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## G = C15⋊8M4(2)  order 240 = 24·3·5

### 1st semidirect product of C15 and M4(2) acting via M4(2)/C22=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C15⋊8M4(2)
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C15⋊C8 — C15⋊8M4(2)
 Lower central C15 — C30 — C15⋊8M4(2)
 Upper central C1 — C2 — C22

Generators and relations for C158M4(2)
G = < a,b,c | a15=b8=c2=1, bab-1=a2, ac=ca, cbc=b5 >

Character table of C158M4(2)

 class 1 2A 2B 3 4A 4B 4C 5 6A 6B 6C 8A 8B 8C 8D 10A 10B 10C 12A 12B 12C 12D 15A 15B 30A 30B 30C 30D 30E 30F size 1 1 2 2 5 5 10 4 2 2 2 30 30 30 30 4 4 4 10 10 10 10 4 4 4 4 4 4 4 4 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 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 -1 1 -1 -1 1 1 -1 -1 1 -i i i -i -1 -1 1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 1 linear of order 4 ρ7 1 1 -1 1 -1 -1 1 1 -1 -1 1 i -i -i i -1 -1 1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 1 linear of order 4 ρ8 1 1 1 1 -1 -1 -1 1 1 1 1 i i -i -i 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 4 ρ9 2 2 -2 -1 2 2 -2 2 1 1 -1 0 0 0 0 -2 -2 2 -1 -1 1 1 -1 -1 1 1 -1 1 1 -1 orthogonal lifted from D6 ρ10 2 2 2 -1 2 2 2 2 -1 -1 -1 0 0 0 0 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 -1 -2 -2 -2 2 -1 -1 -1 0 0 0 0 2 2 2 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ12 2 2 -2 -1 -2 -2 2 2 1 1 -1 0 0 0 0 -2 -2 2 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 symplectic lifted from Dic3, Schur index 2 ρ13 2 -2 0 2 -2i 2i 0 2 0 0 -2 0 0 0 0 0 0 -2 2i -2i 0 0 2 2 0 0 -2 0 0 -2 complex lifted from M4(2) ρ14 2 -2 0 2 2i -2i 0 2 0 0 -2 0 0 0 0 0 0 -2 -2i 2i 0 0 2 2 0 0 -2 0 0 -2 complex lifted from M4(2) ρ15 2 -2 0 -1 -2i 2i 0 2 √-3 -√-3 1 0 0 0 0 0 0 -2 -i i √3 -√3 -1 -1 √-3 √-3 1 -√-3 -√-3 1 complex lifted from C4.Dic3 ρ16 2 -2 0 -1 -2i 2i 0 2 -√-3 √-3 1 0 0 0 0 0 0 -2 -i i -√3 √3 -1 -1 -√-3 -√-3 1 √-3 √-3 1 complex lifted from C4.Dic3 ρ17 2 -2 0 -1 2i -2i 0 2 √-3 -√-3 1 0 0 0 0 0 0 -2 i -i -√3 √3 -1 -1 √-3 √-3 1 -√-3 -√-3 1 complex lifted from C4.Dic3 ρ18 2 -2 0 -1 2i -2i 0 2 -√-3 √-3 1 0 0 0 0 0 0 -2 i -i √3 -√3 -1 -1 -√-3 -√-3 1 √-3 √-3 1 complex lifted from C4.Dic3 ρ19 4 4 -4 4 0 0 0 -1 -4 -4 4 0 0 0 0 1 1 -1 0 0 0 0 -1 -1 1 1 -1 1 1 -1 orthogonal lifted from C2×F5 ρ20 4 4 4 4 0 0 0 -1 4 4 4 0 0 0 0 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ21 4 -4 0 4 0 0 0 -1 0 0 -4 0 0 0 0 -√5 √5 1 0 0 0 0 -1 -1 -√5 √5 1 -√5 √5 1 symplectic lifted from C22.F5, Schur index 2 ρ22 4 -4 0 4 0 0 0 -1 0 0 -4 0 0 0 0 √5 -√5 1 0 0 0 0 -1 -1 √5 -√5 1 √5 -√5 1 symplectic lifted from C22.F5, Schur index 2 ρ23 4 4 4 -2 0 0 0 -1 -2 -2 -2 0 0 0 0 -1 -1 -1 0 0 0 0 1+√-15/2 1-√-15/2 1-√-15/2 1+√-15/2 1+√-15/2 1+√-15/2 1-√-15/2 1-√-15/2 complex lifted from C3⋊F5 ρ24 4 -4 0 -2 0 0 0 -1 -2√-3 2√-3 2 0 0 0 0 √5 -√5 1 0 0 0 0 1+√-15/2 1-√-15/2 -ζ32+ζ53+ζ52 -ζ32+ζ54+ζ5 -1-√-15/2 -ζ3+ζ53+ζ52 -ζ3+ζ54+ζ5 -1+√-15/2 complex faithful ρ25 4 -4 0 -2 0 0 0 -1 2√-3 -2√-3 2 0 0 0 0 -√5 √5 1 0 0 0 0 1+√-15/2 1-√-15/2 -ζ3+ζ54+ζ5 -ζ3+ζ53+ζ52 -1-√-15/2 -ζ32+ζ54+ζ5 -ζ32+ζ53+ζ52 -1+√-15/2 complex faithful ρ26 4 -4 0 -2 0 0 0 -1 -2√-3 2√-3 2 0 0 0 0 -√5 √5 1 0 0 0 0 1-√-15/2 1+√-15/2 -ζ32+ζ54+ζ5 -ζ32+ζ53+ζ52 -1+√-15/2 -ζ3+ζ54+ζ5 -ζ3+ζ53+ζ52 -1-√-15/2 complex faithful ρ27 4 -4 0 -2 0 0 0 -1 2√-3 -2√-3 2 0 0 0 0 √5 -√5 1 0 0 0 0 1-√-15/2 1+√-15/2 -ζ3+ζ53+ζ52 -ζ3+ζ54+ζ5 -1+√-15/2 -ζ32+ζ53+ζ52 -ζ32+ζ54+ζ5 -1-√-15/2 complex faithful ρ28 4 4 4 -2 0 0 0 -1 -2 -2 -2 0 0 0 0 -1 -1 -1 0 0 0 0 1-√-15/2 1+√-15/2 1+√-15/2 1-√-15/2 1-√-15/2 1-√-15/2 1+√-15/2 1+√-15/2 complex lifted from C3⋊F5 ρ29 4 4 -4 -2 0 0 0 -1 2 2 -2 0 0 0 0 1 1 -1 0 0 0 0 1+√-15/2 1-√-15/2 -1+√-15/2 -1-√-15/2 1+√-15/2 -1-√-15/2 -1+√-15/2 1-√-15/2 complex lifted from C2×C3⋊F5 ρ30 4 4 -4 -2 0 0 0 -1 2 2 -2 0 0 0 0 1 1 -1 0 0 0 0 1-√-15/2 1+√-15/2 -1-√-15/2 -1+√-15/2 1-√-15/2 -1+√-15/2 -1-√-15/2 1+√-15/2 complex lifted from C2×C3⋊F5

Smallest permutation representation of C158M4(2)
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 96 40 87 28 106 53 61)(2 104 44 89 29 114 57 63)(3 97 33 76 30 107 46 65)(4 105 37 78 16 115 50 67)(5 98 41 80 17 108 54 69)(6 91 45 82 18 116 58 71)(7 99 34 84 19 109 47 73)(8 92 38 86 20 117 51 75)(9 100 42 88 21 110 55 62)(10 93 31 90 22 118 59 64)(11 101 35 77 23 111 48 66)(12 94 39 79 24 119 52 68)(13 102 43 81 25 112 56 70)(14 95 32 83 26 120 60 72)(15 103 36 85 27 113 49 74)
(61 87)(62 88)(63 89)(64 90)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(91 116)(92 117)(93 118)(94 119)(95 120)(96 106)(97 107)(98 108)(99 109)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)```

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

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

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

C158M4(2) is a maximal subgroup of
Dic5.D12  Dic5.4D12  (C2×C60).C4  C5⋊(C12.D4)  C5⋊C8.D6  S3×C22.F5  D152M4(2)  C60.59(C2×C4)  Dic10.Dic3
C158M4(2) is a maximal quotient of
C30.11C42  Dic5.13D12  C30.22M4(2)

Matrix representation of C158M4(2) in GL6(𝔽241)

 240 1 0 0 0 0 240 0 0 0 0 0 0 0 51 189 0 0 0 0 51 0 0 0 0 0 0 0 240 1 0 0 0 0 50 190
,
 8 22 0 0 0 0 30 233 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 237 159 0 0 0 0 203 4 0 0
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240

`G:=sub<GL(6,GF(241))| [240,240,0,0,0,0,1,0,0,0,0,0,0,0,51,51,0,0,0,0,189,0,0,0,0,0,0,0,240,50,0,0,0,0,1,190],[8,30,0,0,0,0,22,233,0,0,0,0,0,0,0,0,237,203,0,0,0,0,159,4,0,0,1,0,0,0,0,0,0,1,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;`

C158M4(2) in GAP, Magma, Sage, TeX

`C_{15}\rtimes_8M_4(2)`
`% in TeX`

`G:=Group("C15:8M4(2)");`
`// GroupNames label`

`G:=SmallGroup(240,123);`
`// by ID`

`G=gap.SmallGroup(240,123);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,50,964,5189,1745]);`
`// Polycyclic`

`G:=Group<a,b,c|a^15=b^8=c^2=1,b*a*b^-1=a^2,a*c=c*a,c*b*c=b^5>;`
`// generators/relations`

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