Copied to
clipboard

## G = C62.C32order 324 = 22·34

### 4th non-split extension by C62 of C32 acting faithfully

Aliases: C62.4C32, 3- 1+2.A4, (C2×C6).8He3, C32.4(C3×A4), C3.9(C32⋊A4), C32.A4.2C3, C222(C3.He3), (C22×3- 1+2).1C3, (C3×C3.A4).2C3, SmallGroup(324,56)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.C32
 Chief series C1 — C22 — C2×C6 — C62 — C3×C3.A4 — C62.C32
 Lower central C22 — C2×C6 — C62 — C62.C32
 Upper central C1 — C3 — C32 — 3- 1+2

Generators and relations for C62.C32
G = < a,b,c,d | a6=b6=1, c3=d3=b2, ab=ba, cac-1=ab3, dad-1=ab2, cbc-1=a3b4, bd=db, dcd-1=a2b4c >

Character table of C62.C32

 class 1 2 3A 3B 3C 3D 6A 6B 6C 6D 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 9K 9L 18A 18B 18C 18D 18E 18F size 1 3 1 1 3 3 3 3 9 9 9 9 12 12 12 12 12 12 36 36 36 36 9 9 9 9 9 9 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 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 linear of order 3 ρ3 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 linear of order 3 ρ5 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ6 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 1 ζ32 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ7 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 1 ζ3 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ8 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ9 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ10 3 -1 3 3 3 3 -1 -1 -1 -1 3 3 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ11 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 -1 -1 ζ65 ζ6 0 0 0 0 0 0 0 0 0 0 0 0 -1-√-3 -1+√-3 2 2 -1+√-3 -1-√-3 complex lifted from C32⋊A4 ρ12 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 -1 -1 ζ65 ζ6 0 0 0 0 0 0 0 0 0 0 0 0 2 2 -1-√-3 -1+√-3 -1-√-3 -1+√-3 complex lifted from C32⋊A4 ρ13 3 -1 3 3 3 3 -1 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 ζ65 ζ6 ζ6 ζ65 ζ65 ζ6 complex lifted from C3×A4 ρ14 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 -1 -1 ζ6 ζ65 0 0 0 0 0 0 0 0 0 0 0 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 2 2 complex lifted from C32⋊A4 ρ15 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 -1 -1 ζ6 ζ65 0 0 0 0 0 0 0 0 0 0 0 0 2 2 -1+√-3 -1-√-3 -1+√-3 -1-√-3 complex lifted from C32⋊A4 ρ16 3 3 3 3 -3+3√-3/2 -3-3√-3/2 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3 ρ17 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 -1 -1 ζ65 ζ6 0 0 0 0 0 0 0 0 0 0 0 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 2 2 complex lifted from C32⋊A4 ρ18 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 -1 -1 ζ6 ζ65 0 0 0 0 0 0 0 0 0 0 0 0 -1+√-3 -1-√-3 2 2 -1-√-3 -1+√-3 complex lifted from C32⋊A4 ρ19 3 3 3 3 -3-3√-3/2 -3+3√-3/2 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3 ρ20 3 -1 3 3 3 3 -1 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 ζ6 ζ65 ζ65 ζ6 ζ6 ζ65 complex lifted from C3×A4 ρ21 3 3 -3+3√-3/2 -3-3√-3/2 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ95+2ζ92 2ζ97+ζ94 ζ97+2ζ9 2ζ94+ζ9 ζ98+2ζ95 2ζ98+ζ92 0 0 0 0 0 0 0 0 0 0 complex lifted from C3.He3 ρ22 3 3 -3-3√-3/2 -3+3√-3/2 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ97+2ζ9 2ζ98+ζ92 ζ98+2ζ95 ζ95+2ζ92 2ζ97+ζ94 2ζ94+ζ9 0 0 0 0 0 0 0 0 0 0 complex lifted from C3.He3 ρ23 3 3 -3+3√-3/2 -3-3√-3/2 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ98+2ζ95 2ζ94+ζ9 2ζ97+ζ94 ζ97+2ζ9 2ζ98+ζ92 ζ95+2ζ92 0 0 0 0 0 0 0 0 0 0 complex lifted from C3.He3 ρ24 3 3 -3+3√-3/2 -3-3√-3/2 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 2ζ98+ζ92 ζ97+2ζ9 2ζ94+ζ9 2ζ97+ζ94 ζ95+2ζ92 ζ98+2ζ95 0 0 0 0 0 0 0 0 0 0 complex lifted from C3.He3 ρ25 3 3 -3-3√-3/2 -3+3√-3/2 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 2ζ94+ζ9 ζ98+2ζ95 ζ95+2ζ92 2ζ98+ζ92 ζ97+2ζ9 2ζ97+ζ94 0 0 0 0 0 0 0 0 0 0 complex lifted from C3.He3 ρ26 3 3 -3-3√-3/2 -3+3√-3/2 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 2ζ97+ζ94 ζ95+2ζ92 2ζ98+ζ92 ζ98+2ζ95 2ζ94+ζ9 ζ97+2ζ9 0 0 0 0 0 0 0 0 0 0 complex lifted from C3.He3 ρ27 9 -3 -9-9√-3/2 -9+9√-3/2 0 0 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ28 9 -3 -9+9√-3/2 -9-9√-3/2 0 0 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C62.C32
On 54 points
Generators in S54
```(2 14)(3 15)(5 17)(6 18)(8 11)(9 12)(19 48 22 51 25 54)(20 49 23 52 26 46)(21 27 24)(28 31 34)(29 38 35 44 32 41)(30 39 36 45 33 42)(37 40 43)(47 53 50)
(1 10 4 13 7 16)(2 8 5)(3 12 6 15 9 18)(11 17 14)(19 25 22)(20 49 23 52 26 46)(21 50 24 53 27 47)(28 40 31 43 34 37)(29 35 32)(30 42 33 45 36 39)(38 44 41)(48 54 51)
(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)
(1 28 27 4 31 21 7 34 24)(2 32 19 5 35 22 8 29 25)(3 36 20 6 30 23 9 33 26)(10 40 47 13 43 50 16 37 53)(11 44 48 14 38 51 17 41 54)(12 39 49 15 42 52 18 45 46)```

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

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

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

Matrix representation of C62.C32 in GL6(𝔽19)

 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 3 7 0 0 0 0 10 3 11
,
 18 0 0 0 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 13 13 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 2 0 0 0 0 8 17 5 0 0 0 1 0 1

`G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,3,10,0,0,0,0,7,3,0,0,0,0,0,11],[18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,13,0,0,0,0,5,13,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,8,1,0,0,0,2,17,0,0,0,0,0,5,1] >;`

C62.C32 in GAP, Magma, Sage, TeX

`C_6^2.C_3^2`
`% in TeX`

`G:=Group("C6^2.C3^2");`
`// GroupNames label`

`G:=SmallGroup(324,56);`
`// by ID`

`G=gap.SmallGroup(324,56);`
`# by ID`

`G:=PCGroup([6,-3,-3,-3,-3,-2,2,162,145,115,224,4864,8753]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^3=d^3=b^2,a*b=b*a,c*a*c^-1=a*b^3,d*a*d^-1=a*b^2,c*b*c^-1=a^3*b^4,b*d=d*b,d*c*d^-1=a^2*b^4*c>;`
`// generators/relations`

Export

׿
×
𝔽