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 [×2], C3, C4, C4 [×3], C22, C5, C6, C6 [×2], C8 [×4], C2×C4 [×3], D5 [×2], C10, Dic3 [×2], C12, C12, C2×C6, C15, C42, C2×C8 [×2], Dic5, C20, F5 [×2], D10, C3⋊C8, C3⋊C8, C24 [×2], C2×Dic3 [×2], C2×C12, C3×D5 [×2], C30, C8⋊C4, C5⋊2C8, C40, C5⋊C8 [×2], C4×D5, C2×F5 [×2], C2×C3⋊C8, C4×Dic3, C2×C24, C3×Dic5, C60, C3⋊F5 [×2], C6×D5, C8×D5, D5⋊C8, C4×F5, C24⋊C4, C5×C3⋊C8, C15⋊3C8, C3×C5⋊C8 [×2], D5×C12, C2×C3⋊F5 [×2], C8⋊F5, D5×C3⋊C8, C3×D5⋊C8, C4×C3⋊F5, C30.4C42
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C2×C4 [×3], Dic3 [×2], D6, C42, M4(2) [×2], F5, C4×S3 [×2], C2×Dic3, C8⋊C4, C2×F5, C8⋊S3 [×2], C4×Dic3, C4×F5, C24⋊C4, S3×F5, C8⋊F5, Dic3×F5, C30.4C42
►On 120 points
(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 76 46 105 16 61 31 120)(2 83 35 118 17 68 50 103)(3 90 54 101 18 75 39 116)(4 67 43 114 19 82 58 99)(5 74 32 97 20 89 47 112)(6 81 51 110 21 66 36 95)(7 88 40 93 22 73 55 108)(8 65 59 106 23 80 44 91)(9 72 48 119 24 87 33 104)(10 79 37 102 25 64 52 117)(11 86 56 115 26 71 41 100)(12 63 45 98 27 78 60 113)(13 70 34 111 28 85 49 96)(14 77 53 94 29 62 38 109)(15 84 42 107 30 69 57 92)
(1 105 46 61 16 120 31 76)(2 116 47 72 17 101 32 87)(3 97 48 83 18 112 33 68)(4 108 49 64 19 93 34 79)(5 119 50 75 20 104 35 90)(6 100 51 86 21 115 36 71)(7 111 52 67 22 96 37 82)(8 92 53 78 23 107 38 63)(9 103 54 89 24 118 39 74)(10 114 55 70 25 99 40 85)(11 95 56 81 26 110 41 66)(12 106 57 62 27 91 42 77)(13 117 58 73 28 102 43 88)(14 98 59 84 29 113 44 69)(15 109 60 65 30 94 45 80)
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), (1,76,46,105,16,61,31,120)(2,83,35,118,17,68,50,103)(3,90,54,101,18,75,39,116)(4,67,43,114,19,82,58,99)(5,74,32,97,20,89,47,112)(6,81,51,110,21,66,36,95)(7,88,40,93,22,73,55,108)(8,65,59,106,23,80,44,91)(9,72,48,119,24,87,33,104)(10,79,37,102,25,64,52,117)(11,86,56,115,26,71,41,100)(12,63,45,98,27,78,60,113)(13,70,34,111,28,85,49,96)(14,77,53,94,29,62,38,109)(15,84,42,107,30,69,57,92), (1,105,46,61,16,120,31,76)(2,116,47,72,17,101,32,87)(3,97,48,83,18,112,33,68)(4,108,49,64,19,93,34,79)(5,119,50,75,20,104,35,90)(6,100,51,86,21,115,36,71)(7,111,52,67,22,96,37,82)(8,92,53,78,23,107,38,63)(9,103,54,89,24,118,39,74)(10,114,55,70,25,99,40,85)(11,95,56,81,26,110,41,66)(12,106,57,62,27,91,42,77)(13,117,58,73,28,102,43,88)(14,98,59,84,29,113,44,69)(15,109,60,65,30,94,45,80)>;
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), (1,76,46,105,16,61,31,120)(2,83,35,118,17,68,50,103)(3,90,54,101,18,75,39,116)(4,67,43,114,19,82,58,99)(5,74,32,97,20,89,47,112)(6,81,51,110,21,66,36,95)(7,88,40,93,22,73,55,108)(8,65,59,106,23,80,44,91)(9,72,48,119,24,87,33,104)(10,79,37,102,25,64,52,117)(11,86,56,115,26,71,41,100)(12,63,45,98,27,78,60,113)(13,70,34,111,28,85,49,96)(14,77,53,94,29,62,38,109)(15,84,42,107,30,69,57,92), (1,105,46,61,16,120,31,76)(2,116,47,72,17,101,32,87)(3,97,48,83,18,112,33,68)(4,108,49,64,19,93,34,79)(5,119,50,75,20,104,35,90)(6,100,51,86,21,115,36,71)(7,111,52,67,22,96,37,82)(8,92,53,78,23,107,38,63)(9,103,54,89,24,118,39,74)(10,114,55,70,25,99,40,85)(11,95,56,81,26,110,41,66)(12,106,57,62,27,91,42,77)(13,117,58,73,28,102,43,88)(14,98,59,84,29,113,44,69)(15,109,60,65,30,94,45,80) );
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)], [(1,76,46,105,16,61,31,120),(2,83,35,118,17,68,50,103),(3,90,54,101,18,75,39,116),(4,67,43,114,19,82,58,99),(5,74,32,97,20,89,47,112),(6,81,51,110,21,66,36,95),(7,88,40,93,22,73,55,108),(8,65,59,106,23,80,44,91),(9,72,48,119,24,87,33,104),(10,79,37,102,25,64,52,117),(11,86,56,115,26,71,41,100),(12,63,45,98,27,78,60,113),(13,70,34,111,28,85,49,96),(14,77,53,94,29,62,38,109),(15,84,42,107,30,69,57,92)], [(1,105,46,61,16,120,31,76),(2,116,47,72,17,101,32,87),(3,97,48,83,18,112,33,68),(4,108,49,64,19,93,34,79),(5,119,50,75,20,104,35,90),(6,100,51,86,21,115,36,71),(7,111,52,67,22,96,37,82),(8,92,53,78,23,107,38,63),(9,103,54,89,24,118,39,74),(10,114,55,70,25,99,40,85),(11,95,56,81,26,110,41,66),(12,106,57,62,27,91,42,77),(13,117,58,73,28,102,43,88),(14,98,59,84,29,113,44,69),(15,109,60,65,30,94,45,80)])
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