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## G = D30.C2order 120 = 23·3·5

### The non-split extension by D30 of C2 acting faithfully

Aliases: D30.C2, D152C4, C10.3D6, C6.3D10, Dic3Dic5, Dic52S3, Dic32D5, C30.3C22, C52(C4×S3), C31(C4×D5), C156(C2×C4), C2.3(S3×D5), (C5×Dic3)⋊2C2, (C3×Dic5)⋊2C2, SmallGroup(120,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — D30.C2
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — D30.C2
 Lower central C15 — D30.C2
 Upper central C1 — C2

Generators and relations for D30.C2
G = < a,b,c | a30=b2=1, c2=a15, bab=a-1, cac-1=a19, cbc-1=a18b >

Character table of D30.C2

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 6 10A 10B 12A 12B 15A 15B 20A 20B 20C 20D 30A 30B size 1 1 15 15 2 3 3 5 5 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 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 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 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 linear of order 2 ρ5 1 -1 1 -1 1 -i i -i i 1 1 -1 -1 -1 -i i 1 1 -i i i -i -1 -1 linear of order 4 ρ6 1 -1 -1 1 1 -i i i -i 1 1 -1 -1 -1 i -i 1 1 -i i i -i -1 -1 linear of order 4 ρ7 1 -1 1 -1 1 i -i i -i 1 1 -1 -1 -1 i -i 1 1 i -i -i i -1 -1 linear of order 4 ρ8 1 -1 -1 1 1 i -i -i i 1 1 -1 -1 -1 -i i 1 1 i -i -i i -1 -1 linear of order 4 ρ9 2 2 0 0 -1 0 0 -2 -2 2 2 -1 2 2 1 1 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from D6 ρ10 2 2 0 0 -1 0 0 2 2 2 2 -1 2 2 -1 -1 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ11 2 2 0 0 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 ρ12 2 2 0 0 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 ρ13 2 2 0 0 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 ρ14 2 2 0 0 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 ρ15 2 -2 0 0 -1 0 0 -2i 2i 2 2 1 -2 -2 i -i -1 -1 0 0 0 0 1 1 complex lifted from C4×S3 ρ16 2 -2 0 0 -1 0 0 2i -2i 2 2 1 -2 -2 -i i -1 -1 0 0 0 0 1 1 complex lifted from C4×S3 ρ17 2 -2 0 0 2 2i -2i 0 0 -1+√5/2 -1-√5/2 -2 1+√5/2 1-√5/2 0 0 -1-√5/2 -1+√5/2 ζ4ζ54+ζ4ζ5 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ4ζ53+ζ4ζ52 1-√5/2 1+√5/2 complex lifted from C4×D5 ρ18 2 -2 0 0 2 -2i 2i 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 ζ4ζ54+ζ4ζ5 ζ43ζ53+ζ43ζ52 1-√5/2 1+√5/2 complex lifted from C4×D5 ρ19 2 -2 0 0 2 -2i 2i 0 0 -1-√5/2 -1+√5/2 -2 1-√5/2 1+√5/2 0 0 -1+√5/2 -1-√5/2 ζ43ζ53+ζ43ζ52 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ43ζ54+ζ43ζ5 1+√5/2 1-√5/2 complex lifted from C4×D5 ρ20 2 -2 0 0 2 2i -2i 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 ζ43ζ53+ζ43ζ52 ζ4ζ54+ζ4ζ5 1+√5/2 1-√5/2 complex lifted from C4×D5 ρ21 4 4 0 0 -2 0 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 ρ22 4 -4 0 0 -2 0 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 faithful ρ23 4 -4 0 0 -2 0 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 faithful ρ24 4 4 0 0 -2 0 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

Smallest permutation representation of D30.C2
On 60 points
Generators in S60
```(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)
(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 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 60)(52 59)(53 58)(54 57)(55 56)
(1 41 16 56)(2 60 17 45)(3 49 18 34)(4 38 19 53)(5 57 20 42)(6 46 21 31)(7 35 22 50)(8 54 23 39)(9 43 24 58)(10 32 25 47)(11 51 26 36)(12 40 27 55)(13 59 28 44)(14 48 29 33)(15 37 30 52)```

`G:=sub<Sym(60)| (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), (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)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,60)(52,59)(53,58)(54,57)(55,56), (1,41,16,56)(2,60,17,45)(3,49,18,34)(4,38,19,53)(5,57,20,42)(6,46,21,31)(7,35,22,50)(8,54,23,39)(9,43,24,58)(10,32,25,47)(11,51,26,36)(12,40,27,55)(13,59,28,44)(14,48,29,33)(15,37,30,52)>;`

`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), (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)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,60)(52,59)(53,58)(54,57)(55,56), (1,41,16,56)(2,60,17,45)(3,49,18,34)(4,38,19,53)(5,57,20,42)(6,46,21,31)(7,35,22,50)(8,54,23,39)(9,43,24,58)(10,32,25,47)(11,51,26,36)(12,40,27,55)(13,59,28,44)(14,48,29,33)(15,37,30,52) );`

`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)], [(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),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(51,60),(52,59),(53,58),(54,57),(55,56)], [(1,41,16,56),(2,60,17,45),(3,49,18,34),(4,38,19,53),(5,57,20,42),(6,46,21,31),(7,35,22,50),(8,54,23,39),(9,43,24,58),(10,32,25,47),(11,51,26,36),(12,40,27,55),(13,59,28,44),(14,48,29,33),(15,37,30,52)]])`

D30.C2 is a maximal subgroup of
D15⋊C8  Dic3.F5  D60⋊C2  D15⋊Q8  C12.28D10  C4×S3×D5  Dic5.D6  Dic3.D10  D10⋊D6  D90.C2  C30.D6  C6.D30  D30.S3  Dic5.7S4  Dic52S4
D30.C2 is a maximal quotient of
D152C8  D30.5C4  Dic3×Dic5  D304C4  Dic155C4  D90.C2  C30.D6  C6.D30  D30.S3  Dic52S4

Matrix representation of D30.C2 in GL4(𝔽61) generated by

 0 60 0 0 1 1 0 0 0 0 0 60 0 0 1 17
,
 0 60 0 0 60 0 0 0 0 0 44 60 0 0 44 17
,
 50 0 0 0 0 50 0 0 0 0 44 17 0 0 1 17
`G:=sub<GL(4,GF(61))| [0,1,0,0,60,1,0,0,0,0,0,1,0,0,60,17],[0,60,0,0,60,0,0,0,0,0,44,44,0,0,60,17],[50,0,0,0,0,50,0,0,0,0,44,1,0,0,17,17] >;`

D30.C2 in GAP, Magma, Sage, TeX

`D_{30}.C_2`
`% in TeX`

`G:=Group("D30.C2");`
`// GroupNames label`

`G:=SmallGroup(120,10);`
`// by ID`

`G=gap.SmallGroup(120,10);`
`# by ID`

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

`G:=Group<a,b,c|a^30=b^2=1,c^2=a^15,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^18*b>;`
`// generators/relations`

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