metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.7D20, C30.6D4, C15⋊3SD16, D20.1S3, C12.5D10, C20.23D6, Dic30⋊5C2, C60.9C22, C3⋊C8⋊2D5, C4.2(S3×D5), C3⋊3(C40⋊C2), C5⋊1(D4.S3), (C3×D20).1C2, C10.2(C3⋊D4), C2.5(C3⋊D20), (C5×C3⋊C8)⋊2C2, SmallGroup(240,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.D20
G = < a,b,c | a6=1, b20=c2=a3, bab-1=cac-1=a-1, cbc-1=b19 >
Smallest permutation representation of C6.D20
►On 120 points
Generators in S120
(1 71 119 21 51 99)(2 100 52 22 120 72)(3 73 81 23 53 101)(4 102 54 24 82 74)(5 75 83 25 55 103)(6 104 56 26 84 76)(7 77 85 27 57 105)(8 106 58 28 86 78)(9 79 87 29 59 107)(10 108 60 30 88 80)(11 41 89 31 61 109)(12 110 62 32 90 42)(13 43 91 33 63 111)(14 112 64 34 92 44)(15 45 93 35 65 113)(16 114 66 36 94 46)(17 47 95 37 67 115)(18 116 68 38 96 48)(19 49 97 39 69 117)(20 118 70 40 98 50)
(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 20 21 40)(2 39 22 19)(3 18 23 38)(4 37 24 17)(5 16 25 36)(6 35 26 15)(7 14 27 34)(8 33 28 13)(9 12 29 32)(10 31 30 11)(41 80 61 60)(42 59 62 79)(43 78 63 58)(44 57 64 77)(45 76 65 56)(46 55 66 75)(47 74 67 54)(48 53 68 73)(49 72 69 52)(50 51 70 71)(81 96 101 116)(82 115 102 95)(83 94 103 114)(84 113 104 93)(85 92 105 112)(86 111 106 91)(87 90 107 110)(88 109 108 89)(97 120 117 100)(98 99 118 119)
G:=sub<Sym(120)| (1,71,119,21,51,99)(2,100,52,22,120,72)(3,73,81,23,53,101)(4,102,54,24,82,74)(5,75,83,25,55,103)(6,104,56,26,84,76)(7,77,85,27,57,105)(8,106,58,28,86,78)(9,79,87,29,59,107)(10,108,60,30,88,80)(11,41,89,31,61,109)(12,110,62,32,90,42)(13,43,91,33,63,111)(14,112,64,34,92,44)(15,45,93,35,65,113)(16,114,66,36,94,46)(17,47,95,37,67,115)(18,116,68,38,96,48)(19,49,97,39,69,117)(20,118,70,40,98,50), (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), (1,20,21,40)(2,39,22,19)(3,18,23,38)(4,37,24,17)(5,16,25,36)(6,35,26,15)(7,14,27,34)(8,33,28,13)(9,12,29,32)(10,31,30,11)(41,80,61,60)(42,59,62,79)(43,78,63,58)(44,57,64,77)(45,76,65,56)(46,55,66,75)(47,74,67,54)(48,53,68,73)(49,72,69,52)(50,51,70,71)(81,96,101,116)(82,115,102,95)(83,94,103,114)(84,113,104,93)(85,92,105,112)(86,111,106,91)(87,90,107,110)(88,109,108,89)(97,120,117,100)(98,99,118,119)>;
G:=Group( (1,71,119,21,51,99)(2,100,52,22,120,72)(3,73,81,23,53,101)(4,102,54,24,82,74)(5,75,83,25,55,103)(6,104,56,26,84,76)(7,77,85,27,57,105)(8,106,58,28,86,78)(9,79,87,29,59,107)(10,108,60,30,88,80)(11,41,89,31,61,109)(12,110,62,32,90,42)(13,43,91,33,63,111)(14,112,64,34,92,44)(15,45,93,35,65,113)(16,114,66,36,94,46)(17,47,95,37,67,115)(18,116,68,38,96,48)(19,49,97,39,69,117)(20,118,70,40,98,50), (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), (1,20,21,40)(2,39,22,19)(3,18,23,38)(4,37,24,17)(5,16,25,36)(6,35,26,15)(7,14,27,34)(8,33,28,13)(9,12,29,32)(10,31,30,11)(41,80,61,60)(42,59,62,79)(43,78,63,58)(44,57,64,77)(45,76,65,56)(46,55,66,75)(47,74,67,54)(48,53,68,73)(49,72,69,52)(50,51,70,71)(81,96,101,116)(82,115,102,95)(83,94,103,114)(84,113,104,93)(85,92,105,112)(86,111,106,91)(87,90,107,110)(88,109,108,89)(97,120,117,100)(98,99,118,119) );
G=PermutationGroup([[(1,71,119,21,51,99),(2,100,52,22,120,72),(3,73,81,23,53,101),(4,102,54,24,82,74),(5,75,83,25,55,103),(6,104,56,26,84,76),(7,77,85,27,57,105),(8,106,58,28,86,78),(9,79,87,29,59,107),(10,108,60,30,88,80),(11,41,89,31,61,109),(12,110,62,32,90,42),(13,43,91,33,63,111),(14,112,64,34,92,44),(15,45,93,35,65,113),(16,114,66,36,94,46),(17,47,95,37,67,115),(18,116,68,38,96,48),(19,49,97,39,69,117),(20,118,70,40,98,50)], [(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)], [(1,20,21,40),(2,39,22,19),(3,18,23,38),(4,37,24,17),(5,16,25,36),(6,35,26,15),(7,14,27,34),(8,33,28,13),(9,12,29,32),(10,31,30,11),(41,80,61,60),(42,59,62,79),(43,78,63,58),(44,57,64,77),(45,76,65,56),(46,55,66,75),(47,74,67,54),(48,53,68,73),(49,72,69,52),(50,51,70,71),(81,96,101,116),(82,115,102,95),(83,94,103,114),(84,113,104,93),(85,92,105,112),(86,111,106,91),(87,90,107,110),(88,109,108,89),(97,120,117,100),(98,99,118,119)]])
C6.D20 is a maximal subgroup of
S3×C40⋊C2 D40⋊S3 D40⋊7S3 C40.2D6 D20.31D6 D60⋊30C22 C60.63D4 D30.8D4 D5×D4.S3 D12⋊10D10 D20.10D6 D15⋊SD16 D20.13D6 D20.14D6 D20.17D6
C6.D20 is a maximal quotient of
C6.D40 Dic30⋊12C4 C60.Q8
36 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 12 | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 40A | ··· | 40H | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 40 | ··· | 40 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 20 | 2 | 2 | 60 | 2 | 2 | 2 | 20 | 20 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | SD16 | D10 | C3⋊D4 | D20 | C40⋊C2 | D4.S3 | S3×D5 | C3⋊D20 | C6.D20 |
| kernel | C6.D20 | C5×C3⋊C8 | C3×D20 | Dic30 | D20 | C30 | C3⋊C8 | C20 | C15 | C12 | C10 | C6 | C3 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C6.D20 ►in GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 239 | 49 |
| 0 | 0 | 177 | 1 |
| 94 | 106 | 0 | 0 |
| 166 | 200 | 0 | 0 |
| 0 | 0 | 145 | 41 |
| 0 | 0 | 28 | 96 |
| 109 | 226 | 0 | 0 |
| 37 | 132 | 0 | 0 |
| 0 | 0 | 145 | 41 |
| 0 | 0 | 28 | 96 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,239,177,0,0,49,1],[94,166,0,0,106,200,0,0,0,0,145,28,0,0,41,96],[109,37,0,0,226,132,0,0,0,0,145,28,0,0,41,96] >;
C6.D20 in GAP, Magma, Sage, TeX
C_6.D_{20} % in TeX
G:=Group("C6.D20"); // GroupNames label
G:=SmallGroup(240,18);
// by ID
G=gap.SmallGroup(240,18);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,73,31,218,50,490,6917]);
// Polycyclic
G:=Group<a,b,c|a^6=1,b^20=c^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^19>;
// generators/relations
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