direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×D12, C20⋊9D6, C30⋊6D4, C60⋊11C22, C30.50C23, C6⋊1(C5×D4), C3⋊1(D4×C10), C4⋊2(S3×C10), (C2×C20)⋊6S3, C15⋊12(C2×D4), (C2×C60)⋊10C2, C12⋊2(C2×C10), (C2×C12)⋊3C10, D6⋊1(C2×C10), (C2×C10).38D6, (C22×S3)⋊1C10, (S3×C10)⋊9C22, C6.3(C22×C10), C10.40(C22×S3), C22.10(S3×C10), (C2×C30).49C22, (S3×C2×C10)⋊5C2, (C2×C4)⋊2(C5×S3), C2.4(S3×C2×C10), (C2×C6).10(C2×C10), SmallGroup(240,167)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×D12
G = < a,b,c | a10=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 248 in 108 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C10, C10, C10, C12, D6, D6, C2×C6, C15, C2×D4, C20, C2×C10, C2×C10, D12, C2×C12, C22×S3, C5×S3, C30, C30, C2×C20, C5×D4, C22×C10, C2×D12, C60, S3×C10, S3×C10, C2×C30, D4×C10, C5×D12, C2×C60, S3×C2×C10, C10×D12
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, D12, C22×S3, C5×S3, C5×D4, C22×C10, C2×D12, S3×C10, D4×C10, C5×D12, S3×C2×C10, C10×D12
►On 120 points
Generators in S120
(1 71 30 22 96 74 108 39 50 113)(2 72 31 23 85 75 97 40 51 114)(3 61 32 24 86 76 98 41 52 115)(4 62 33 13 87 77 99 42 53 116)(5 63 34 14 88 78 100 43 54 117)(6 64 35 15 89 79 101 44 55 118)(7 65 36 16 90 80 102 45 56 119)(8 66 25 17 91 81 103 46 57 120)(9 67 26 18 92 82 104 47 58 109)(10 68 27 19 93 83 105 48 59 110)(11 69 28 20 94 84 106 37 60 111)(12 70 29 21 95 73 107 38 49 112)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 34)(26 33)(27 32)(28 31)(29 30)(35 36)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 50)(51 60)(52 59)(53 58)(54 57)(55 56)(61 68)(62 67)(63 66)(64 65)(69 72)(70 71)(73 74)(75 84)(76 83)(77 82)(78 81)(79 80)(85 94)(86 93)(87 92)(88 91)(89 90)(95 96)(97 106)(98 105)(99 104)(100 103)(101 102)(107 108)(109 116)(110 115)(111 114)(112 113)(117 120)(118 119)
G:=sub<Sym(120)| (1,71,30,22,96,74,108,39,50,113)(2,72,31,23,85,75,97,40,51,114)(3,61,32,24,86,76,98,41,52,115)(4,62,33,13,87,77,99,42,53,116)(5,63,34,14,88,78,100,43,54,117)(6,64,35,15,89,79,101,44,55,118)(7,65,36,16,90,80,102,45,56,119)(8,66,25,17,91,81,103,46,57,120)(9,67,26,18,92,82,104,47,58,109)(10,68,27,19,93,83,105,48,59,110)(11,69,28,20,94,84,106,37,60,111)(12,70,29,21,95,73,107,38,49,112), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,34)(26,33)(27,32)(28,31)(29,30)(35,36)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,50)(51,60)(52,59)(53,58)(54,57)(55,56)(61,68)(62,67)(63,66)(64,65)(69,72)(70,71)(73,74)(75,84)(76,83)(77,82)(78,81)(79,80)(85,94)(86,93)(87,92)(88,91)(89,90)(95,96)(97,106)(98,105)(99,104)(100,103)(101,102)(107,108)(109,116)(110,115)(111,114)(112,113)(117,120)(118,119)>;
G:=Group( (1,71,30,22,96,74,108,39,50,113)(2,72,31,23,85,75,97,40,51,114)(3,61,32,24,86,76,98,41,52,115)(4,62,33,13,87,77,99,42,53,116)(5,63,34,14,88,78,100,43,54,117)(6,64,35,15,89,79,101,44,55,118)(7,65,36,16,90,80,102,45,56,119)(8,66,25,17,91,81,103,46,57,120)(9,67,26,18,92,82,104,47,58,109)(10,68,27,19,93,83,105,48,59,110)(11,69,28,20,94,84,106,37,60,111)(12,70,29,21,95,73,107,38,49,112), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(25,34)(26,33)(27,32)(28,31)(29,30)(35,36)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,50)(51,60)(52,59)(53,58)(54,57)(55,56)(61,68)(62,67)(63,66)(64,65)(69,72)(70,71)(73,74)(75,84)(76,83)(77,82)(78,81)(79,80)(85,94)(86,93)(87,92)(88,91)(89,90)(95,96)(97,106)(98,105)(99,104)(100,103)(101,102)(107,108)(109,116)(110,115)(111,114)(112,113)(117,120)(118,119) );
G=PermutationGroup([[(1,71,30,22,96,74,108,39,50,113),(2,72,31,23,85,75,97,40,51,114),(3,61,32,24,86,76,98,41,52,115),(4,62,33,13,87,77,99,42,53,116),(5,63,34,14,88,78,100,43,54,117),(6,64,35,15,89,79,101,44,55,118),(7,65,36,16,90,80,102,45,56,119),(8,66,25,17,91,81,103,46,57,120),(9,67,26,18,92,82,104,47,58,109),(10,68,27,19,93,83,105,48,59,110),(11,69,28,20,94,84,106,37,60,111),(12,70,29,21,95,73,107,38,49,112)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22),(25,34),(26,33),(27,32),(28,31),(29,30),(35,36),(37,40),(38,39),(41,48),(42,47),(43,46),(44,45),(49,50),(51,60),(52,59),(53,58),(54,57),(55,56),(61,68),(62,67),(63,66),(64,65),(69,72),(70,71),(73,74),(75,84),(76,83),(77,82),(78,81),(79,80),(85,94),(86,93),(87,92),(88,91),(89,90),(95,96),(97,106),(98,105),(99,104),(100,103),(101,102),(107,108),(109,116),(110,115),(111,114),(112,113),(117,120),(118,119)]])
C10×D12 is a maximal subgroup of
C20.5D12 D12⋊Dic5 C10.D24 D20⋊21D6 D60⋊36C22 C60.89D4 C60.69D4 Dic5⋊D12 (C2×D12).D5 Dic15⋊8D4 C60⋊D4 C20⋊D12 Dic15⋊2D4 C60⋊10D4 C20⋊2D12 D30⋊4D4 D20⋊26D6 S3×D4×C10
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 10M | ··· | 10AB | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D4 | D6 | D6 | D12 | C5×S3 | C5×D4 | S3×C10 | S3×C10 | C5×D12 |
| kernel | C10×D12 | C5×D12 | C2×C60 | S3×C2×C10 | C2×D12 | D12 | C2×C12 | C22×S3 | C2×C20 | C30 | C20 | C2×C10 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 4 | 16 | 4 | 8 | 1 | 2 | 2 | 1 | 4 | 4 | 8 | 8 | 4 | 16 |
Matrix representation of C10×D12 ►in GL4(𝔽61) generated by
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 23 | 38 |
| 0 | 0 | 23 | 46 |
| 1 | 1 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 38 | 23 |
| 0 | 0 | 46 | 23 |
G:=sub<GL(4,GF(61))| [27,0,0,0,0,27,0,0,0,0,9,0,0,0,0,9],[1,60,0,0,1,0,0,0,0,0,23,23,0,0,38,46],[1,0,0,0,1,60,0,0,0,0,38,46,0,0,23,23] >;
C10×D12 in GAP, Magma, Sage, TeX
C_{10}\times D_{12} % in TeX
G:=Group("C10xD12"); // GroupNames label
G:=SmallGroup(240,167);
// by ID
G=gap.SmallGroup(240,167);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-3,794,194,5765]);
// Polycyclic
G:=Group<a,b,c|a^10=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations