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G = C72⋊S3order 294 = 2·3·72

The semidirect product of C72 and S3 acting faithfully

Aliases: C72⋊S3, C723C31C2, SmallGroup(294,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C72⋊3C3 — C72⋊S3
 Chief series C1 — C72 — C72⋊3C3 — C72⋊S3
 Lower central C72⋊3C3 — C72⋊S3
 Upper central C1

Generators and relations for C72⋊S3
G = < a,b,c,d | a7=b7=c3=d2=1, ab=ba, cac-1=a2, dad=b, cbc-1=b4, dbd=a, dcd=c-1 >

21C2
49C3
2C7
3C7
3C7
49S3
3D7
21C14
14C7⋊C3

Character table of C72⋊S3

 class 1 2 3 7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K 14A 14B 14C 14D 14E 14F size 1 21 98 3 3 3 3 3 3 6 6 6 6 6 21 21 21 21 21 21 ρ1 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 linear of order 2 ρ3 2 0 -1 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 orthogonal lifted from S3 ρ4 3 1 0 2ζ73+ζ7 2ζ75+ζ74 ζ76+2ζ74 2ζ76+ζ72 ζ73+2ζ72 ζ75+2ζ7 -1+√-7/2 ζ75+ζ72+1 -1-√-7/2 ζ74+ζ73+1 ζ76+ζ7+1 ζ74 ζ76 ζ72 ζ75 ζ7 ζ73 complex faithful ρ5 3 -1 0 ζ76+2ζ74 ζ73+2ζ72 2ζ73+ζ7 ζ75+2ζ7 2ζ75+ζ74 2ζ76+ζ72 -1-√-7/2 ζ75+ζ72+1 -1+√-7/2 ζ74+ζ73+1 ζ76+ζ7+1 -ζ73 -ζ7 -ζ75 -ζ72 -ζ76 -ζ74 complex faithful ρ6 3 1 0 2ζ76+ζ72 2ζ73+ζ7 ζ75+2ζ7 2ζ75+ζ74 ζ76+2ζ74 ζ73+2ζ72 -1+√-7/2 ζ74+ζ73+1 -1-√-7/2 ζ76+ζ7+1 ζ75+ζ72+1 ζ7 ζ75 ζ74 ζ73 ζ72 ζ76 complex faithful ρ7 3 1 0 2ζ75+ζ74 2ζ76+ζ72 ζ73+2ζ72 2ζ73+ζ7 ζ75+2ζ7 ζ76+2ζ74 -1+√-7/2 ζ76+ζ7+1 -1-√-7/2 ζ75+ζ72+1 ζ74+ζ73+1 ζ72 ζ73 ζ7 ζ76 ζ74 ζ75 complex faithful ρ8 3 -1 0 2ζ73+ζ7 2ζ75+ζ74 ζ76+2ζ74 2ζ76+ζ72 ζ73+2ζ72 ζ75+2ζ7 -1+√-7/2 ζ75+ζ72+1 -1-√-7/2 ζ74+ζ73+1 ζ76+ζ7+1 -ζ74 -ζ76 -ζ72 -ζ75 -ζ7 -ζ73 complex faithful ρ9 3 -1 0 2ζ75+ζ74 2ζ76+ζ72 ζ73+2ζ72 2ζ73+ζ7 ζ75+2ζ7 ζ76+2ζ74 -1+√-7/2 ζ76+ζ7+1 -1-√-7/2 ζ75+ζ72+1 ζ74+ζ73+1 -ζ72 -ζ73 -ζ7 -ζ76 -ζ74 -ζ75 complex faithful ρ10 3 1 0 ζ73+2ζ72 ζ75+2ζ7 2ζ75+ζ74 ζ76+2ζ74 2ζ76+ζ72 2ζ73+ζ7 -1-√-7/2 ζ76+ζ7+1 -1+√-7/2 ζ75+ζ72+1 ζ74+ζ73+1 ζ75 ζ74 ζ76 ζ7 ζ73 ζ72 complex faithful ρ11 3 -1 0 2ζ76+ζ72 2ζ73+ζ7 ζ75+2ζ7 2ζ75+ζ74 ζ76+2ζ74 ζ73+2ζ72 -1+√-7/2 ζ74+ζ73+1 -1-√-7/2 ζ76+ζ7+1 ζ75+ζ72+1 -ζ7 -ζ75 -ζ74 -ζ73 -ζ72 -ζ76 complex faithful ρ12 3 -1 0 ζ75+2ζ7 ζ76+2ζ74 2ζ76+ζ72 ζ73+2ζ72 2ζ73+ζ7 2ζ75+ζ74 -1-√-7/2 ζ74+ζ73+1 -1+√-7/2 ζ76+ζ7+1 ζ75+ζ72+1 -ζ76 -ζ72 -ζ73 -ζ74 -ζ75 -ζ7 complex faithful ρ13 3 1 0 ζ76+2ζ74 ζ73+2ζ72 2ζ73+ζ7 ζ75+2ζ7 2ζ75+ζ74 2ζ76+ζ72 -1-√-7/2 ζ75+ζ72+1 -1+√-7/2 ζ74+ζ73+1 ζ76+ζ7+1 ζ73 ζ7 ζ75 ζ72 ζ76 ζ74 complex faithful ρ14 3 -1 0 ζ73+2ζ72 ζ75+2ζ7 2ζ75+ζ74 ζ76+2ζ74 2ζ76+ζ72 2ζ73+ζ7 -1-√-7/2 ζ76+ζ7+1 -1+√-7/2 ζ75+ζ72+1 ζ74+ζ73+1 -ζ75 -ζ74 -ζ76 -ζ7 -ζ73 -ζ72 complex faithful ρ15 3 1 0 ζ75+2ζ7 ζ76+2ζ74 2ζ76+ζ72 ζ73+2ζ72 2ζ73+ζ7 2ζ75+ζ74 -1-√-7/2 ζ74+ζ73+1 -1+√-7/2 ζ76+ζ7+1 ζ75+ζ72+1 ζ76 ζ72 ζ73 ζ74 ζ75 ζ7 complex faithful ρ16 6 0 0 2ζ76+2ζ7+2 2ζ74+2ζ73+2 2ζ76+2ζ7+2 2ζ75+2ζ72+2 2ζ74+2ζ73+2 2ζ75+2ζ72+2 -1 2ζ76+ζ75+ζ72+2ζ7 -1 2ζ75+ζ74+ζ73+2ζ72 ζ76+2ζ74+2ζ73+ζ7 0 0 0 0 0 0 orthogonal faithful ρ17 6 0 0 2ζ75+2ζ72+2 2ζ76+2ζ7+2 2ζ75+2ζ72+2 2ζ74+2ζ73+2 2ζ76+2ζ7+2 2ζ74+2ζ73+2 -1 2ζ75+ζ74+ζ73+2ζ72 -1 ζ76+2ζ74+2ζ73+ζ7 2ζ76+ζ75+ζ72+2ζ7 0 0 0 0 0 0 orthogonal faithful ρ18 6 0 0 2ζ74+2ζ73+2 2ζ75+2ζ72+2 2ζ74+2ζ73+2 2ζ76+2ζ7+2 2ζ75+2ζ72+2 2ζ76+2ζ7+2 -1 ζ76+2ζ74+2ζ73+ζ7 -1 2ζ76+ζ75+ζ72+2ζ7 2ζ75+ζ74+ζ73+2ζ72 0 0 0 0 0 0 orthogonal faithful ρ19 6 0 0 -1+√-7 -1+√-7 -1-√-7 -1+√-7 -1-√-7 -1-√-7 5+√-7/2 -1 5-√-7/2 -1 -1 0 0 0 0 0 0 complex faithful ρ20 6 0 0 -1-√-7 -1-√-7 -1+√-7 -1-√-7 -1+√-7 -1+√-7 5-√-7/2 -1 5+√-7/2 -1 -1 0 0 0 0 0 0 complex faithful

Permutation representations of C72⋊S3
On 14 points - transitive group 14T15
Generators in S14
```(8 9 10 11 12 13 14)
(1 7 6 5 4 3 2)
(2 3 5)(4 7 6)(8 14 10)(9 11 12)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(7 14)```

`G:=sub<Sym(14)| (8,9,10,11,12,13,14), (1,7,6,5,4,3,2), (2,3,5)(4,7,6)(8,14,10)(9,11,12), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(7,14)>;`

`G:=Group( (8,9,10,11,12,13,14), (1,7,6,5,4,3,2), (2,3,5)(4,7,6)(8,14,10)(9,11,12), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(7,14) );`

`G=PermutationGroup([[(8,9,10,11,12,13,14)], [(1,7,6,5,4,3,2)], [(2,3,5),(4,7,6),(8,14,10),(9,11,12)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(7,14)]])`

`G:=TransitiveGroup(14,15);`

On 21 points - transitive group 21T17
Generators in S21
```(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(1 7 6 5 4 3 2)(8 13 11 9 14 12 10)(15 18 21 17 20 16 19)
(1 11 16)(2 8 18)(3 12 20)(4 9 15)(5 13 17)(6 10 19)(7 14 21)
(2 7)(3 6)(4 5)(8 21)(9 17)(10 20)(11 16)(12 19)(13 15)(14 18)```

`G:=sub<Sym(21)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,7,6,5,4,3,2)(8,13,11,9,14,12,10)(15,18,21,17,20,16,19), (1,11,16)(2,8,18)(3,12,20)(4,9,15)(5,13,17)(6,10,19)(7,14,21), (2,7)(3,6)(4,5)(8,21)(9,17)(10,20)(11,16)(12,19)(13,15)(14,18)>;`

`G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,7,6,5,4,3,2)(8,13,11,9,14,12,10)(15,18,21,17,20,16,19), (1,11,16)(2,8,18)(3,12,20)(4,9,15)(5,13,17)(6,10,19)(7,14,21), (2,7)(3,6)(4,5)(8,21)(9,17)(10,20)(11,16)(12,19)(13,15)(14,18) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,7,6,5,4,3,2),(8,13,11,9,14,12,10),(15,18,21,17,20,16,19)], [(1,11,16),(2,8,18),(3,12,20),(4,9,15),(5,13,17),(6,10,19),(7,14,21)], [(2,7),(3,6),(4,5),(8,21),(9,17),(10,20),(11,16),(12,19),(13,15),(14,18)]])`

`G:=TransitiveGroup(21,17);`

On 21 points - transitive group 21T18
Generators in S21
```(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(1 2 3 4 5 6 7)(8 10 12 14 9 11 13)(15 19 16 20 17 21 18)
(1 10 19)(2 14 21)(3 11 16)(4 8 18)(5 12 20)(6 9 15)(7 13 17)
(8 18)(9 15)(10 19)(11 16)(12 20)(13 17)(14 21)```

`G:=sub<Sym(21)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,2,3,4,5,6,7)(8,10,12,14,9,11,13)(15,19,16,20,17,21,18), (1,10,19)(2,14,21)(3,11,16)(4,8,18)(5,12,20)(6,9,15)(7,13,17), (8,18)(9,15)(10,19)(11,16)(12,20)(13,17)(14,21)>;`

`G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,2,3,4,5,6,7)(8,10,12,14,9,11,13)(15,19,16,20,17,21,18), (1,10,19)(2,14,21)(3,11,16)(4,8,18)(5,12,20)(6,9,15)(7,13,17), (8,18)(9,15)(10,19)(11,16)(12,20)(13,17)(14,21) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,2,3,4,5,6,7),(8,10,12,14,9,11,13),(15,19,16,20,17,21,18)], [(1,10,19),(2,14,21),(3,11,16),(4,8,18),(5,12,20),(6,9,15),(7,13,17)], [(8,18),(9,15),(10,19),(11,16),(12,20),(13,17),(14,21)]])`

`G:=TransitiveGroup(21,18);`

Polynomial with Galois group C72⋊S3 over ℚ
actionf(x)Disc(f)
14T15x14-105x12-147x11+5271x10+19838x9-94150x8-607634x7+570164x6+12260920x5+42847770x4+95169270x3+197804880x2+348280352x+336238208-224·724·317·1312·3172·47377272·49178134212·10963085927262635337784632

Matrix representation of C72⋊S3 in GL3(𝔽43) generated by

 20 27 6 15 26 6 34 7 21
,
 42 0 42 24 24 25 1 1 1
,
 7 28 8 15 8 30 33 16 28
,
 42 11 16 0 25 30 0 5 18
`G:=sub<GL(3,GF(43))| [20,15,34,27,26,7,6,6,21],[42,24,1,0,24,1,42,25,1],[7,15,33,28,8,16,8,30,28],[42,0,0,11,25,5,16,30,18] >;`

C72⋊S3 in GAP, Magma, Sage, TeX

`C_7^2\rtimes S_3`
`% in TeX`

`G:=Group("C7^2:S3");`
`// GroupNames label`

`G:=SmallGroup(294,7);`
`// by ID`

`G=gap.SmallGroup(294,7);`
`# by ID`

`G:=PCGroup([4,-2,-3,-7,7,33,506,78,99,1351]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^7=b^7=c^3=d^2=1,a*b=b*a,c*a*c^-1=a^2,d*a*d=b,c*b*c^-1=b^4,d*b*d=a,d*c*d=c^-1>;`
`// generators/relations`

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