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## G = D15⋊C8order 240 = 24·3·5

### The semidirect product of D15 and C8 acting via C8/C2=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — D15⋊C8
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C3×C5⋊C8 — D15⋊C8
 Lower central C15 — D15⋊C8
 Upper central C1 — C2

Generators and relations for D15⋊C8
G = < a,b,c | a15=b2=c8=1, bab=a-1, cac-1=a13, cbc-1=a12b >

Character table of D15⋊C8

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5 6 8A 8B 8C 8D 8E 8F 8G 8H 10 12A 12B 15 20A 20B 24A 24B 24C 24D 30 size 1 1 15 15 2 3 3 5 5 4 2 5 5 5 5 15 15 15 15 4 10 10 8 12 12 10 10 10 10 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 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 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ3 1 1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 linear of order 2 ρ4 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 linear of order 2 ρ5 1 1 -1 -1 1 1 1 -1 -1 1 1 -i i -i i -i i -i i 1 -1 -1 1 1 1 -i -i i i 1 linear of order 4 ρ6 1 1 -1 -1 1 1 1 -1 -1 1 1 i -i i -i i -i i -i 1 -1 -1 1 1 1 i i -i -i 1 linear of order 4 ρ7 1 1 1 1 1 -1 -1 -1 -1 1 1 -i i -i i i -i i -i 1 -1 -1 1 -1 -1 -i -i i i 1 linear of order 4 ρ8 1 1 1 1 1 -1 -1 -1 -1 1 1 i -i i -i -i i -i i 1 -1 -1 1 -1 -1 i i -i -i 1 linear of order 4 ρ9 1 -1 1 -1 1 i -i i -i 1 -1 ζ83 ζ8 ζ87 ζ85 ζ85 ζ83 ζ8 ζ87 -1 -i i 1 i -i ζ83 ζ87 ζ8 ζ85 -1 linear of order 8 ρ10 1 -1 1 -1 1 -i i -i i 1 -1 ζ8 ζ83 ζ85 ζ87 ζ87 ζ8 ζ83 ζ85 -1 i -i 1 -i i ζ8 ζ85 ζ83 ζ87 -1 linear of order 8 ρ11 1 -1 -1 1 1 -i i i -i 1 -1 ζ87 ζ85 ζ83 ζ8 ζ85 ζ83 ζ8 ζ87 -1 -i i 1 -i i ζ87 ζ83 ζ85 ζ8 -1 linear of order 8 ρ12 1 -1 1 -1 1 -i i -i i 1 -1 ζ85 ζ87 ζ8 ζ83 ζ83 ζ85 ζ87 ζ8 -1 i -i 1 -i i ζ85 ζ8 ζ87 ζ83 -1 linear of order 8 ρ13 1 -1 1 -1 1 i -i i -i 1 -1 ζ87 ζ85 ζ83 ζ8 ζ8 ζ87 ζ85 ζ83 -1 -i i 1 i -i ζ87 ζ83 ζ85 ζ8 -1 linear of order 8 ρ14 1 -1 -1 1 1 i -i -i i 1 -1 ζ8 ζ83 ζ85 ζ87 ζ83 ζ85 ζ87 ζ8 -1 i -i 1 i -i ζ8 ζ85 ζ83 ζ87 -1 linear of order 8 ρ15 1 -1 -1 1 1 -i i i -i 1 -1 ζ83 ζ8 ζ87 ζ85 ζ8 ζ87 ζ85 ζ83 -1 -i i 1 -i i ζ83 ζ87 ζ8 ζ85 -1 linear of order 8 ρ16 1 -1 -1 1 1 i -i -i i 1 -1 ζ85 ζ87 ζ8 ζ83 ζ87 ζ8 ζ83 ζ85 -1 i -i 1 i -i ζ85 ζ8 ζ87 ζ83 -1 linear of order 8 ρ17 2 2 0 0 -1 0 0 2 2 2 -1 -2 -2 -2 -2 0 0 0 0 2 -1 -1 -1 0 0 1 1 1 1 -1 orthogonal lifted from D6 ρ18 2 2 0 0 -1 0 0 2 2 2 -1 2 2 2 2 0 0 0 0 2 -1 -1 -1 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ19 2 2 0 0 -1 0 0 -2 -2 2 -1 -2i 2i -2i 2i 0 0 0 0 2 1 1 -1 0 0 i i -i -i -1 complex lifted from C4×S3 ρ20 2 2 0 0 -1 0 0 -2 -2 2 -1 2i -2i 2i -2i 0 0 0 0 2 1 1 -1 0 0 -i -i i i -1 complex lifted from C4×S3 ρ21 2 -2 0 0 -1 0 0 2i -2i 2 1 2ζ83 2ζ8 2ζ87 2ζ85 0 0 0 0 -2 i -i -1 0 0 ζ87 ζ83 ζ85 ζ8 1 complex lifted from S3×C8 ρ22 2 -2 0 0 -1 0 0 -2i 2i 2 1 2ζ8 2ζ83 2ζ85 2ζ87 0 0 0 0 -2 -i i -1 0 0 ζ85 ζ8 ζ87 ζ83 1 complex lifted from S3×C8 ρ23 2 -2 0 0 -1 0 0 2i -2i 2 1 2ζ87 2ζ85 2ζ83 2ζ8 0 0 0 0 -2 i -i -1 0 0 ζ83 ζ87 ζ8 ζ85 1 complex lifted from S3×C8 ρ24 2 -2 0 0 -1 0 0 -2i 2i 2 1 2ζ85 2ζ87 2ζ8 2ζ83 0 0 0 0 -2 -i i -1 0 0 ζ8 ζ85 ζ83 ζ87 1 complex lifted from S3×C8 ρ25 4 4 0 0 4 4 4 0 0 -1 4 0 0 0 0 0 0 0 0 -1 0 0 -1 -1 -1 0 0 0 0 -1 orthogonal lifted from F5 ρ26 4 4 0 0 4 -4 -4 0 0 -1 4 0 0 0 0 0 0 0 0 -1 0 0 -1 1 1 0 0 0 0 -1 orthogonal lifted from C2×F5 ρ27 4 -4 0 0 4 4i -4i 0 0 -1 -4 0 0 0 0 0 0 0 0 1 0 0 -1 -i i 0 0 0 0 1 complex lifted from D5⋊C8, Schur index 2 ρ28 4 -4 0 0 4 -4i 4i 0 0 -1 -4 0 0 0 0 0 0 0 0 1 0 0 -1 i -i 0 0 0 0 1 complex lifted from D5⋊C8, Schur index 2 ρ29 8 -8 0 0 -4 0 0 0 0 -2 4 0 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 0 0 -1 orthogonal faithful, Schur index 2 ρ30 8 8 0 0 -4 0 0 0 0 -2 -4 0 0 0 0 0 0 0 0 -2 0 0 1 0 0 0 0 0 0 1 orthogonal lifted from S3×F5

Smallest permutation representation of D15⋊C8
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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 60)(39 59)(40 58)(41 57)(42 56)(43 55)(44 54)(45 53)(61 80)(62 79)(63 78)(64 77)(65 76)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 120)(104 119)(105 118)
(1 120 59 86 16 104 40 71)(2 112 48 84 17 96 44 69)(3 119 52 82 18 103 33 67)(4 111 56 80 19 95 37 65)(5 118 60 78 20 102 41 63)(6 110 49 76 21 94 45 61)(7 117 53 89 22 101 34 74)(8 109 57 87 23 93 38 72)(9 116 46 85 24 100 42 70)(10 108 50 83 25 92 31 68)(11 115 54 81 26 99 35 66)(12 107 58 79 27 91 39 64)(13 114 47 77 28 98 43 62)(14 106 51 90 29 105 32 75)(15 113 55 88 30 97 36 73)```

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

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

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

D15⋊C8 is a maximal subgroup of   S3×D5⋊C8  D60.C4  D15⋊M4(2)  Dic6.F5  C5⋊C8.D6  D15⋊C8⋊C2  D152M4(2)
D15⋊C8 is a maximal quotient of   D15⋊C16  D30.C8  Dic3×C5⋊C8  D30⋊C8  Dic15⋊C8

Matrix representation of D15⋊C8 in GL6(𝔽241)

 240 1 0 0 0 0 240 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 240 240 240 240 0 0 1 0 0 0
,
 240 0 0 0 0 0 240 1 0 0 0 0 0 0 0 0 240 0 0 0 0 240 0 0 0 0 240 0 0 0 0 0 1 1 1 1
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 141 0 138 138 0 0 138 138 0 141 0 0 103 3 103 0 0 0 100 238 238 100

`G:=sub<GL(6,GF(241))| [240,240,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,1,0,0,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240,0],[240,240,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,1,0,0,0,240,0,1,0,0,240,0,0,1,0,0,0,0,0,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,141,138,103,100,0,0,0,138,3,238,0,0,138,0,103,238,0,0,138,141,0,100] >;`

D15⋊C8 in GAP, Magma, Sage, TeX

`D_{15}\rtimes C_8`
`% in TeX`

`G:=Group("D15:C8");`
`// GroupNames label`

`G:=SmallGroup(240,99);`
`// by ID`

`G=gap.SmallGroup(240,99);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,55,50,490,3461,1745]);`
`// Polycyclic`

`G:=Group<a,b,c|a^15=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^12*b>;`
`// generators/relations`

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