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G = C15⋊Q8order 120 = 23·3·5

The semidirect product of C15 and Q8 acting via Q8/C2=C22

Aliases: C15⋊Q8, C51Dic6, C6.7D10, C10.7D6, C31Dic10, Dic3.D5, C30.7C22, Dic5.1S3, Dic15.2C2, C2.7(S3×D5), (C5×Dic3).1C2, (C3×Dic5).1C2, SmallGroup(120,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C15⋊Q8
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C15⋊Q8
 Lower central C15 — C30 — C15⋊Q8
 Upper central C1 — C2

Generators and relations for C15⋊Q8
G = < a,b,c | a15=b4=1, c2=b2, bab-1=a11, cac-1=a4, cbc-1=b-1 >

Character table of C15⋊Q8

 class 1 2 3 4A 4B 4C 5A 5B 6 10A 10B 12A 12B 15A 15B 20A 20B 20C 20D 30A 30B size 1 1 2 6 10 30 2 2 2 2 2 10 10 4 4 6 6 6 6 4 4 ρ1 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 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 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 linear of order 2 ρ5 2 2 -1 0 -2 0 2 2 -1 2 2 1 1 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from D6 ρ6 2 2 2 2 0 0 -1+√5/2 -1-√5/2 2 -1+√5/2 -1-√5/2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ7 2 2 -1 0 2 0 2 2 -1 2 2 -1 -1 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ8 2 2 2 -2 0 0 -1-√5/2 -1+√5/2 2 -1-√5/2 -1+√5/2 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ9 2 2 2 -2 0 0 -1+√5/2 -1-√5/2 2 -1+√5/2 -1-√5/2 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ10 2 2 2 2 0 0 -1-√5/2 -1+√5/2 2 -1-√5/2 -1+√5/2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ11 2 -2 2 0 0 0 2 2 -2 -2 -2 0 0 2 2 0 0 0 0 -2 -2 symplectic lifted from Q8, Schur index 2 ρ12 2 -2 -1 0 0 0 2 2 1 -2 -2 -√3 √3 -1 -1 0 0 0 0 1 1 symplectic lifted from Dic6, Schur index 2 ρ13 2 -2 -1 0 0 0 2 2 1 -2 -2 √3 -√3 -1 -1 0 0 0 0 1 1 symplectic lifted from Dic6, Schur index 2 ρ14 2 -2 2 0 0 0 -1+√5/2 -1-√5/2 -2 1-√5/2 1+√5/2 0 0 -1-√5/2 -1+√5/2 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 1+√5/2 1-√5/2 symplectic lifted from Dic10, Schur index 2 ρ15 2 -2 2 0 0 0 -1+√5/2 -1-√5/2 -2 1-√5/2 1+√5/2 0 0 -1-√5/2 -1+√5/2 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 1+√5/2 1-√5/2 symplectic lifted from Dic10, Schur index 2 ρ16 2 -2 2 0 0 0 -1-√5/2 -1+√5/2 -2 1+√5/2 1-√5/2 0 0 -1+√5/2 -1-√5/2 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 1-√5/2 1+√5/2 symplectic lifted from Dic10, Schur index 2 ρ17 2 -2 2 0 0 0 -1-√5/2 -1+√5/2 -2 1+√5/2 1-√5/2 0 0 -1+√5/2 -1-√5/2 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 1-√5/2 1+√5/2 symplectic lifted from Dic10, Schur index 2 ρ18 4 4 -2 0 0 0 -1-√5 -1+√5 -2 -1-√5 -1+√5 0 0 1-√5/2 1+√5/2 0 0 0 0 1-√5/2 1+√5/2 orthogonal lifted from S3×D5 ρ19 4 4 -2 0 0 0 -1+√5 -1-√5 -2 -1+√5 -1-√5 0 0 1+√5/2 1-√5/2 0 0 0 0 1+√5/2 1-√5/2 orthogonal lifted from S3×D5 ρ20 4 -4 -2 0 0 0 -1-√5 -1+√5 2 1+√5 1-√5 0 0 1-√5/2 1+√5/2 0 0 0 0 -1+√5/2 -1-√5/2 symplectic faithful, Schur index 2 ρ21 4 -4 -2 0 0 0 -1+√5 -1-√5 2 1-√5 1+√5 0 0 1+√5/2 1-√5/2 0 0 0 0 -1-√5/2 -1+√5/2 symplectic faithful, Schur index 2

Smallest permutation representation of C15⋊Q8
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 46 20 39)(2 57 21 35)(3 53 22 31)(4 49 23 42)(5 60 24 38)(6 56 25 34)(7 52 26 45)(8 48 27 41)(9 59 28 37)(10 55 29 33)(11 51 30 44)(12 47 16 40)(13 58 17 36)(14 54 18 32)(15 50 19 43)(61 96 83 110)(62 92 84 106)(63 103 85 117)(64 99 86 113)(65 95 87 109)(66 91 88 120)(67 102 89 116)(68 98 90 112)(69 94 76 108)(70 105 77 119)(71 101 78 115)(72 97 79 111)(73 93 80 107)(74 104 81 118)(75 100 82 114)
(1 83 20 61)(2 87 21 65)(3 76 22 69)(4 80 23 73)(5 84 24 62)(6 88 25 66)(7 77 26 70)(8 81 27 74)(9 85 28 63)(10 89 29 67)(11 78 30 71)(12 82 16 75)(13 86 17 64)(14 90 18 68)(15 79 19 72)(31 108 53 94)(32 112 54 98)(33 116 55 102)(34 120 56 91)(35 109 57 95)(36 113 58 99)(37 117 59 103)(38 106 60 92)(39 110 46 96)(40 114 47 100)(41 118 48 104)(42 107 49 93)(43 111 50 97)(44 115 51 101)(45 119 52 105)```

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

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

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

C15⋊Q8 is a maximal subgroup of
D5×Dic6  S3×Dic10  D15⋊Q8  D6.D10  Dic5.D6  C30.C23  Dic3.D10  C45⋊Q8  C15⋊Dic6  C3⋊Dic30  C323Dic10  CSU2(𝔽3)⋊D5  A4⋊Dic10
C15⋊Q8 is a maximal quotient of
C30.Q8  Dic155C4  C6.Dic10  C45⋊Q8  C15⋊Dic6  C3⋊Dic30  C323Dic10  A4⋊Dic10

Matrix representation of C15⋊Q8 in GL4(𝔽61) generated by

 60 17 0 0 44 44 0 0 0 0 60 1 0 0 60 0
,
 29 54 0 0 7 32 0 0 0 0 11 50 0 0 0 50
,
 20 58 0 0 32 41 0 0 0 0 23 15 0 0 46 38
`G:=sub<GL(4,GF(61))| [60,44,0,0,17,44,0,0,0,0,60,60,0,0,1,0],[29,7,0,0,54,32,0,0,0,0,11,0,0,0,50,50],[20,32,0,0,58,41,0,0,0,0,23,46,0,0,15,38] >;`

C15⋊Q8 in GAP, Magma, Sage, TeX

`C_{15}\rtimes Q_8`
`% in TeX`

`G:=Group("C15:Q8");`
`// GroupNames label`

`G:=SmallGroup(120,14);`
`// by ID`

`G=gap.SmallGroup(120,14);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-3,-5,20,61,26,168,2404]);`
`// Polycyclic`

`G:=Group<a,b,c|a^15=b^4=1,c^2=b^2,b*a*b^-1=a^11,c*a*c^-1=a^4,c*b*c^-1=b^-1>;`
`// generators/relations`

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