metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.4C42, C3⋊C8⋊4F5, C5⋊C8⋊1Dic3, C15⋊3C8⋊6C4, C6.9(C4×F5), D5⋊C8.2S3, C3⋊2(C8⋊F5), C5⋊1(C24⋊C4), C15⋊2(C8⋊C4), C4.25(S3×F5), C20.25(C4×S3), C60.25(C2×C4), (C4×D5).71D6, D10.8(C4×S3), C12.32(C2×F5), C2.5(Dic3×F5), C10.4(C4×Dic3), D5.1(C8⋊S3), (C3×D5).2M4(2), Dic5.8(C2×Dic3), (D5×C12).63C22, (C5×C3⋊C8)⋊6C4, (C3×C5⋊C8)⋊2C4, (D5×C3⋊C8).9C2, (C4×C3⋊F5).3C2, (C2×C3⋊F5).2C4, (C3×D5⋊C8).3C2, (C6×D5).12(C2×C4), (C3×Dic5).16(C2×C4), SmallGroup(480,226)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.4C42
G = < a,b,c | a30=1, b4=c4=a15, bab-1=a13, cac-1=a11, cbc-1=a15b >
Subgroups: 356 in 80 conjugacy classes, 36 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D5, C10, Dic3, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, F5, D10, C3⋊C8, C3⋊C8, C24, C2×Dic3, C2×C12, C3×D5, C30, C8⋊C4, C5⋊2C8, C40, C5⋊C8, C4×D5, C2×F5, C2×C3⋊C8, C4×Dic3, C2×C24, C3×Dic5, C60, C3⋊F5, C6×D5, C8×D5, D5⋊C8, C4×F5, C24⋊C4, C5×C3⋊C8, C15⋊3C8, C3×C5⋊C8, D5×C12, C2×C3⋊F5, C8⋊F5, D5×C3⋊C8, C3×D5⋊C8, C4×C3⋊F5, C30.4C42
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, M4(2), F5, C4×S3, C2×Dic3, C8⋊C4, C2×F5, C8⋊S3, C4×Dic3, C4×F5, C24⋊C4, S3×F5, C8⋊F5, Dic3×F5, C30.4C42
►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 68 36 106 16 83 51 91)(2 75 55 119 17 90 40 104)(3 82 44 102 18 67 59 117)(4 89 33 115 19 74 48 100)(5 66 52 98 20 81 37 113)(6 73 41 111 21 88 56 96)(7 80 60 94 22 65 45 109)(8 87 49 107 23 72 34 92)(9 64 38 120 24 79 53 105)(10 71 57 103 25 86 42 118)(11 78 46 116 26 63 31 101)(12 85 35 99 27 70 50 114)(13 62 54 112 28 77 39 97)(14 69 43 95 29 84 58 110)(15 76 32 108 30 61 47 93)
(1 106 36 83 16 91 51 68)(2 117 37 64 17 102 52 79)(3 98 38 75 18 113 53 90)(4 109 39 86 19 94 54 71)(5 120 40 67 20 105 55 82)(6 101 41 78 21 116 56 63)(7 112 42 89 22 97 57 74)(8 93 43 70 23 108 58 85)(9 104 44 81 24 119 59 66)(10 115 45 62 25 100 60 77)(11 96 46 73 26 111 31 88)(12 107 47 84 27 92 32 69)(13 118 48 65 28 103 33 80)(14 99 49 76 29 114 34 61)(15 110 50 87 30 95 35 72)
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), (1,68,36,106,16,83,51,91)(2,75,55,119,17,90,40,104)(3,82,44,102,18,67,59,117)(4,89,33,115,19,74,48,100)(5,66,52,98,20,81,37,113)(6,73,41,111,21,88,56,96)(7,80,60,94,22,65,45,109)(8,87,49,107,23,72,34,92)(9,64,38,120,24,79,53,105)(10,71,57,103,25,86,42,118)(11,78,46,116,26,63,31,101)(12,85,35,99,27,70,50,114)(13,62,54,112,28,77,39,97)(14,69,43,95,29,84,58,110)(15,76,32,108,30,61,47,93), (1,106,36,83,16,91,51,68)(2,117,37,64,17,102,52,79)(3,98,38,75,18,113,53,90)(4,109,39,86,19,94,54,71)(5,120,40,67,20,105,55,82)(6,101,41,78,21,116,56,63)(7,112,42,89,22,97,57,74)(8,93,43,70,23,108,58,85)(9,104,44,81,24,119,59,66)(10,115,45,62,25,100,60,77)(11,96,46,73,26,111,31,88)(12,107,47,84,27,92,32,69)(13,118,48,65,28,103,33,80)(14,99,49,76,29,114,34,61)(15,110,50,87,30,95,35,72)>;
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), (1,68,36,106,16,83,51,91)(2,75,55,119,17,90,40,104)(3,82,44,102,18,67,59,117)(4,89,33,115,19,74,48,100)(5,66,52,98,20,81,37,113)(6,73,41,111,21,88,56,96)(7,80,60,94,22,65,45,109)(8,87,49,107,23,72,34,92)(9,64,38,120,24,79,53,105)(10,71,57,103,25,86,42,118)(11,78,46,116,26,63,31,101)(12,85,35,99,27,70,50,114)(13,62,54,112,28,77,39,97)(14,69,43,95,29,84,58,110)(15,76,32,108,30,61,47,93), (1,106,36,83,16,91,51,68)(2,117,37,64,17,102,52,79)(3,98,38,75,18,113,53,90)(4,109,39,86,19,94,54,71)(5,120,40,67,20,105,55,82)(6,101,41,78,21,116,56,63)(7,112,42,89,22,97,57,74)(8,93,43,70,23,108,58,85)(9,104,44,81,24,119,59,66)(10,115,45,62,25,100,60,77)(11,96,46,73,26,111,31,88)(12,107,47,84,27,92,32,69)(13,118,48,65,28,103,33,80)(14,99,49,76,29,114,34,61)(15,110,50,87,30,95,35,72) );
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)], [(1,68,36,106,16,83,51,91),(2,75,55,119,17,90,40,104),(3,82,44,102,18,67,59,117),(4,89,33,115,19,74,48,100),(5,66,52,98,20,81,37,113),(6,73,41,111,21,88,56,96),(7,80,60,94,22,65,45,109),(8,87,49,107,23,72,34,92),(9,64,38,120,24,79,53,105),(10,71,57,103,25,86,42,118),(11,78,46,116,26,63,31,101),(12,85,35,99,27,70,50,114),(13,62,54,112,28,77,39,97),(14,69,43,95,29,84,58,110),(15,76,32,108,30,61,47,93)], [(1,106,36,83,16,91,51,68),(2,117,37,64,17,102,52,79),(3,98,38,75,18,113,53,90),(4,109,39,86,19,94,54,71),(5,120,40,67,20,105,55,82),(6,101,41,78,21,116,56,63),(7,112,42,89,22,97,57,74),(8,93,43,70,23,108,58,85),(9,104,44,81,24,119,59,66),(10,115,45,62,25,100,60,77),(11,96,46,73,26,111,31,88),(12,107,47,84,27,92,32,69),(13,118,48,65,28,103,33,80),(14,99,49,76,29,114,34,61),(15,110,50,87,30,95,35,72)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10 | 12A | 12B | 12C | 12D | 15 | 20A | 20B | 24A | ··· | 24H | 30 | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 12 | 12 | 15 | 20 | 20 | 24 | ··· | 24 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 2 | 1 | 1 | 5 | 5 | 30 | 30 | 30 | 30 | 4 | 2 | 10 | 10 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | 30 | 4 | 2 | 2 | 10 | 10 | 8 | 4 | 4 | 10 | ··· | 10 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | - | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | Dic3 | D6 | M4(2) | C4×S3 | C4×S3 | C8⋊S3 | F5 | C2×F5 | C4×F5 | C8⋊F5 | S3×F5 | Dic3×F5 | C30.4C42 |
| kernel | C30.4C42 | D5×C3⋊C8 | C3×D5⋊C8 | C4×C3⋊F5 | C5×C3⋊C8 | C15⋊3C8 | C3×C5⋊C8 | C2×C3⋊F5 | D5⋊C8 | C5⋊C8 | C4×D5 | C3×D5 | C20 | D10 | D5 | C3⋊C8 | C12 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 4 | 2 | 2 | 8 | 1 | 1 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of C30.4C42 ►in GL6(𝔽241)
| 240 | 240 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 213 | 0 | 185 | 185 |
| 0 | 0 | 185 | 185 | 0 | 213 |
| 0 | 0 | 56 | 28 | 56 | 0 |
| 0 | 0 | 28 | 213 | 213 | 28 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 177 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 136 | 0 | 31 | 31 |
| 0 | 0 | 210 | 105 | 210 | 0 |
| 0 | 0 | 0 | 210 | 105 | 210 |
| 0 | 0 | 31 | 31 | 0 | 136 |
G:=sub<GL(6,GF(241))| [240,1,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,1,0,0,240,0,0,1,0,0,0,240,0,1,0,0,0,0,240,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,213,185,56,28,0,0,0,185,28,213,0,0,185,0,56,213,0,0,185,213,0,28],[64,177,0,0,0,0,0,177,0,0,0,0,0,0,136,210,0,31,0,0,0,105,210,31,0,0,31,210,105,0,0,0,31,0,210,136] >;
C30.4C42 in GAP, Magma, Sage, TeX
C_{30}._4C_4^2 % in TeX
G:=Group("C30.4C4^2"); // GroupNames label
G:=SmallGroup(480,226);
// by ID
G=gap.SmallGroup(480,226);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,100,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c|a^30=1,b^4=c^4=a^15,b*a*b^-1=a^13,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations