(82,98),(83,99),(84,100)], [(1,97),(2,98),(3,99),(4,100),(5,101),(6,102),(7,90),(8,85),(9,86),(10,87),(11,88),(12,89),(13,111),(14,112),(15,113),(16,114),(17,109),(18,110),(19,104),(20,105),(21,106),(22,107),(23,108),(24,103),(25,117),(26,118),(27,119),(28,120),(29,115),(30,116),(31,70),(32,71),(33,72),(34,67),(35,68),(36,69),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,57),(44,58),(45,59),(46,60),(47,55),(48,56),(49,82),(50,83),(51,84),(52,79),(53,80),(54,81),(61,92),(62,93),(63,94),(64,95),(65,96),(66,91)], [(1,29,41,17,59),(2,30,42,18,60),(3,25,37,13,55),(4,26,38,14,56),(5,27,39,15,57),(6,28,40,16,58),(7,54,36,19,94),(8,49,31,20,95),(9,50,32,21,96),(10,51,33,22,91),(11,52,34,23,92),(12,53,35,24,93),(43,101,119,75,113),(44,102,120,76,114),(45,97,115,77,109),(46,98,116,78,110),(47,99,117,73,111),(48,100,118,74,112),(61,88,79,67,108),(62,89,80,68,103),(63,90,81,69,104),(64,85,82,70,105),(65,86,83,71,106),(66,87,84,72,107)], [(1,100),(2,101),(3,102),(4,97),(5,98),(6,99),(7,91,36,22),(8,92,31,23),(9,93,32,24),(10,94,33,19),(11,95,34,20),(12,96,35,21),(13,120,37,44),(14,115,38,45),(15,116,39,46),(16,117,40,47),(17,118,41,48),(18,119,42,43),(25,76,55,114),(26,77,56,109),(27,78,57,110),(28,73,58,111),(29,74,59,112),(30,75,60,113),(49,52),(50,53),(51,54),(61,70,108,85),(62,71,103,86),(63,72,104,87),(64,67,105,88),(65,68,106,89),(66,69,107,90),(79,82),(80,83),(81,84)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 0 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 |
| 60 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 54 | 0 | 47 | 47 |
| 0 | 0 | 14 | 7 | 14 | 0 |
| 0 | 0 | 0 | 14 | 7 | 14 |
| 0 | 0 | 47 | 47 | 0 | 54 |
| 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 |
| 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 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 50 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,14],[60,0,0,0,0,0,2,1,0,0,0,0,0,0,54,14,0,47,0,0,0,7,14,47,0,0,47,14,7,0,0,0,47,0,14,54],[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],[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],[50,50,0,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4H | 5 | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | ··· | 6R | 6S | 6T | 6U | 6V | 10A | ··· | 10G | 12A | ··· | 12P | 15A | 15B | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 1 | 1 | 10 | ··· | 10 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 4 | ··· | 4 | 10 | ··· | 10 | 4 | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D4 | C3×D4 | F5 | C2×F5 | C3×F5 | C22⋊F5 | C6×F5 | C3×C22⋊F5 |
| kernel | C6×C22⋊F5 | C3×C22⋊F5 | C2×C6×F5 | D5×C22×C6 | C2×C22⋊F5 | D5×C2×C6 | C22×C30 | C22⋊F5 | C22×F5 | C23×D5 | C22×D5 | C22×C10 | C6×D5 | D10 | C22×C6 | C2×C6 | C23 | C6 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 6 | 2 | 8 | 4 | 2 | 12 | 4 | 4 | 8 | 1 | 3 | 2 | 4 | 6 | 8 |
In GAP, Magma, Sage, TeX
C_6\times C_2^2\rtimes F_5
% in TeX
G:=Group("C6xC2^2:F5"); // GroupNames label
G:=SmallGroup(480,1059);
// by ID
G=gap.SmallGroup(480,1059);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,1094,9414,818]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^2=c^2=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀