metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.32D6, C15⋊5M4(2), C12.32D10, C60.32C22, D10.1Dic3, Dic5.1Dic3, C3⋊C8⋊4D5, C3⋊3(C8⋊D5), (C6×D5).2C4, (C4×D5).2S3, C15⋊3C8⋊11C2, C4.25(S3×D5), C6.11(C4×D5), C30.25(C2×C4), (D5×C12).3C2, C5⋊3(C4.Dic3), C2.3(D5×Dic3), (C3×Dic5).2C4, C10.9(C2×Dic3), (C5×C3⋊C8)⋊6C2, SmallGroup(240,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.32D6
G = < a,b,c | a20=b6=1, c2=a15, bab-1=cac-1=a9, cbc-1=a10b-1 >
Smallest permutation representation of C20.32D6
►On 120 points
Generators in S120
(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 55 107)(2 44 108 10 56 116)(3 53 109 19 57 105)(4 42 110 8 58 114)(5 51 111 17 59 103)(6 60 112)(7 49 113 15 41 101)(9 47 115 13 43 119)(11 45 117)(12 54 118 20 46 106)(14 52 120 18 48 104)(16 50 102)(21 75 98 31 65 88)(22 64 99 40 66 97)(23 73 100 29 67 86)(24 62 81 38 68 95)(25 71 82 27 69 84)(26 80 83 36 70 93)(28 78 85 34 72 91)(30 76 87 32 74 89)(33 63 90 39 77 96)(35 61 92 37 79 94)
(1 70 16 65 11 80 6 75)(2 79 17 74 12 69 7 64)(3 68 18 63 13 78 8 73)(4 77 19 72 14 67 9 62)(5 66 20 61 15 76 10 71)(21 45 36 60 31 55 26 50)(22 54 37 49 32 44 27 59)(23 43 38 58 33 53 28 48)(24 52 39 47 34 42 29 57)(25 41 40 56 35 51 30 46)(81 104 96 119 91 114 86 109)(82 113 97 108 92 103 87 118)(83 102 98 117 93 112 88 107)(84 111 99 106 94 101 89 116)(85 120 100 115 95 110 90 105)
G:=sub<Sym(120)| (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), (1,55,107)(2,44,108,10,56,116)(3,53,109,19,57,105)(4,42,110,8,58,114)(5,51,111,17,59,103)(6,60,112)(7,49,113,15,41,101)(9,47,115,13,43,119)(11,45,117)(12,54,118,20,46,106)(14,52,120,18,48,104)(16,50,102)(21,75,98,31,65,88)(22,64,99,40,66,97)(23,73,100,29,67,86)(24,62,81,38,68,95)(25,71,82,27,69,84)(26,80,83,36,70,93)(28,78,85,34,72,91)(30,76,87,32,74,89)(33,63,90,39,77,96)(35,61,92,37,79,94), (1,70,16,65,11,80,6,75)(2,79,17,74,12,69,7,64)(3,68,18,63,13,78,8,73)(4,77,19,72,14,67,9,62)(5,66,20,61,15,76,10,71)(21,45,36,60,31,55,26,50)(22,54,37,49,32,44,27,59)(23,43,38,58,33,53,28,48)(24,52,39,47,34,42,29,57)(25,41,40,56,35,51,30,46)(81,104,96,119,91,114,86,109)(82,113,97,108,92,103,87,118)(83,102,98,117,93,112,88,107)(84,111,99,106,94,101,89,116)(85,120,100,115,95,110,90,105)>;
G:=Group( (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), (1,55,107)(2,44,108,10,56,116)(3,53,109,19,57,105)(4,42,110,8,58,114)(5,51,111,17,59,103)(6,60,112)(7,49,113,15,41,101)(9,47,115,13,43,119)(11,45,117)(12,54,118,20,46,106)(14,52,120,18,48,104)(16,50,102)(21,75,98,31,65,88)(22,64,99,40,66,97)(23,73,100,29,67,86)(24,62,81,38,68,95)(25,71,82,27,69,84)(26,80,83,36,70,93)(28,78,85,34,72,91)(30,76,87,32,74,89)(33,63,90,39,77,96)(35,61,92,37,79,94), (1,70,16,65,11,80,6,75)(2,79,17,74,12,69,7,64)(3,68,18,63,13,78,8,73)(4,77,19,72,14,67,9,62)(5,66,20,61,15,76,10,71)(21,45,36,60,31,55,26,50)(22,54,37,49,32,44,27,59)(23,43,38,58,33,53,28,48)(24,52,39,47,34,42,29,57)(25,41,40,56,35,51,30,46)(81,104,96,119,91,114,86,109)(82,113,97,108,92,103,87,118)(83,102,98,117,93,112,88,107)(84,111,99,106,94,101,89,116)(85,120,100,115,95,110,90,105) );
G=PermutationGroup([[(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)], [(1,55,107),(2,44,108,10,56,116),(3,53,109,19,57,105),(4,42,110,8,58,114),(5,51,111,17,59,103),(6,60,112),(7,49,113,15,41,101),(9,47,115,13,43,119),(11,45,117),(12,54,118,20,46,106),(14,52,120,18,48,104),(16,50,102),(21,75,98,31,65,88),(22,64,99,40,66,97),(23,73,100,29,67,86),(24,62,81,38,68,95),(25,71,82,27,69,84),(26,80,83,36,70,93),(28,78,85,34,72,91),(30,76,87,32,74,89),(33,63,90,39,77,96),(35,61,92,37,79,94)], [(1,70,16,65,11,80,6,75),(2,79,17,74,12,69,7,64),(3,68,18,63,13,78,8,73),(4,77,19,72,14,67,9,62),(5,66,20,61,15,76,10,71),(21,45,36,60,31,55,26,50),(22,54,37,49,32,44,27,59),(23,43,38,58,33,53,28,48),(24,52,39,47,34,42,29,57),(25,41,40,56,35,51,30,46),(81,104,96,119,91,114,86,109),(82,113,97,108,92,103,87,118),(83,102,98,117,93,112,88,107),(84,111,99,106,94,101,89,116),(85,120,100,115,95,110,90,105)]])
C20.32D6 is a maximal subgroup of
S3×C8⋊D5 C40⋊D6 C40.54D6 C40.34D6 D5×C4.Dic3 D20.3Dic3 D20.2Dic3 Dic10⋊3D6 C60.8C23 D12⋊10D10 D20.9D6 D20⋊D6 D20.13D6 D12.27D10 C60.39C23
C20.32D6 is a maximal quotient of
C30.21C42 C60.93D4 C60.13Q8
42 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 40A | ··· | 40H | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 40 | ··· | 40 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 10 | 2 | 1 | 1 | 10 | 2 | 2 | 2 | 10 | 10 | 6 | 6 | 30 | 30 | 2 | 2 | 2 | 2 | 10 | 10 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D5 | Dic3 | D6 | Dic3 | M4(2) | D10 | C4×D5 | C4.Dic3 | C8⋊D5 | S3×D5 | D5×Dic3 | C20.32D6 |
| kernel | C20.32D6 | C5×C3⋊C8 | C15⋊3C8 | D5×C12 | C3×Dic5 | C6×D5 | C4×D5 | C3⋊C8 | Dic5 | C20 | D10 | C15 | C12 | C6 | C5 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 2 | 2 | 4 |
Matrix representation of C20.32D6 ►in GL4(𝔽241) generated by
| 51 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 |
| 51 | 240 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 51 | 240 | 0 | 0 |
| 0 | 0 | 0 | 225 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(241))| [51,1,0,0,240,0,0,0,0,0,64,0,0,0,0,64],[1,51,0,0,0,240,0,0,0,0,15,0,0,0,0,16],[1,51,0,0,0,240,0,0,0,0,0,4,0,0,225,0] >;
C20.32D6 in GAP, Magma, Sage, TeX
C_{20}._{32}D_6 % in TeX
G:=Group("C20.32D6"); // GroupNames label
G:=SmallGroup(240,10);
// by ID
G=gap.SmallGroup(240,10);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,31,50,490,6917]);
// Polycyclic
G:=Group<a,b,c|a^20=b^6=1,c^2=a^15,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^10*b^-1>;
// generators/relations
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