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## G = C22⋊D36order 288 = 25·32

### The semidirect product of C22 and D36 acting via D36/C12=S3

Aliases: C22⋊D36, C12.2S4, C23.3D18, C3.(C4⋊S4), C4⋊(C3.S4), (C2×C6).D12, C3.A41D4, C6.18(C2×S4), (C22×C4)⋊2D9, (C22×C12).3S3, (C22×C6).15D6, (C2×C3.S4)⋊1C2, (C4×C3.A4)⋊1C2, C2.4(C2×C3.S4), (C2×C3.A4).3C22, SmallGroup(288,334)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×C3.A4 — C22⋊D36
 Chief series C1 — C22 — C2×C6 — C3.A4 — C2×C3.A4 — C2×C3.S4 — C22⋊D36
 Lower central C3.A4 — C2×C3.A4 — C22⋊D36
 Upper central C1 — C2 — C4

Generators and relations for C22⋊D36
G = < a,b,c,d | a2=b2=c36=d2=1, dad=cbc-1=ab=ba, cac-1=b, bd=db, dcd=c-1 >

Subgroups: 668 in 96 conjugacy classes, 18 normal (16 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×3], C22, C22 [×8], S3 [×2], C6, C6 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C9, Dic3 [×2], C12, C12, D6 [×6], C2×C6, C2×C6 [×2], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], D9 [×2], C18, D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C22×S3 [×2], C22×C6, C4⋊D4, C36, C3.A4, D18 [×2], C4⋊Dic3, D6⋊C4 [×2], C2×D12, C2×C3⋊D4 [×2], C22×C12, D36, C3.S4 [×2], C2×C3.A4, C127D4, C4×C3.A4, C2×C3.S4 [×2], C22⋊D36
Quotients: C1, C2 [×3], C22, S3, D4, D6, D9, D12, S4, D18, C2×S4, D36, C3.S4, C4⋊S4, C2×C3.S4, C22⋊D36

Character table of C22⋊D36

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 9A 9B 9C 12A 12B 12C 12D 18A 18B 18C 36A 36B 36C 36D 36E 36F size 1 1 3 3 36 36 2 2 6 36 36 2 6 6 8 8 8 2 2 6 6 8 8 8 8 8 8 8 8 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 2 2 2 2 0 0 2 2 2 0 0 2 2 2 -1 -1 -1 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 2 2 0 0 2 -2 -2 0 0 2 2 2 -1 -1 -1 -2 -2 -2 -2 -1 -1 -1 1 1 1 1 1 1 orthogonal lifted from D6 ρ7 2 -2 2 -2 0 0 2 0 0 0 0 -2 -2 2 2 2 2 0 0 0 0 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ8 2 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 1 1 1 1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ97-ζ92 orthogonal lifted from D18 ρ9 2 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 1 1 1 1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ98-ζ9 orthogonal lifted from D18 ρ10 2 2 2 2 0 0 -1 2 2 0 0 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -1 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 orthogonal lifted from D9 ρ11 2 2 2 2 0 0 -1 2 2 0 0 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -1 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 orthogonal lifted from D9 ρ12 2 2 2 2 0 0 -1 2 2 0 0 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -1 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 orthogonal lifted from D9 ρ13 2 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 1 1 1 1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ95-ζ94 orthogonal lifted from D18 ρ14 2 -2 2 -2 0 0 2 0 0 0 0 -2 -2 2 -1 -1 -1 0 0 0 0 1 1 1 -√3 -√3 √3 √3 √3 -√3 orthogonal lifted from D12 ρ15 2 -2 2 -2 0 0 2 0 0 0 0 -2 -2 2 -1 -1 -1 0 0 0 0 1 1 1 √3 √3 -√3 -√3 -√3 √3 orthogonal lifted from D12 ρ16 2 -2 2 -2 0 0 -1 0 0 0 0 1 1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 √3 -√3 √3 -√3 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 ζ4ζ97-ζ4ζ92 ζ43ζ98-ζ43ζ9 -ζ4ζ97+ζ4ζ92 ζ4ζ95-ζ4ζ94 -ζ43ζ98+ζ43ζ9 -ζ4ζ95+ζ4ζ94 orthogonal lifted from D36 ρ17 2 -2 2 -2 0 0 -1 0 0 0 0 1 1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -√3 √3 -√3 √3 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ4ζ97+ζ4ζ92 -ζ43ζ98+ζ43ζ9 ζ4ζ97-ζ4ζ92 -ζ4ζ95+ζ4ζ94 ζ43ζ98-ζ43ζ9 ζ4ζ95-ζ4ζ94 orthogonal lifted from D36 ρ18 2 -2 2 -2 0 0 -1 0 0 0 0 1 1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 √3 -√3 √3 -√3 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ4ζ95+ζ4ζ94 ζ4ζ97-ζ4ζ92 ζ4ζ95-ζ4ζ94 -ζ43ζ98+ζ43ζ9 -ζ4ζ97+ζ4ζ92 ζ43ζ98-ζ43ζ9 orthogonal lifted from D36 ρ19 2 -2 2 -2 0 0 -1 0 0 0 0 1 1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 √3 -√3 √3 -√3 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 ζ43ζ98-ζ43ζ9 -ζ4ζ95+ζ4ζ94 -ζ43ζ98+ζ43ζ9 -ζ4ζ97+ζ4ζ92 ζ4ζ95-ζ4ζ94 ζ4ζ97-ζ4ζ92 orthogonal lifted from D36 ρ20 2 -2 2 -2 0 0 -1 0 0 0 0 1 1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -√3 √3 -√3 √3 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ43ζ98+ζ43ζ9 ζ4ζ95-ζ4ζ94 ζ43ζ98-ζ43ζ9 ζ4ζ97-ζ4ζ92 -ζ4ζ95+ζ4ζ94 -ζ4ζ97+ζ4ζ92 orthogonal lifted from D36 ρ21 2 -2 2 -2 0 0 -1 0 0 0 0 1 1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -√3 √3 -√3 √3 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 ζ4ζ95-ζ4ζ94 -ζ4ζ97+ζ4ζ92 -ζ4ζ95+ζ4ζ94 ζ43ζ98-ζ43ζ9 ζ4ζ97-ζ4ζ92 -ζ43ζ98+ζ43ζ9 orthogonal lifted from D36 ρ22 3 3 -1 -1 -1 1 3 -3 1 1 -1 3 -1 -1 0 0 0 -3 -3 1 1 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ23 3 3 -1 -1 1 1 3 3 -1 -1 -1 3 -1 -1 0 0 0 3 3 -1 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ24 3 3 -1 -1 -1 -1 3 3 -1 1 1 3 -1 -1 0 0 0 3 3 -1 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ25 3 3 -1 -1 1 -1 3 -3 1 -1 1 3 -1 -1 0 0 0 -3 -3 1 1 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ26 6 -6 -2 2 0 0 6 0 0 0 0 -6 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4⋊S4 ρ27 6 6 -2 -2 0 0 -3 6 -2 0 0 -3 1 1 0 0 0 -3 -3 1 1 0 0 0 0 0 0 0 0 0 orthogonal lifted from C3.S4 ρ28 6 6 -2 -2 0 0 -3 -6 2 0 0 -3 1 1 0 0 0 3 3 -1 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C3.S4 ρ29 6 -6 -2 2 0 0 -3 0 0 0 0 3 -1 1 0 0 0 -3√3 3√3 √3 -√3 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ30 6 -6 -2 2 0 0 -3 0 0 0 0 3 -1 1 0 0 0 3√3 -3√3 -√3 √3 0 0 0 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of C22⋊D36
On 36 points
Generators in S36
```(1 19)(2 20)(4 22)(5 23)(7 25)(8 26)(10 28)(11 29)(13 31)(14 32)(16 34)(17 35)
(1 19)(3 21)(4 22)(6 24)(7 25)(9 27)(10 28)(12 30)(13 31)(15 33)(16 34)(18 36)
(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)
(1 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 15)(28 36)(29 35)(30 34)(31 33)```

`G:=sub<Sym(36)| (1,19)(2,20)(4,22)(5,23)(7,25)(8,26)(10,28)(11,29)(13,31)(14,32)(16,34)(17,35), (1,19)(3,21)(4,22)(6,24)(7,25)(9,27)(10,28)(12,30)(13,31)(15,33)(16,34)(18,36), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(28,36)(29,35)(30,34)(31,33)>;`

`G:=Group( (1,19)(2,20)(4,22)(5,23)(7,25)(8,26)(10,28)(11,29)(13,31)(14,32)(16,34)(17,35), (1,19)(3,21)(4,22)(6,24)(7,25)(9,27)(10,28)(12,30)(13,31)(15,33)(16,34)(18,36), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(28,36)(29,35)(30,34)(31,33) );`

`G=PermutationGroup([(1,19),(2,20),(4,22),(5,23),(7,25),(8,26),(10,28),(11,29),(13,31),(14,32),(16,34),(17,35)], [(1,19),(3,21),(4,22),(6,24),(7,25),(9,27),(10,28),(12,30),(13,31),(15,33),(16,34),(18,36)], [(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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16),(13,15),(28,36),(29,35),(30,34),(31,33)])`

Matrix representation of C22⋊D36 in GL5(𝔽37)

 1 0 0 0 0 0 1 0 0 0 0 0 0 36 1 0 0 0 36 0 0 0 1 36 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 36 0 1 0 0 36 1 0
,
 27 7 0 0 0 30 27 0 0 0 0 0 36 1 0 0 0 36 0 0 0 0 36 0 1
,
 14 29 0 0 0 29 23 0 0 0 0 0 36 0 0 0 0 36 1 0 0 0 36 0 1

`G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,36,36,36,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,36,36,36,0,0,0,0,1,0,0,0,1,0],[27,30,0,0,0,7,27,0,0,0,0,0,36,36,36,0,0,1,0,0,0,0,0,0,1],[14,29,0,0,0,29,23,0,0,0,0,0,36,36,36,0,0,0,1,0,0,0,0,0,1] >;`

C22⋊D36 in GAP, Magma, Sage, TeX

`C_2^2\rtimes D_{36}`
`% in TeX`

`G:=Group("C2^2:D36");`
`// GroupNames label`

`G:=SmallGroup(288,334);`
`// by ID`

`G=gap.SmallGroup(288,334);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,36,1123,192,1684,6053,782,3534,1350]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^2=b^2=c^36=d^2=1,d*a*d=c*b*c^-1=a*b=b*a,c*a*c^-1=b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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