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G = C15⋊C8order 120 = 23·3·5

1st semidirect product of C15 and C8 acting via C8/C2=C4

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C151C8, 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

C1C15 — C15⋊C8
C1C5C15C30C3×Dic5 — C15⋊C8
C15 — C15⋊C8
C1C2

Generators and relations for C15⋊C8
 G = < a,b | a15=b8=1, bab-1=a2 >

5C4
15C8
5C12
5C3⋊C8
3C5⋊C8

Character table of C15⋊C8

 class 1234A4B568A8B8C8D1012A12B15A15B30A30B
 size 112554215151515410104444
ρ1111111111111111111    trivial
ρ21111111-1-1-1-11111111    linear of order 2
ρ3111-1-111i-ii-i1-1-11111    linear of order 4
ρ4111-1-111-ii-ii1-1-11111    linear of order 4
ρ51-11i-i1-1ζ87ζ85ζ83ζ8-1i-i11-1-1    linear of order 8
ρ61-11i-i1-1ζ83ζ8ζ87ζ85-1i-i11-1-1    linear of order 8
ρ71-11-ii1-1ζ8ζ83ζ85ζ87-1-ii11-1-1    linear of order 8
ρ81-11-ii1-1ζ85ζ87ζ8ζ83-1-ii11-1-1    linear of order 8
ρ922-1222-100002-1-1-1-1-1-1    orthogonal lifted from S3
ρ1022-1-2-22-10000211-1-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ112-2-12i-2i210000-2-ii-1-111    complex lifted from C3⋊C8
ρ122-2-1-2i2i210000-2i-i-1-111    complex lifted from C3⋊C8
ρ1344400-140000-100-1-1-1-1    orthogonal lifted from F5
ρ144-4400-1-40000100-1-111    symplectic lifted from C5⋊C8, Schur index 2
ρ154-4-200-1200001001+-15/21--15/2-1+-15/2-1--15/2    complex faithful
ρ1644-200-1-20000-1001+-15/21--15/21--15/21+-15/2    complex lifted from C3⋊F5
ρ1744-200-1-20000-1001--15/21+-15/21+-15/21--15/2    complex lifted from C3⋊F5
ρ184-4-200-1200001001--15/21+-15/2-1--15/2-1+-15/2    complex faithful

Smallest permutation representation of C15⋊C8
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)
(1 106 57 83 24 96 40 64)(2 114 46 85 25 104 44 66)(3 107 50 87 26 97 33 68)(4 115 54 89 27 105 37 70)(5 108 58 76 28 98 41 72)(6 116 47 78 29 91 45 74)(7 109 51 80 30 99 34 61)(8 117 55 82 16 92 38 63)(9 110 59 84 17 100 42 65)(10 118 48 86 18 93 31 67)(11 111 52 88 19 101 35 69)(12 119 56 90 20 94 39 71)(13 112 60 77 21 102 43 73)(14 120 49 79 22 95 32 75)(15 113 53 81 23 103 36 62)

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), (1,106,57,83,24,96,40,64)(2,114,46,85,25,104,44,66)(3,107,50,87,26,97,33,68)(4,115,54,89,27,105,37,70)(5,108,58,76,28,98,41,72)(6,116,47,78,29,91,45,74)(7,109,51,80,30,99,34,61)(8,117,55,82,16,92,38,63)(9,110,59,84,17,100,42,65)(10,118,48,86,18,93,31,67)(11,111,52,88,19,101,35,69)(12,119,56,90,20,94,39,71)(13,112,60,77,21,102,43,73)(14,120,49,79,22,95,32,75)(15,113,53,81,23,103,36,62)>;

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), (1,106,57,83,24,96,40,64)(2,114,46,85,25,104,44,66)(3,107,50,87,26,97,33,68)(4,115,54,89,27,105,37,70)(5,108,58,76,28,98,41,72)(6,116,47,78,29,91,45,74)(7,109,51,80,30,99,34,61)(8,117,55,82,16,92,38,63)(9,110,59,84,17,100,42,65)(10,118,48,86,18,93,31,67)(11,111,52,88,19,101,35,69)(12,119,56,90,20,94,39,71)(13,112,60,77,21,102,43,73)(14,120,49,79,22,95,32,75)(15,113,53,81,23,103,36,62) );

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)], [(1,106,57,83,24,96,40,64),(2,114,46,85,25,104,44,66),(3,107,50,87,26,97,33,68),(4,115,54,89,27,105,37,70),(5,108,58,76,28,98,41,72),(6,116,47,78,29,91,45,74),(7,109,51,80,30,99,34,61),(8,117,55,82,16,92,38,63),(9,110,59,84,17,100,42,65),(10,118,48,86,18,93,31,67),(11,111,52,88,19,101,35,69),(12,119,56,90,20,94,39,71),(13,112,60,77,21,102,43,73),(14,120,49,79,22,95,32,75),(15,113,53,81,23,103,36,62)])

C15⋊C8 is a maximal subgroup of
S3×C5⋊C8  D15⋊C8  D6.F5  Dic3.F5  C60.C4  C12.F5  C158M4(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)

24010000
24000000
0022912711412
002291150126
00011522912
001141272290
,
1691240000
52720000
0021167132143
00113237102210
001801284139
00109987130

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

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