metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C15⋊1C8, C6.F5, C30.1C4, C10.Dic3, Dic5.2S3, C5⋊(C3⋊C8), C3⋊(C5⋊C8), C2.(C3⋊F5), (C3×Dic5).3C2, SmallGroup(120,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C15⋊C8 |
Generators and relations for C15⋊C8
G = < a,b | a15=b8=1, bab-1=a2 >
Character table of C15⋊C8
| class | 1 | 2 | 3 | 4A | 4B | 5 | 6 | 8A | 8B | 8C | 8D | 10 | 12A | 12B | 15A | 15B | 30A | 30B | |
| size | 1 | 1 | 2 | 5 | 5 | 4 | 2 | 15 | 15 | 15 | 15 | 4 | 10 | 10 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 1 | -1 | 1 | i | -i | 1 | -1 | ζ87 | ζ85 | ζ83 | ζ8 | -1 | i | -i | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ6 | 1 | -1 | 1 | i | -i | 1 | -1 | ζ83 | ζ8 | ζ87 | ζ85 | -1 | i | -i | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ7 | 1 | -1 | 1 | -i | i | 1 | -1 | ζ8 | ζ83 | ζ85 | ζ87 | -1 | -i | i | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ8 | 1 | -1 | 1 | -i | i | 1 | -1 | ζ85 | ζ87 | ζ8 | ζ83 | -1 | -i | i | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ9 | 2 | 2 | -1 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -1 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 2 | 1 | 1 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ11 | 2 | -2 | -1 | 2i | -2i | 2 | 1 | 0 | 0 | 0 | 0 | -2 | -i | i | -1 | -1 | 1 | 1 | complex lifted from C3⋊C8 |
| ρ12 | 2 | -2 | -1 | -2i | 2i | 2 | 1 | 0 | 0 | 0 | 0 | -2 | i | -i | -1 | -1 | 1 | 1 | complex lifted from C3⋊C8 |
| ρ13 | 4 | 4 | 4 | 0 | 0 | -1 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ14 | 4 | -4 | 4 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | -1 | 1 | 1 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ15 | 4 | -4 | -2 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1+√-15/2 | 1-√-15/2 | -1+√-15/2 | -1-√-15/2 | complex faithful |
| ρ16 | 4 | 4 | -2 | 0 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | complex lifted from C3⋊F5 |
| ρ17 | 4 | 4 | -2 | 0 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | complex lifted from C3⋊F5 |
| ρ18 | 4 | -4 | -2 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1-√-15/2 | 1+√-15/2 | -1-√-15/2 | -1+√-15/2 | complex faithful |
►Regular action 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)
(1 111 55 78 24 91 42 66)(2 119 59 80 25 99 31 68)(3 112 48 82 26 92 35 70)(4 120 52 84 27 100 39 72)(5 113 56 86 28 93 43 74)(6 106 60 88 29 101 32 61)(7 114 49 90 30 94 36 63)(8 107 53 77 16 102 40 65)(9 115 57 79 17 95 44 67)(10 108 46 81 18 103 33 69)(11 116 50 83 19 96 37 71)(12 109 54 85 20 104 41 73)(13 117 58 87 21 97 45 75)(14 110 47 89 22 105 34 62)(15 118 51 76 23 98 38 64)
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), (1,111,55,78,24,91,42,66)(2,119,59,80,25,99,31,68)(3,112,48,82,26,92,35,70)(4,120,52,84,27,100,39,72)(5,113,56,86,28,93,43,74)(6,106,60,88,29,101,32,61)(7,114,49,90,30,94,36,63)(8,107,53,77,16,102,40,65)(9,115,57,79,17,95,44,67)(10,108,46,81,18,103,33,69)(11,116,50,83,19,96,37,71)(12,109,54,85,20,104,41,73)(13,117,58,87,21,97,45,75)(14,110,47,89,22,105,34,62)(15,118,51,76,23,98,38,64)>;
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), (1,111,55,78,24,91,42,66)(2,119,59,80,25,99,31,68)(3,112,48,82,26,92,35,70)(4,120,52,84,27,100,39,72)(5,113,56,86,28,93,43,74)(6,106,60,88,29,101,32,61)(7,114,49,90,30,94,36,63)(8,107,53,77,16,102,40,65)(9,115,57,79,17,95,44,67)(10,108,46,81,18,103,33,69)(11,116,50,83,19,96,37,71)(12,109,54,85,20,104,41,73)(13,117,58,87,21,97,45,75)(14,110,47,89,22,105,34,62)(15,118,51,76,23,98,38,64) );
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)], [(1,111,55,78,24,91,42,66),(2,119,59,80,25,99,31,68),(3,112,48,82,26,92,35,70),(4,120,52,84,27,100,39,72),(5,113,56,86,28,93,43,74),(6,106,60,88,29,101,32,61),(7,114,49,90,30,94,36,63),(8,107,53,77,16,102,40,65),(9,115,57,79,17,95,44,67),(10,108,46,81,18,103,33,69),(11,116,50,83,19,96,37,71),(12,109,54,85,20,104,41,73),(13,117,58,87,21,97,45,75),(14,110,47,89,22,105,34,62),(15,118,51,76,23,98,38,64)]])
C15⋊C8 is a maximal subgroup of
S3×C5⋊C8 D15⋊C8 D6.F5 Dic3.F5 C60.C4 C12.F5 C15⋊8M4(2) C45⋊C8 C30.Dic3 C5⋊U2(𝔽3) Dic5.S4
C15⋊C8 is a maximal quotient of
C15⋊C16 C45⋊C8 C30.Dic3 Dic5.S4
Matrix representation of C15⋊C8 ►in GL6(𝔽241)
| 240 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 229 | 127 | 114 | 12 |
| 0 | 0 | 229 | 115 | 0 | 126 |
| 0 | 0 | 0 | 115 | 229 | 12 |
| 0 | 0 | 114 | 127 | 229 | 0 |
| 169 | 124 | 0 | 0 | 0 | 0 |
| 52 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 211 | 67 | 132 | 143 |
| 0 | 0 | 113 | 237 | 102 | 210 |
| 0 | 0 | 180 | 128 | 4 | 139 |
| 0 | 0 | 109 | 98 | 71 | 30 |
G:=sub<GL(6,GF(241))| [240,240,0,0,0,0,1,0,0,0,0,0,0,0,229,229,0,114,0,0,127,115,115,127,0,0,114,0,229,229,0,0,12,126,12,0],[169,52,0,0,0,0,124,72,0,0,0,0,0,0,211,113,180,109,0,0,67,237,128,98,0,0,132,102,4,71,0,0,143,210,139,30] >;
C15⋊C8 in GAP, Magma, Sage, TeX
C_{15}\rtimes C_8 % in TeX
G:=Group("C15:C8"); // GroupNames label
G:=SmallGroup(120,7);
// by ID
G=gap.SmallGroup(120,7);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-5,10,26,323,1804,1209]);
// Polycyclic
G:=Group<a,b|a^15=b^8=1,b*a*b^-1=a^2>;
// generators/relations
Export
Subgroup lattice of C15⋊C8 in TeX
Character table of C15⋊C8 in TeX