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