direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D4⋊D6, C60.227D4, C60.233C23, D4⋊S3⋊6C10, D4⋊4(S3×C10), (C5×D4)⋊26D6, (C5×Q8)⋊26D6, Q8⋊5(S3×C10), C6.59(D4×C10), C12.49(C5×D4), (C2×C30).95D4, (C10×D12)⋊26C2, (C2×D12)⋊10C10, C15⋊39(C8⋊C22), Q8⋊2S3⋊6C10, C30.442(C2×D4), (C2×C20).248D6, C4.Dic3⋊9C10, D12.11(C2×C10), (D4×C15)⋊35C22, (Q8×C15)⋊31C22, C20.117(C3⋊D4), C12.17(C22×C10), (C2×C60).370C22, C20.206(C22×S3), (C5×D12).50C22, C3⋊C8⋊4(C2×C10), C3⋊5(C5×C8⋊C22), C4○D4⋊3(C5×S3), C4.17(S3×C2×C10), (C2×C6).8(C5×D4), (C5×D4⋊S3)⋊14C2, (C5×C4○D4)⋊10S3, (C3×C4○D4)⋊1C10, (C3×D4)⋊4(C2×C10), (C5×C3⋊C8)⋊26C22, (C3×Q8)⋊4(C2×C10), C4.24(C5×C3⋊D4), (C15×C4○D4)⋊11C2, (C2×C4).20(S3×C10), C2.23(C10×C3⋊D4), C22.5(C5×C3⋊D4), (C2×C12).44(C2×C10), (C5×Q8⋊2S3)⋊14C2, C10.144(C2×C3⋊D4), (C5×C4.Dic3)⋊21C2, (C2×C10).41(C3⋊D4), SmallGroup(480,828)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4⋊D6
G = < a,b,c,d,e | a5=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ece=b-1c, ede=d-1 >
Subgroups: 372 in 136 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C12, C12, D6, C2×C6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, C2×C10, C2×C10, C3⋊C8, D12, D12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C5×S3, C30, C30, C8⋊C22, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, C4.Dic3, D4⋊S3, Q8⋊2S3, C2×D12, C3×C4○D4, C60, C60, S3×C10, C2×C30, C2×C30, C5×M4(2), C5×D8, C5×SD16, D4×C10, C5×C4○D4, D4⋊D6, C5×C3⋊C8, C5×D12, C5×D12, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, S3×C2×C10, C5×C8⋊C22, C5×C4.Dic3, C5×D4⋊S3, C5×Q8⋊2S3, C10×D12, C15×C4○D4, C5×D4⋊D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C8⋊C22, C5×D4, C22×C10, C2×C3⋊D4, S3×C10, D4×C10, D4⋊D6, C5×C3⋊D4, S3×C2×C10, C5×C8⋊C22, C10×C3⋊D4, C5×D4⋊D6
►On 120 points
Generators in S120
(1 50 38 26 14)(2 51 39 27 15)(3 49 37 25 13)(4 52 40 28 16)(5 53 41 29 17)(6 54 42 30 18)(7 55 43 31 19)(8 56 44 32 20)(9 57 45 33 21)(10 60 46 34 22)(11 58 47 35 23)(12 59 48 36 24)(61 113 101 89 77)(62 114 102 90 78)(63 109 97 85 73)(64 110 98 86 74)(65 111 99 87 75)(66 112 100 88 76)(67 115 105 91 79)(68 116 106 92 80)(69 117 107 93 81)(70 118 108 94 82)(71 119 103 95 83)(72 120 104 96 84)
(1 11 5 8)(2 12 6 9)(3 10 4 7)(13 22 16 19)(14 23 17 20)(15 24 18 21)(25 34 28 31)(26 35 29 32)(27 36 30 33)(37 46 40 43)(38 47 41 44)(39 48 42 45)(49 60 52 55)(50 58 53 56)(51 59 54 57)(61 71 64 68)(62 72 65 69)(63 67 66 70)(73 79 76 82)(74 80 77 83)(75 81 78 84)(85 91 88 94)(86 92 89 95)(87 93 90 96)(97 105 100 108)(98 106 101 103)(99 107 102 104)(109 115 112 118)(110 116 113 119)(111 117 114 120)
(1 68)(2 72)(3 70)(4 67)(5 71)(6 69)(7 63)(8 61)(9 65)(10 66)(11 64)(12 62)(13 82)(14 80)(15 84)(16 79)(17 83)(18 81)(19 73)(20 77)(21 75)(22 76)(23 74)(24 78)(25 94)(26 92)(27 96)(28 91)(29 95)(30 93)(31 85)(32 89)(33 87)(34 88)(35 86)(36 90)(37 108)(38 106)(39 104)(40 105)(41 103)(42 107)(43 97)(44 101)(45 99)(46 100)(47 98)(48 102)(49 118)(50 116)(51 120)(52 115)(53 119)(54 117)(55 109)(56 113)(57 111)(58 110)(59 114)(60 112)
(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 3)(4 5)(7 11)(8 10)(9 12)(13 14)(16 17)(19 23)(20 22)(21 24)(25 26)(28 29)(31 35)(32 34)(33 36)(37 38)(40 41)(43 47)(44 46)(45 48)(49 50)(52 53)(55 58)(56 60)(57 59)(61 70)(62 69)(63 68)(64 67)(65 72)(66 71)(73 80)(74 79)(75 84)(76 83)(77 82)(78 81)(85 92)(86 91)(87 96)(88 95)(89 94)(90 93)(97 106)(98 105)(99 104)(100 103)(101 108)(102 107)(109 116)(110 115)(111 120)(112 119)(113 118)(114 117)
G:=sub<Sym(120)| (1,50,38,26,14)(2,51,39,27,15)(3,49,37,25,13)(4,52,40,28,16)(5,53,41,29,17)(6,54,42,30,18)(7,55,43,31,19)(8,56,44,32,20)(9,57,45,33,21)(10,60,46,34,22)(11,58,47,35,23)(12,59,48,36,24)(61,113,101,89,77)(62,114,102,90,78)(63,109,97,85,73)(64,110,98,86,74)(65,111,99,87,75)(66,112,100,88,76)(67,115,105,91,79)(68,116,106,92,80)(69,117,107,93,81)(70,118,108,94,82)(71,119,103,95,83)(72,120,104,96,84), (1,11,5,8)(2,12,6,9)(3,10,4,7)(13,22,16,19)(14,23,17,20)(15,24,18,21)(25,34,28,31)(26,35,29,32)(27,36,30,33)(37,46,40,43)(38,47,41,44)(39,48,42,45)(49,60,52,55)(50,58,53,56)(51,59,54,57)(61,71,64,68)(62,72,65,69)(63,67,66,70)(73,79,76,82)(74,80,77,83)(75,81,78,84)(85,91,88,94)(86,92,89,95)(87,93,90,96)(97,105,100,108)(98,106,101,103)(99,107,102,104)(109,115,112,118)(110,116,113,119)(111,117,114,120), (1,68)(2,72)(3,70)(4,67)(5,71)(6,69)(7,63)(8,61)(9,65)(10,66)(11,64)(12,62)(13,82)(14,80)(15,84)(16,79)(17,83)(18,81)(19,73)(20,77)(21,75)(22,76)(23,74)(24,78)(25,94)(26,92)(27,96)(28,91)(29,95)(30,93)(31,85)(32,89)(33,87)(34,88)(35,86)(36,90)(37,108)(38,106)(39,104)(40,105)(41,103)(42,107)(43,97)(44,101)(45,99)(46,100)(47,98)(48,102)(49,118)(50,116)(51,120)(52,115)(53,119)(54,117)(55,109)(56,113)(57,111)(58,110)(59,114)(60,112), (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), (1,3)(4,5)(7,11)(8,10)(9,12)(13,14)(16,17)(19,23)(20,22)(21,24)(25,26)(28,29)(31,35)(32,34)(33,36)(37,38)(40,41)(43,47)(44,46)(45,48)(49,50)(52,53)(55,58)(56,60)(57,59)(61,70)(62,69)(63,68)(64,67)(65,72)(66,71)(73,80)(74,79)(75,84)(76,83)(77,82)(78,81)(85,92)(86,91)(87,96)(88,95)(89,94)(90,93)(97,106)(98,105)(99,104)(100,103)(101,108)(102,107)(109,116)(110,115)(111,120)(112,119)(113,118)(114,117)>;
G:=Group( (1,50,38,26,14)(2,51,39,27,15)(3,49,37,25,13)(4,52,40,28,16)(5,53,41,29,17)(6,54,42,30,18)(7,55,43,31,19)(8,56,44,32,20)(9,57,45,33,21)(10,60,46,34,22)(11,58,47,35,23)(12,59,48,36,24)(61,113,101,89,77)(62,114,102,90,78)(63,109,97,85,73)(64,110,98,86,74)(65,111,99,87,75)(66,112,100,88,76)(67,115,105,91,79)(68,116,106,92,80)(69,117,107,93,81)(70,118,108,94,82)(71,119,103,95,83)(72,120,104,96,84), (1,11,5,8)(2,12,6,9)(3,10,4,7)(13,22,16,19)(14,23,17,20)(15,24,18,21)(25,34,28,31)(26,35,29,32)(27,36,30,33)(37,46,40,43)(38,47,41,44)(39,48,42,45)(49,60,52,55)(50,58,53,56)(51,59,54,57)(61,71,64,68)(62,72,65,69)(63,67,66,70)(73,79,76,82)(74,80,77,83)(75,81,78,84)(85,91,88,94)(86,92,89,95)(87,93,90,96)(97,105,100,108)(98,106,101,103)(99,107,102,104)(109,115,112,118)(110,116,113,119)(111,117,114,120), (1,68)(2,72)(3,70)(4,67)(5,71)(6,69)(7,63)(8,61)(9,65)(10,66)(11,64)(12,62)(13,82)(14,80)(15,84)(16,79)(17,83)(18,81)(19,73)(20,77)(21,75)(22,76)(23,74)(24,78)(25,94)(26,92)(27,96)(28,91)(29,95)(30,93)(31,85)(32,89)(33,87)(34,88)(35,86)(36,90)(37,108)(38,106)(39,104)(40,105)(41,103)(42,107)(43,97)(44,101)(45,99)(46,100)(47,98)(48,102)(49,118)(50,116)(51,120)(52,115)(53,119)(54,117)(55,109)(56,113)(57,111)(58,110)(59,114)(60,112), (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), (1,3)(4,5)(7,11)(8,10)(9,12)(13,14)(16,17)(19,23)(20,22)(21,24)(25,26)(28,29)(31,35)(32,34)(33,36)(37,38)(40,41)(43,47)(44,46)(45,48)(49,50)(52,53)(55,58)(56,60)(57,59)(61,70)(62,69)(63,68)(64,67)(65,72)(66,71)(73,80)(74,79)(75,84)(76,83)(77,82)(78,81)(85,92)(86,91)(87,96)(88,95)(89,94)(90,93)(97,106)(98,105)(99,104)(100,103)(101,108)(102,107)(109,116)(110,115)(111,120)(112,119)(113,118)(114,117) );
G=PermutationGroup([[(1,50,38,26,14),(2,51,39,27,15),(3,49,37,25,13),(4,52,40,28,16),(5,53,41,29,17),(6,54,42,30,18),(7,55,43,31,19),(8,56,44,32,20),(9,57,45,33,21),(10,60,46,34,22),(11,58,47,35,23),(12,59,48,36,24),(61,113,101,89,77),(62,114,102,90,78),(63,109,97,85,73),(64,110,98,86,74),(65,111,99,87,75),(66,112,100,88,76),(67,115,105,91,79),(68,116,106,92,80),(69,117,107,93,81),(70,118,108,94,82),(71,119,103,95,83),(72,120,104,96,84)], [(1,11,5,8),(2,12,6,9),(3,10,4,7),(13,22,16,19),(14,23,17,20),(15,24,18,21),(25,34,28,31),(26,35,29,32),(27,36,30,33),(37,46,40,43),(38,47,41,44),(39,48,42,45),(49,60,52,55),(50,58,53,56),(51,59,54,57),(61,71,64,68),(62,72,65,69),(63,67,66,70),(73,79,76,82),(74,80,77,83),(75,81,78,84),(85,91,88,94),(86,92,89,95),(87,93,90,96),(97,105,100,108),(98,106,101,103),(99,107,102,104),(109,115,112,118),(110,116,113,119),(111,117,114,120)], [(1,68),(2,72),(3,70),(4,67),(5,71),(6,69),(7,63),(8,61),(9,65),(10,66),(11,64),(12,62),(13,82),(14,80),(15,84),(16,79),(17,83),(18,81),(19,73),(20,77),(21,75),(22,76),(23,74),(24,78),(25,94),(26,92),(27,96),(28,91),(29,95),(30,93),(31,85),(32,89),(33,87),(34,88),(35,86),(36,90),(37,108),(38,106),(39,104),(40,105),(41,103),(42,107),(43,97),(44,101),(45,99),(46,100),(47,98),(48,102),(49,118),(50,116),(51,120),(52,115),(53,119),(54,117),(55,109),(56,113),(57,111),(58,110),(59,114),(60,112)], [(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)], [(1,3),(4,5),(7,11),(8,10),(9,12),(13,14),(16,17),(19,23),(20,22),(21,24),(25,26),(28,29),(31,35),(32,34),(33,36),(37,38),(40,41),(43,47),(44,46),(45,48),(49,50),(52,53),(55,58),(56,60),(57,59),(61,70),(62,69),(63,68),(64,67),(65,72),(66,71),(73,80),(74,79),(75,84),(76,83),(77,82),(78,81),(85,92),(86,91),(87,96),(88,95),(89,94),(90,93),(97,106),(98,105),(99,104),(100,103),(101,108),(102,107),(109,116),(110,115),(111,120),(112,119),(113,118),(114,117)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | ··· | 10T | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 40A | ··· | 40H | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | D6 | C3⋊D4 | C3⋊D4 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | S3×C10 | C5×C3⋊D4 | C5×C3⋊D4 | C8⋊C22 | D4⋊D6 | C5×C8⋊C22 | C5×D4⋊D6 |
| kernel | C5×D4⋊D6 | C5×C4.Dic3 | C5×D4⋊S3 | C5×Q8⋊2S3 | C10×D12 | C15×C4○D4 | D4⋊D6 | C4.Dic3 | D4⋊S3 | Q8⋊2S3 | C2×D12 | C3×C4○D4 | C5×C4○D4 | C60 | C2×C30 | C2×C20 | C5×D4 | C5×Q8 | C20 | C2×C10 | C4○D4 | C12 | C2×C6 | C2×C4 | D4 | Q8 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 8 | 8 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C5×D4⋊D6 ►in GL4(𝔽241) generated by
| 87 | 0 | 0 | 0 |
| 0 | 87 | 0 | 0 |
| 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 87 |
| 99 | 198 | 0 | 0 |
| 43 | 142 | 0 | 0 |
| 0 | 0 | 142 | 43 |
| 0 | 0 | 198 | 99 |
| 0 | 0 | 142 | 43 |
| 0 | 0 | 198 | 99 |
| 99 | 198 | 0 | 0 |
| 43 | 142 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 1 |
| 240 | 240 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 142 | 99 |
| 0 | 0 | 198 | 99 |
G:=sub<GL(4,GF(241))| [87,0,0,0,0,87,0,0,0,0,87,0,0,0,0,87],[99,43,0,0,198,142,0,0,0,0,142,198,0,0,43,99],[0,0,99,43,0,0,198,142,142,198,0,0,43,99,0,0],[0,240,0,0,1,240,0,0,0,0,0,1,0,0,240,1],[240,0,0,0,240,1,0,0,0,0,142,198,0,0,99,99] >;
C5×D4⋊D6 in GAP, Magma, Sage, TeX
C_5\times D_4\rtimes D_6
% in TeX
G:=Group("C5xD4:D6"); // GroupNames label
G:=SmallGroup(480,828);
// by ID
G=gap.SmallGroup(480,828);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,926,891,4204,1068,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations