non-abelian, soluble, monomial, A-group
Aliases: C72⋊S3, C72⋊3C3⋊1C2, SmallGroup(294,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C72⋊3C3 — C72⋊S3 |
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 >
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 |
►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
| action | f(x) | Disc(f) |
|---|---|---|
| 14T15 | x14-105x12-147x11+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
Export
Subgroup lattice of C72⋊S3 in TeX
Character table of C72⋊S3 in TeX