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## G = C62.C22order 144 = 24·32

### 5th non-split extension by C62 of C22 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.C22
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C62.C22
 Lower central C32 — C3×C6 — C62.C22
 Upper central C1 — C22

Generators and relations for C62.C22
G = < a,b,c,d | a6=b6=1, c2=d2=a3, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 160 in 60 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C3×Dic3, C3⋊Dic3, C62, Dic3⋊C4, C6×Dic3, C2×C3⋊Dic3, C62.C22
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, Dic6, C4×S3, C3⋊D4, S32, Dic3⋊C4, C6.D6, D6⋊S3, C322Q8, C62.C22

Character table of C62.C22

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 1 1 2 2 4 6 6 6 6 18 18 2 2 2 2 2 2 4 4 4 6 6 6 6 6 6 6 6 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 i -i -i i -1 1 -1 -1 1 -1 -1 1 -1 1 -1 i -i -i i i i -i -i linear of order 4 ρ6 1 -1 1 -1 1 1 1 i i -i -i 1 -1 -1 -1 1 -1 -1 1 -1 1 -1 -i -i -i -i i i i i linear of order 4 ρ7 1 -1 1 -1 1 1 1 -i -i i i 1 -1 -1 -1 1 -1 -1 1 -1 1 -1 i i i i -i -i -i -i linear of order 4 ρ8 1 -1 1 -1 1 1 1 -i i i -i -1 1 -1 -1 1 -1 -1 1 -1 1 -1 -i i i -i -i -i i i linear of order 4 ρ9 2 2 2 2 2 -1 -1 -2 0 -2 0 0 0 2 -1 2 2 -1 -1 -1 -1 -1 0 1 1 0 1 1 0 0 orthogonal lifted from D6 ρ10 2 2 -2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -1 2 -1 0 2 0 2 0 0 -1 2 -1 -1 2 2 -1 -1 -1 -1 0 0 -1 0 0 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 2 -1 2 -1 0 -2 0 -2 0 0 -1 2 -1 -1 2 2 -1 -1 -1 1 0 0 1 0 0 1 1 orthogonal lifted from D6 ρ13 2 2 2 2 2 -1 -1 2 0 2 0 0 0 2 -1 2 2 -1 -1 -1 -1 -1 0 -1 -1 0 -1 -1 0 0 orthogonal lifted from S3 ρ14 2 -2 -2 2 2 2 2 0 0 0 0 0 0 2 2 -2 -2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ15 2 -2 -2 2 2 -1 -1 0 0 0 0 0 0 2 -1 -2 -2 1 1 -1 1 1 0 -√3 √3 0 √3 -√3 0 0 symplectic lifted from Dic6, Schur index 2 ρ16 2 -2 -2 2 -1 2 -1 0 0 0 0 0 0 -1 2 1 1 -2 -2 -1 1 1 -√3 0 0 √3 0 0 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ17 2 -2 -2 2 2 -1 -1 0 0 0 0 0 0 2 -1 -2 -2 1 1 -1 1 1 0 √3 -√3 0 -√3 √3 0 0 symplectic lifted from Dic6, Schur index 2 ρ18 2 -2 -2 2 -1 2 -1 0 0 0 0 0 0 -1 2 1 1 -2 -2 -1 1 1 √3 0 0 -√3 0 0 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ19 2 2 -2 -2 2 -1 -1 0 0 0 0 0 0 -2 1 -2 2 -1 1 1 1 -1 0 -√-3 √-3 0 -√-3 √-3 0 0 complex lifted from C3⋊D4 ρ20 2 2 -2 -2 2 -1 -1 0 0 0 0 0 0 -2 1 -2 2 -1 1 1 1 -1 0 √-3 -√-3 0 √-3 -√-3 0 0 complex lifted from C3⋊D4 ρ21 2 2 -2 -2 -1 2 -1 0 0 0 0 0 0 1 -2 1 -1 2 -2 1 1 -1 -√-3 0 0 √-3 0 0 -√-3 √-3 complex lifted from C3⋊D4 ρ22 2 2 -2 -2 -1 2 -1 0 0 0 0 0 0 1 -2 1 -1 2 -2 1 1 -1 √-3 0 0 -√-3 0 0 √-3 -√-3 complex lifted from C3⋊D4 ρ23 2 -2 2 -2 -1 2 -1 0 -2i 0 2i 0 0 1 -2 -1 1 -2 2 1 -1 1 -i 0 0 -i 0 0 i i complex lifted from C4×S3 ρ24 2 -2 2 -2 2 -1 -1 -2i 0 2i 0 0 0 -2 1 2 -2 1 -1 1 -1 1 0 -i -i 0 i i 0 0 complex lifted from C4×S3 ρ25 2 -2 2 -2 2 -1 -1 2i 0 -2i 0 0 0 -2 1 2 -2 1 -1 1 -1 1 0 i i 0 -i -i 0 0 complex lifted from C4×S3 ρ26 2 -2 2 -2 -1 2 -1 0 2i 0 -2i 0 0 1 -2 -1 1 -2 2 1 -1 1 i 0 0 i 0 0 -i -i complex lifted from C4×S3 ρ27 4 4 4 4 -2 -2 1 0 0 0 0 0 0 -2 -2 -2 -2 -2 -2 1 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ28 4 -4 4 -4 -2 -2 1 0 0 0 0 0 0 2 2 -2 2 2 -2 -1 1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C6.D6 ρ29 4 4 -4 -4 -2 -2 1 0 0 0 0 0 0 2 2 2 -2 -2 2 -1 -1 1 0 0 0 0 0 0 0 0 symplectic lifted from D6⋊S3, Schur index 2 ρ30 4 -4 -4 4 -2 -2 1 0 0 0 0 0 0 -2 -2 2 2 2 2 1 -1 -1 0 0 0 0 0 0 0 0 symplectic lifted from C32⋊2Q8, Schur index 2

Smallest permutation representation of C62.C22
On 48 points
Generators in S48
```(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)
(1 17 5 15 3 13)(2 18 6 16 4 14)(7 47 11 45 9 43)(8 48 12 46 10 44)(19 29 21 25 23 27)(20 30 22 26 24 28)(31 38 33 40 35 42)(32 39 34 41 36 37)
(1 33 4 36)(2 34 5 31)(3 35 6 32)(7 27 10 30)(8 28 11 25)(9 29 12 26)(13 40 16 37)(14 41 17 38)(15 42 18 39)(19 46 22 43)(20 47 23 44)(21 48 24 45)
(1 21 4 24)(2 20 5 23)(3 19 6 22)(7 33 10 36)(8 32 11 35)(9 31 12 34)(13 29 16 26)(14 28 17 25)(15 27 18 30)(37 47 40 44)(38 46 41 43)(39 45 42 48)```

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

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

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

Matrix representation of C62.C22 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 7 3 0 0 0 0 10 6 0 0 0 0 0 0 8 0 0 0 0 0 5 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,10,0,0,0,0,3,6,0,0,0,0,0,0,8,5,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C62.C22 in GAP, Magma, Sage, TeX

`C_6^2.C_2^2`
`% in TeX`

`G:=Group("C6^2.C2^2");`
`// GroupNames label`

`G:=SmallGroup(144,67);`
`// by ID`

`G=gap.SmallGroup(144,67);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,31,490,3461]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^2=d^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;`
`// generators/relations`

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