metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.3C42, C3⋊C8⋊3F5, C15⋊3C8⋊5C4, C15⋊C8⋊2C4, C6.8(C4×F5), C3⋊3(C8⋊F5), C15⋊1(C8⋊C4), (C2×F5).Dic3, (C4×F5).2S3, (C6×F5).2C4, C4.24(S3×F5), C20.24(C4×S3), C60.24(C2×C4), (C4×D5).70D6, (C12×F5).3C2, C12.31(C2×F5), C2.4(Dic3×F5), C10.3(C4×Dic3), D5.(C4.Dic3), C60.C4.3C2, C5⋊1(C42.S3), D10.6(C2×Dic3), Dic5.10(C4×S3), (C3×D5).1M4(2), (D5×C12).62C22, (C5×C3⋊C8)⋊5C4, (D5×C3⋊C8).8C2, (C6×D5).11(C2×C4), (C3×Dic5).15(C2×C4), SmallGroup(480,225)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.3C42
G = < a,b,c | a30=b4=1, c4=a15, bab-1=a13, cac-1=a11, cbc-1=a15b >
Subgroups: 308 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, C12, C12 [×3], C2×C6, C15, C42, C2×C8 [×2], Dic5, C20, F5 [×2], D10, C3⋊C8, C3⋊C8 [×3], C2×C12 [×3], C3×D5 [×2], C30, C8⋊C4, C5⋊2C8, C40, C5⋊C8 [×2], C4×D5, C2×F5 [×2], C2×C3⋊C8 [×2], C4×C12, C3×Dic5, C60, C3×F5 [×2], C6×D5, C8×D5, D5⋊C8, C4×F5, C42.S3, C5×C3⋊C8, C15⋊3C8, C15⋊C8 [×2], D5×C12, C6×F5 [×2], C8⋊F5, D5×C3⋊C8, C12×F5, C60.C4, C30.3C42
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, C4.Dic3 [×2], C4×Dic3, C4×F5, C42.S3, S3×F5, C8⋊F5, Dic3×F5, C30.3C42
►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)
(2 8 20 14)(3 15 9 27)(4 22 28 10)(5 29 17 23)(7 13 25 19)(12 18 30 24)(31 49 55 37)(32 56 44 50)(34 40 52 46)(35 47 41 59)(36 54 60 42)(39 45 57 51)(61 70 73 64)(62 77)(63 84 81 90)(65 68 89 86)(66 75 78 69)(67 82)(71 80 83 74)(72 87)(76 85 88 79)(91 100 103 94)(92 107)(93 114 111 120)(95 98 119 116)(96 105 108 99)(97 112)(101 110 113 104)(102 117)(106 115 118 109)
(1 112 53 67 16 97 38 82)(2 93 54 78 17 108 39 63)(3 104 55 89 18 119 40 74)(4 115 56 70 19 100 41 85)(5 96 57 81 20 111 42 66)(6 107 58 62 21 92 43 77)(7 118 59 73 22 103 44 88)(8 99 60 84 23 114 45 69)(9 110 31 65 24 95 46 80)(10 91 32 76 25 106 47 61)(11 102 33 87 26 117 48 72)(12 113 34 68 27 98 49 83)(13 94 35 79 28 109 50 64)(14 105 36 90 29 120 51 75)(15 116 37 71 30 101 52 86)
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), (2,8,20,14)(3,15,9,27)(4,22,28,10)(5,29,17,23)(7,13,25,19)(12,18,30,24)(31,49,55,37)(32,56,44,50)(34,40,52,46)(35,47,41,59)(36,54,60,42)(39,45,57,51)(61,70,73,64)(62,77)(63,84,81,90)(65,68,89,86)(66,75,78,69)(67,82)(71,80,83,74)(72,87)(76,85,88,79)(91,100,103,94)(92,107)(93,114,111,120)(95,98,119,116)(96,105,108,99)(97,112)(101,110,113,104)(102,117)(106,115,118,109), (1,112,53,67,16,97,38,82)(2,93,54,78,17,108,39,63)(3,104,55,89,18,119,40,74)(4,115,56,70,19,100,41,85)(5,96,57,81,20,111,42,66)(6,107,58,62,21,92,43,77)(7,118,59,73,22,103,44,88)(8,99,60,84,23,114,45,69)(9,110,31,65,24,95,46,80)(10,91,32,76,25,106,47,61)(11,102,33,87,26,117,48,72)(12,113,34,68,27,98,49,83)(13,94,35,79,28,109,50,64)(14,105,36,90,29,120,51,75)(15,116,37,71,30,101,52,86)>;
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), (2,8,20,14)(3,15,9,27)(4,22,28,10)(5,29,17,23)(7,13,25,19)(12,18,30,24)(31,49,55,37)(32,56,44,50)(34,40,52,46)(35,47,41,59)(36,54,60,42)(39,45,57,51)(61,70,73,64)(62,77)(63,84,81,90)(65,68,89,86)(66,75,78,69)(67,82)(71,80,83,74)(72,87)(76,85,88,79)(91,100,103,94)(92,107)(93,114,111,120)(95,98,119,116)(96,105,108,99)(97,112)(101,110,113,104)(102,117)(106,115,118,109), (1,112,53,67,16,97,38,82)(2,93,54,78,17,108,39,63)(3,104,55,89,18,119,40,74)(4,115,56,70,19,100,41,85)(5,96,57,81,20,111,42,66)(6,107,58,62,21,92,43,77)(7,118,59,73,22,103,44,88)(8,99,60,84,23,114,45,69)(9,110,31,65,24,95,46,80)(10,91,32,76,25,106,47,61)(11,102,33,87,26,117,48,72)(12,113,34,68,27,98,49,83)(13,94,35,79,28,109,50,64)(14,105,36,90,29,120,51,75)(15,116,37,71,30,101,52,86) );
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)], [(2,8,20,14),(3,15,9,27),(4,22,28,10),(5,29,17,23),(7,13,25,19),(12,18,30,24),(31,49,55,37),(32,56,44,50),(34,40,52,46),(35,47,41,59),(36,54,60,42),(39,45,57,51),(61,70,73,64),(62,77),(63,84,81,90),(65,68,89,86),(66,75,78,69),(67,82),(71,80,83,74),(72,87),(76,85,88,79),(91,100,103,94),(92,107),(93,114,111,120),(95,98,119,116),(96,105,108,99),(97,112),(101,110,113,104),(102,117),(106,115,118,109)], [(1,112,53,67,16,97,38,82),(2,93,54,78,17,108,39,63),(3,104,55,89,18,119,40,74),(4,115,56,70,19,100,41,85),(5,96,57,81,20,111,42,66),(6,107,58,62,21,92,43,77),(7,118,59,73,22,103,44,88),(8,99,60,84,23,114,45,69),(9,110,31,65,24,95,46,80),(10,91,32,76,25,106,47,61),(11,102,33,87,26,117,48,72),(12,113,34,68,27,98,49,83),(13,94,35,79,28,109,50,64),(14,105,36,90,29,120,51,75),(15,116,37,71,30,101,52,86)])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 6A | 6B | 6C | 8A | 8B | 8C | ··· | 8H | 10 | 12A | 12B | 12C | ··· | 12L | 15 | 20A | 20B | 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 | 10 | 12 | 12 | 12 | ··· | 12 | 15 | 20 | 20 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 2 | 1 | 1 | 5 | 5 | 10 | 10 | 10 | 10 | 4 | 2 | 10 | 10 | 6 | 6 | 30 | ··· | 30 | 4 | 2 | 2 | 10 | ··· | 10 | 8 | 4 | 4 | 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 | D6 | Dic3 | M4(2) | C4×S3 | C4×S3 | C4.Dic3 | F5 | C2×F5 | C4×F5 | C8⋊F5 | S3×F5 | Dic3×F5 | C30.3C42 |
| kernel | C30.3C42 | D5×C3⋊C8 | C12×F5 | C60.C4 | C5×C3⋊C8 | C15⋊3C8 | C15⋊C8 | C6×F5 | C4×F5 | C4×D5 | C2×F5 | C3×D5 | Dic5 | C20 | D5 | C3⋊C8 | C12 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 2 | 2 | 8 | 1 | 1 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of C30.3C42 ►in GL6(𝔽241)
| 0 | 240 | 0 | 0 | 0 | 0 |
| 1 | 240 | 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 213 | 0 | 185 | 185 |
| 0 | 0 | 56 | 28 | 56 | 0 |
| 0 | 0 | 0 | 56 | 28 | 56 |
| 0 | 0 | 185 | 185 | 0 | 213 |
G:=sub<GL(6,GF(241))| [0,1,0,0,0,0,240,240,0,0,0,0,0,0,0,0,0,1,0,0,240,0,0,1,0,0,0,240,0,1,0,0,0,0,240,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0],[240,240,0,0,0,0,0,1,0,0,0,0,0,0,213,56,0,185,0,0,0,28,56,185,0,0,185,56,28,0,0,0,185,0,56,213] >;
C30.3C42 in GAP, Magma, Sage, TeX
C_{30}._3C_4^2 % in TeX
G:=Group("C30.3C4^2"); // GroupNames label
G:=SmallGroup(480,225);
// by ID
G=gap.SmallGroup(480,225);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c|a^30=b^4=1,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