direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D6⋊C4, D6⋊C20, C30.46D4, C10.17D12, (C2×C20)⋊1S3, (C2×C60)⋊2C2, C6.6(C5×D4), (S3×C10)⋊5C4, (C2×C12)⋊1C10, C2.5(S3×C20), C6.4(C2×C20), C2.2(C5×D12), C10.26(C4×S3), C15⋊9(C22⋊C4), C30.49(C2×C4), (C2×C10).34D6, (C22×S3).C10, (C10×Dic3)⋊7C2, (C2×Dic3)⋊1C10, C22.6(S3×C10), C10.22(C3⋊D4), (C2×C30).45C22, (C2×C4)⋊1(C5×S3), C3⋊1(C5×C22⋊C4), (S3×C2×C10).3C2, C2.2(C5×C3⋊D4), (C2×C6).6(C2×C10), SmallGroup(240,59)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D6⋊C4
G = < a,b,c,d | a5=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >
Subgroups: 152 in 68 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C22⋊C4, C20, C2×C10, C2×C10, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C2×C20, C2×C20, C22×C10, D6⋊C4, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C5×C22⋊C4, C10×Dic3, C2×C60, S3×C2×C10, C5×D6⋊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, C2×C20, C5×D4, D6⋊C4, S3×C10, C5×C22⋊C4, S3×C20, C5×D12, C5×C3⋊D4, C5×D6⋊C4
►On 120 points
Generators in S120
(1 53 41 29 17)(2 54 42 30 18)(3 49 37 25 13)(4 50 38 26 14)(5 51 39 27 15)(6 52 40 28 16)(7 55 43 31 19)(8 56 44 32 20)(9 57 45 33 21)(10 58 46 34 22)(11 59 47 35 23)(12 60 48 36 24)(61 109 97 85 73)(62 110 98 86 74)(63 111 99 87 75)(64 112 100 88 76)(65 113 101 89 77)(66 114 102 90 78)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)
(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 6)(2 5)(3 4)(7 8)(9 12)(10 11)(13 14)(15 18)(16 17)(19 20)(21 24)(22 23)(25 26)(27 30)(28 29)(31 32)(33 36)(34 35)(37 38)(39 42)(40 41)(43 44)(45 48)(46 47)(49 50)(51 54)(52 53)(55 56)(57 60)(58 59)(61 65)(62 64)(67 71)(68 70)(73 77)(74 76)(79 83)(80 82)(85 89)(86 88)(91 95)(92 94)(97 101)(98 100)(103 107)(104 106)(109 113)(110 112)(115 119)(116 118)
(1 71 11 65)(2 72 12 66)(3 67 7 61)(4 68 8 62)(5 69 9 63)(6 70 10 64)(13 79 19 73)(14 80 20 74)(15 81 21 75)(16 82 22 76)(17 83 23 77)(18 84 24 78)(25 91 31 85)(26 92 32 86)(27 93 33 87)(28 94 34 88)(29 95 35 89)(30 96 36 90)(37 103 43 97)(38 104 44 98)(39 105 45 99)(40 106 46 100)(41 107 47 101)(42 108 48 102)(49 115 55 109)(50 116 56 110)(51 117 57 111)(52 118 58 112)(53 119 59 113)(54 120 60 114)
G:=sub<Sym(120)| (1,53,41,29,17)(2,54,42,30,18)(3,49,37,25,13)(4,50,38,26,14)(5,51,39,27,15)(6,52,40,28,16)(7,55,43,31,19)(8,56,44,32,20)(9,57,45,33,21)(10,58,46,34,22)(11,59,47,35,23)(12,60,48,36,24)(61,109,97,85,73)(62,110,98,86,74)(63,111,99,87,75)(64,112,100,88,76)(65,113,101,89,77)(66,114,102,90,78)(67,115,103,91,79)(68,116,104,92,80)(69,117,105,93,81)(70,118,106,94,82)(71,119,107,95,83)(72,120,108,96,84), (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), (1,6)(2,5)(3,4)(7,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32)(33,36)(34,35)(37,38)(39,42)(40,41)(43,44)(45,48)(46,47)(49,50)(51,54)(52,53)(55,56)(57,60)(58,59)(61,65)(62,64)(67,71)(68,70)(73,77)(74,76)(79,83)(80,82)(85,89)(86,88)(91,95)(92,94)(97,101)(98,100)(103,107)(104,106)(109,113)(110,112)(115,119)(116,118), (1,71,11,65)(2,72,12,66)(3,67,7,61)(4,68,8,62)(5,69,9,63)(6,70,10,64)(13,79,19,73)(14,80,20,74)(15,81,21,75)(16,82,22,76)(17,83,23,77)(18,84,24,78)(25,91,31,85)(26,92,32,86)(27,93,33,87)(28,94,34,88)(29,95,35,89)(30,96,36,90)(37,103,43,97)(38,104,44,98)(39,105,45,99)(40,106,46,100)(41,107,47,101)(42,108,48,102)(49,115,55,109)(50,116,56,110)(51,117,57,111)(52,118,58,112)(53,119,59,113)(54,120,60,114)>;
G:=Group( (1,53,41,29,17)(2,54,42,30,18)(3,49,37,25,13)(4,50,38,26,14)(5,51,39,27,15)(6,52,40,28,16)(7,55,43,31,19)(8,56,44,32,20)(9,57,45,33,21)(10,58,46,34,22)(11,59,47,35,23)(12,60,48,36,24)(61,109,97,85,73)(62,110,98,86,74)(63,111,99,87,75)(64,112,100,88,76)(65,113,101,89,77)(66,114,102,90,78)(67,115,103,91,79)(68,116,104,92,80)(69,117,105,93,81)(70,118,106,94,82)(71,119,107,95,83)(72,120,108,96,84), (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), (1,6)(2,5)(3,4)(7,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32)(33,36)(34,35)(37,38)(39,42)(40,41)(43,44)(45,48)(46,47)(49,50)(51,54)(52,53)(55,56)(57,60)(58,59)(61,65)(62,64)(67,71)(68,70)(73,77)(74,76)(79,83)(80,82)(85,89)(86,88)(91,95)(92,94)(97,101)(98,100)(103,107)(104,106)(109,113)(110,112)(115,119)(116,118), (1,71,11,65)(2,72,12,66)(3,67,7,61)(4,68,8,62)(5,69,9,63)(6,70,10,64)(13,79,19,73)(14,80,20,74)(15,81,21,75)(16,82,22,76)(17,83,23,77)(18,84,24,78)(25,91,31,85)(26,92,32,86)(27,93,33,87)(28,94,34,88)(29,95,35,89)(30,96,36,90)(37,103,43,97)(38,104,44,98)(39,105,45,99)(40,106,46,100)(41,107,47,101)(42,108,48,102)(49,115,55,109)(50,116,56,110)(51,117,57,111)(52,118,58,112)(53,119,59,113)(54,120,60,114) );
G=PermutationGroup([[(1,53,41,29,17),(2,54,42,30,18),(3,49,37,25,13),(4,50,38,26,14),(5,51,39,27,15),(6,52,40,28,16),(7,55,43,31,19),(8,56,44,32,20),(9,57,45,33,21),(10,58,46,34,22),(11,59,47,35,23),(12,60,48,36,24),(61,109,97,85,73),(62,110,98,86,74),(63,111,99,87,75),(64,112,100,88,76),(65,113,101,89,77),(66,114,102,90,78),(67,115,103,91,79),(68,116,104,92,80),(69,117,105,93,81),(70,118,106,94,82),(71,119,107,95,83),(72,120,108,96,84)], [(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)], [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14),(15,18),(16,17),(19,20),(21,24),(22,23),(25,26),(27,30),(28,29),(31,32),(33,36),(34,35),(37,38),(39,42),(40,41),(43,44),(45,48),(46,47),(49,50),(51,54),(52,53),(55,56),(57,60),(58,59),(61,65),(62,64),(67,71),(68,70),(73,77),(74,76),(79,83),(80,82),(85,89),(86,88),(91,95),(92,94),(97,101),(98,100),(103,107),(104,106),(109,113),(110,112),(115,119),(116,118)], [(1,71,11,65),(2,72,12,66),(3,67,7,61),(4,68,8,62),(5,69,9,63),(6,70,10,64),(13,79,19,73),(14,80,20,74),(15,81,21,75),(16,82,22,76),(17,83,23,77),(18,84,24,78),(25,91,31,85),(26,92,32,86),(27,93,33,87),(28,94,34,88),(29,95,35,89),(30,96,36,90),(37,103,43,97),(38,104,44,98),(39,105,45,99),(40,106,46,100),(41,107,47,101),(42,108,48,102),(49,115,55,109),(50,116,56,110),(51,117,57,111),(52,118,58,112),(53,119,59,113),(54,120,60,114)]])
C5×D6⋊C4 is a maximal subgroup of
(C2×C20).D6 D6⋊C4.D5 C60⋊5C4⋊C2 Dic5.8D12 D30.35D4 D6⋊Dic5.C2 C5⋊(C42⋊3S3) D6.(C4×D5) (S3×Dic5)⋊C4 Dic5⋊4D12 Dic15⋊14D4 Dic15⋊D4 D6⋊1Dic10 D10.17D12 D6⋊2Dic10 D30⋊D4 D6.D20 D6⋊3Dic10 D6⋊4Dic10 D30.7D4 D6⋊(C4×D5) Dic15⋊9D4 D6⋊C4⋊D5 D10⋊D12 D6.9D20 D30⋊12D4 Dic15.10D4 Dic15.31D4 D30.27D4 D6⋊4D20 D30⋊5D4 C20×D12 C5×S3×C22⋊C4 C20×C3⋊D4
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 10M | ··· | 10T | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | D4 | D6 | C4×S3 | D12 | C3⋊D4 | C5×S3 | C5×D4 | S3×C10 | S3×C20 | C5×D12 | C5×C3⋊D4 |
| kernel | C5×D6⋊C4 | C10×Dic3 | C2×C60 | S3×C2×C10 | S3×C10 | D6⋊C4 | C2×Dic3 | C2×C12 | C22×S3 | D6 | C2×C20 | C30 | C2×C10 | C10 | C10 | C10 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 8 | 4 | 8 | 8 | 8 |
Matrix representation of C5×D6⋊C4 ►in GL4(𝔽61) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 1 |
| 60 | 0 | 0 | 0 |
| 23 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 60 | 0 |
| 53 | 55 | 0 | 0 |
| 21 | 8 | 0 | 0 |
| 0 | 0 | 52 | 43 |
| 0 | 0 | 18 | 9 |
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[60,0,0,0,0,60,0,0,0,0,0,1,0,0,60,1],[60,23,0,0,0,1,0,0,0,0,0,60,0,0,60,0],[53,21,0,0,55,8,0,0,0,0,52,18,0,0,43,9] >;
C5×D6⋊C4 in GAP, Magma, Sage, TeX
C_5\times D_6\rtimes C_4
% in TeX
G:=Group("C5xD6:C4"); // GroupNames label
G:=SmallGroup(240,59);
// by ID
G=gap.SmallGroup(240,59);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-3,505,127,5765]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^6=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations