direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D12⋊C4, D12⋊4C20, Dic6⋊4C20, C60.239D4, C15⋊20C4≀C2, C4.3(S3×C20), (C5×D12)⋊16C4, C20.78(C4×S3), C12.6(C2×C20), (C2×C30).88D4, C12.54(C5×D4), C60.176(C2×C4), C4○D12.2C10, (C5×Dic6)⋊16C4, (C4×Dic3)⋊1C10, (C2×C20).349D6, (C2×C10).25D12, M4(2)⋊4(C5×S3), (C5×M4(2))⋊8S3, C22.3(C5×D12), C10.57(D6⋊C4), (Dic3×C20)⋊13C2, (C3×M4(2))⋊8C10, C20.122(C3⋊D4), C30.99(C22⋊C4), (C15×M4(2))⋊18C2, (C2×C60).344C22, C3⋊2(C5×C4≀C2), (C2×C6).1(C5×D4), C2.11(C5×D6⋊C4), C4.29(C5×C3⋊D4), (C2×C4).37(S3×C10), (C5×C4○D12).8C2, C6.10(C5×C22⋊C4), (C2×C12).14(C2×C10), SmallGroup(480,144)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D12⋊C4
G = < a,b,c,d | a5=b12=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b7c >
Subgroups: 228 in 88 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C10, C10, Dic3, C12, D6, C2×C6, C15, C42, M4(2), C4○D4, C20, C20, C2×C10, C2×C10, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C5×S3, C30, C30, C4≀C2, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C4×Dic3, C3×M4(2), C4○D12, C5×Dic3, C60, S3×C10, C2×C30, C4×C20, C5×M4(2), C5×C4○D4, D12⋊C4, C120, C5×Dic6, S3×C20, C5×D12, C10×Dic3, C5×C3⋊D4, C2×C60, C5×C4≀C2, Dic3×C20, C15×M4(2), C5×C4○D12, C5×D12⋊C4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C10, D6, C22⋊C4, C20, C2×C10, C4×S3, D12, C3⋊D4, C5×S3, C4≀C2, C2×C20, C5×D4, D6⋊C4, S3×C10, C5×C22⋊C4, D12⋊C4, S3×C20, C5×D12, C5×C3⋊D4, C5×C4≀C2, C5×D6⋊C4, C5×D12⋊C4
►On 120 points
Generators in S120
(1 51 47 25 20)(2 52 48 26 21)(3 53 37 27 22)(4 54 38 28 23)(5 55 39 29 24)(6 56 40 30 13)(7 57 41 31 14)(8 58 42 32 15)(9 59 43 33 16)(10 60 44 34 17)(11 49 45 35 18)(12 50 46 36 19)(61 110 103 96 81)(62 111 104 85 82)(63 112 105 86 83)(64 113 106 87 84)(65 114 107 88 73)(66 115 108 89 74)(67 116 97 90 75)(68 117 98 91 76)(69 118 99 92 77)(70 119 100 93 78)(71 120 101 94 79)(72 109 102 95 80)
(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 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 72)(10 71)(11 70)(12 69)(13 83)(14 82)(15 81)(16 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 74)(23 73)(24 84)(25 91)(26 90)(27 89)(28 88)(29 87)(30 86)(31 85)(32 96)(33 95)(34 94)(35 93)(36 92)(37 108)(38 107)(39 106)(40 105)(41 104)(42 103)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 119)(50 118)(51 117)(52 116)(53 115)(54 114)(55 113)(56 112)(57 111)(58 110)(59 109)(60 120)
(1 4 7 10)(2 9 8 3)(5 12 11 6)(13 24 19 18)(14 17 20 23)(15 22 21 16)(25 28 31 34)(26 33 32 27)(29 36 35 30)(37 48 43 42)(38 41 44 47)(39 46 45 40)(49 56 55 50)(51 54 57 60)(52 59 58 53)(61 65)(62 70)(64 68)(67 71)(73 81)(75 79)(76 84)(78 82)(85 93)(87 91)(88 96)(90 94)(97 101)(98 106)(100 104)(103 107)(110 114)(111 119)(113 117)(116 120)
G:=sub<Sym(120)| (1,51,47,25,20)(2,52,48,26,21)(3,53,37,27,22)(4,54,38,28,23)(5,55,39,29,24)(6,56,40,30,13)(7,57,41,31,14)(8,58,42,32,15)(9,59,43,33,16)(10,60,44,34,17)(11,49,45,35,18)(12,50,46,36,19)(61,110,103,96,81)(62,111,104,85,82)(63,112,105,86,83)(64,113,106,87,84)(65,114,107,88,73)(66,115,108,89,74)(67,116,97,90,75)(68,117,98,91,76)(69,118,99,92,77)(70,119,100,93,78)(71,120,101,94,79)(72,109,102,95,80), (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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,72)(10,71)(11,70)(12,69)(13,83)(14,82)(15,81)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,74)(23,73)(24,84)(25,91)(26,90)(27,89)(28,88)(29,87)(30,86)(31,85)(32,96)(33,95)(34,94)(35,93)(36,92)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,119)(50,118)(51,117)(52,116)(53,115)(54,114)(55,113)(56,112)(57,111)(58,110)(59,109)(60,120), (1,4,7,10)(2,9,8,3)(5,12,11,6)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,28,31,34)(26,33,32,27)(29,36,35,30)(37,48,43,42)(38,41,44,47)(39,46,45,40)(49,56,55,50)(51,54,57,60)(52,59,58,53)(61,65)(62,70)(64,68)(67,71)(73,81)(75,79)(76,84)(78,82)(85,93)(87,91)(88,96)(90,94)(97,101)(98,106)(100,104)(103,107)(110,114)(111,119)(113,117)(116,120)>;
G:=Group( (1,51,47,25,20)(2,52,48,26,21)(3,53,37,27,22)(4,54,38,28,23)(5,55,39,29,24)(6,56,40,30,13)(7,57,41,31,14)(8,58,42,32,15)(9,59,43,33,16)(10,60,44,34,17)(11,49,45,35,18)(12,50,46,36,19)(61,110,103,96,81)(62,111,104,85,82)(63,112,105,86,83)(64,113,106,87,84)(65,114,107,88,73)(66,115,108,89,74)(67,116,97,90,75)(68,117,98,91,76)(69,118,99,92,77)(70,119,100,93,78)(71,120,101,94,79)(72,109,102,95,80), (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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,72)(10,71)(11,70)(12,69)(13,83)(14,82)(15,81)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,74)(23,73)(24,84)(25,91)(26,90)(27,89)(28,88)(29,87)(30,86)(31,85)(32,96)(33,95)(34,94)(35,93)(36,92)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,119)(50,118)(51,117)(52,116)(53,115)(54,114)(55,113)(56,112)(57,111)(58,110)(59,109)(60,120), (1,4,7,10)(2,9,8,3)(5,12,11,6)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,28,31,34)(26,33,32,27)(29,36,35,30)(37,48,43,42)(38,41,44,47)(39,46,45,40)(49,56,55,50)(51,54,57,60)(52,59,58,53)(61,65)(62,70)(64,68)(67,71)(73,81)(75,79)(76,84)(78,82)(85,93)(87,91)(88,96)(90,94)(97,101)(98,106)(100,104)(103,107)(110,114)(111,119)(113,117)(116,120) );
G=PermutationGroup([[(1,51,47,25,20),(2,52,48,26,21),(3,53,37,27,22),(4,54,38,28,23),(5,55,39,29,24),(6,56,40,30,13),(7,57,41,31,14),(8,58,42,32,15),(9,59,43,33,16),(10,60,44,34,17),(11,49,45,35,18),(12,50,46,36,19),(61,110,103,96,81),(62,111,104,85,82),(63,112,105,86,83),(64,113,106,87,84),(65,114,107,88,73),(66,115,108,89,74),(67,116,97,90,75),(68,117,98,91,76),(69,118,99,92,77),(70,119,100,93,78),(71,120,101,94,79),(72,109,102,95,80)], [(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)], [(1,68),(2,67),(3,66),(4,65),(5,64),(6,63),(7,62),(8,61),(9,72),(10,71),(11,70),(12,69),(13,83),(14,82),(15,81),(16,80),(17,79),(18,78),(19,77),(20,76),(21,75),(22,74),(23,73),(24,84),(25,91),(26,90),(27,89),(28,88),(29,87),(30,86),(31,85),(32,96),(33,95),(34,94),(35,93),(36,92),(37,108),(38,107),(39,106),(40,105),(41,104),(42,103),(43,102),(44,101),(45,100),(46,99),(47,98),(48,97),(49,119),(50,118),(51,117),(52,116),(53,115),(54,114),(55,113),(56,112),(57,111),(58,110),(59,109),(60,120)], [(1,4,7,10),(2,9,8,3),(5,12,11,6),(13,24,19,18),(14,17,20,23),(15,22,21,16),(25,28,31,34),(26,33,32,27),(29,36,35,30),(37,48,43,42),(38,41,44,47),(39,46,45,40),(49,56,55,50),(51,54,57,60),(52,59,58,53),(61,65),(62,70),(64,68),(67,71),(73,81),(75,79),(76,84),(78,82),(85,93),(87,91),(88,96),(90,94),(97,101),(98,106),(100,104),(103,107),(110,114),(111,119),(113,117),(116,120)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20AB | 20AC | 20AD | 20AE | 20AF | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 12 | 2 | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 12 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
120 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 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | S3 | D4 | D4 | D6 | C4×S3 | C3⋊D4 | D12 | C5×S3 | C4≀C2 | C5×D4 | C5×D4 | S3×C10 | S3×C20 | C5×C3⋊D4 | C5×D12 | C5×C4≀C2 | D12⋊C4 | C5×D12⋊C4 |
| kernel | C5×D12⋊C4 | Dic3×C20 | C15×M4(2) | C5×C4○D12 | C5×Dic6 | C5×D12 | D12⋊C4 | C4×Dic3 | C3×M4(2) | C4○D12 | Dic6 | D12 | C5×M4(2) | C60 | C2×C30 | C2×C20 | C20 | C20 | C2×C10 | M4(2) | C15 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C3 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 2 | 8 |
Matrix representation of C5×D12⋊C4 ►in GL4(𝔽241) generated by
| 91 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 |
| 0 | 0 | 205 | 0 |
| 0 | 0 | 0 | 205 |
| 0 | 240 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 177 |
| 43 | 142 | 0 | 0 |
| 99 | 198 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 240 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,205,0,0,0,0,205],[0,1,0,0,240,1,0,0,0,0,64,0,0,0,0,177],[43,99,0,0,142,198,0,0,0,0,0,1,0,0,1,0],[240,1,0,0,0,1,0,0,0,0,177,0,0,0,0,1] >;
C5×D12⋊C4 in GAP, Magma, Sage, TeX
C_5\times D_{12}\rtimes C_4 % in TeX
G:=Group("C5xD12:C4"); // GroupNames label
G:=SmallGroup(480,144);
// by ID
G=gap.SmallGroup(480,144);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,589,148,136,4204,2111,102,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^12=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^7*c>;
// generators/relations