metabelian, soluble, monomial, A-group
Aliases: C32⋊Dic7, C7⋊(C32⋊C4), C3⋊S3.D7, (C3×C21)⋊2C4, (C7×C3⋊S3).2C2, SmallGroup(252,32)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C21 — C32⋊Dic7 |
Generators and relations for C32⋊Dic7
G = < a,b,c,d | a3=b3=c14=1, d2=c7, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a-1b-1, dcd-1=c-1 >
Character table of C32⋊Dic7
| class | 1 | 2 | 3A | 3B | 4A | 4B | 7A | 7B | 7C | 14A | 14B | 14C | 21A | 21B | 21C | 21D | 21E | 21F | 21G | 21H | 21I | 21J | 21K | 21L | |
| size | 1 | 9 | 4 | 4 | 63 | 63 | 2 | 2 | 2 | 18 | 18 | 18 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | 1 | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | 1 | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ6 | 2 | 2 | 2 | 2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ7 | 2 | 2 | 2 | 2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ8 | 2 | -2 | 2 | 2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | symplectic lifted from Dic7, Schur index 2 |
| ρ9 | 2 | -2 | 2 | 2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | symplectic lifted from Dic7, Schur index 2 |
| ρ10 | 2 | -2 | 2 | 2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | symplectic lifted from Dic7, Schur index 2 |
| ρ11 | 4 | 0 | -2 | 1 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | 1 | -2 | -2 | -2 | -2 | -2 | 1 | 1 | 1 | 1 | -2 | 1 | orthogonal lifted from C32⋊C4 |
| ρ12 | 4 | 0 | 1 | -2 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | 1 | 1 | -2 | -2 | -2 | -2 | 1 | -2 | orthogonal lifted from C32⋊C4 |
| ρ13 | 4 | 0 | -2 | 1 | 0 | 0 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | 0 | 2ζ76-ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | 2ζ75-ζ72 | -ζ76+2ζ7 | 2ζ74-ζ73 | -ζ75+2ζ72 | -ζ74-ζ73 | -ζ74+2ζ73 | complex faithful |
| ρ14 | 4 | 0 | 1 | -2 | 0 | 0 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | 0 | -ζ75-ζ72 | 2ζ75-ζ72 | 2ζ74-ζ73 | -ζ74+2ζ73 | -ζ75+2ζ72 | -ζ76+2ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | 2ζ76-ζ7 | -ζ76-ζ7 | complex faithful |
| ρ15 | 4 | 0 | -2 | 1 | 0 | 0 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | 0 | 2ζ74-ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76+2ζ7 | -ζ74+2ζ73 | 2ζ75-ζ72 | 2ζ76-ζ7 | -ζ75-ζ72 | -ζ75+2ζ72 | complex faithful |
| ρ16 | 4 | 0 | -2 | 1 | 0 | 0 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | 0 | -ζ75+2ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | 2ζ74-ζ73 | 2ζ75-ζ72 | 2ζ76-ζ7 | -ζ74+2ζ73 | -ζ76-ζ7 | -ζ76+2ζ7 | complex faithful |
| ρ17 | 4 | 0 | 1 | -2 | 0 | 0 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ74+2ζ73 | -ζ76+2ζ7 | 2ζ76-ζ7 | 2ζ74-ζ73 | -ζ75+2ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | 2ζ75-ζ72 | -ζ75-ζ72 | complex faithful |
| ρ18 | 4 | 0 | 1 | -2 | 0 | 0 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | 0 | -ζ74-ζ73 | 2ζ74-ζ73 | 2ζ76-ζ7 | -ζ76+2ζ7 | -ζ74+2ζ73 | 2ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75+2ζ72 | -ζ75-ζ72 | complex faithful |
| ρ19 | 4 | 0 | 1 | -2 | 0 | 0 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ75+2ζ72 | -ζ74+2ζ73 | 2ζ74-ζ73 | 2ζ75-ζ72 | 2ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76+2ζ7 | -ζ76-ζ7 | complex faithful |
| ρ20 | 4 | 0 | 1 | -2 | 0 | 0 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | 0 | -ζ76-ζ7 | 2ζ76-ζ7 | -ζ75+2ζ72 | 2ζ75-ζ72 | -ζ76+2ζ7 | 2ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74+2ζ73 | -ζ74-ζ73 | complex faithful |
| ρ21 | 4 | 0 | -2 | 1 | 0 | 0 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | 0 | 2ζ75-ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74+2ζ73 | -ζ75+2ζ72 | -ζ76+2ζ7 | 2ζ74-ζ73 | -ζ76-ζ7 | 2ζ76-ζ7 | complex faithful |
| ρ22 | 4 | 0 | 1 | -2 | 0 | 0 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ76+2ζ7 | 2ζ75-ζ72 | -ζ75+2ζ72 | 2ζ76-ζ7 | -ζ74+2ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | 2ζ74-ζ73 | -ζ74-ζ73 | complex faithful |
| ρ23 | 4 | 0 | -2 | 1 | 0 | 0 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | 0 | -ζ76+2ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75+2ζ72 | 2ζ76-ζ7 | -ζ74+2ζ73 | 2ζ75-ζ72 | -ζ74-ζ73 | 2ζ74-ζ73 | complex faithful |
| ρ24 | 4 | 0 | -2 | 1 | 0 | 0 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | 0 | -ζ74+2ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | 2ζ76-ζ7 | 2ζ74-ζ73 | -ζ75+2ζ72 | -ζ76+2ζ7 | -ζ75-ζ72 | 2ζ75-ζ72 | complex faithful |
►On 42 points
Generators in S42
(8 42 35)(9 36 29)(10 30 37)(11 38 31)(12 32 39)(13 40 33)(14 34 41)
(1 16 23)(2 24 17)(3 18 25)(4 26 19)(5 20 27)(6 28 21)(7 22 15)(8 42 35)(9 36 29)(10 30 37)(11 38 31)(12 32 39)(13 40 33)(14 34 41)
(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)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 40 22 33)(16 39 23 32)(17 38 24 31)(18 37 25 30)(19 36 26 29)(20 35 27 42)(21 34 28 41)
G:=sub<Sym(42)| (8,42,35)(9,36,29)(10,30,37)(11,38,31)(12,32,39)(13,40,33)(14,34,41), (1,16,23)(2,24,17)(3,18,25)(4,26,19)(5,20,27)(6,28,21)(7,22,15)(8,42,35)(9,36,29)(10,30,37)(11,38,31)(12,32,39)(13,40,33)(14,34,41), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,40,22,33)(16,39,23,32)(17,38,24,31)(18,37,25,30)(19,36,26,29)(20,35,27,42)(21,34,28,41)>;
G:=Group( (8,42,35)(9,36,29)(10,30,37)(11,38,31)(12,32,39)(13,40,33)(14,34,41), (1,16,23)(2,24,17)(3,18,25)(4,26,19)(5,20,27)(6,28,21)(7,22,15)(8,42,35)(9,36,29)(10,30,37)(11,38,31)(12,32,39)(13,40,33)(14,34,41), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,40,22,33)(16,39,23,32)(17,38,24,31)(18,37,25,30)(19,36,26,29)(20,35,27,42)(21,34,28,41) );
G=PermutationGroup([[(8,42,35),(9,36,29),(10,30,37),(11,38,31),(12,32,39),(13,40,33),(14,34,41)], [(1,16,23),(2,24,17),(3,18,25),(4,26,19),(5,20,27),(6,28,21),(7,22,15),(8,42,35),(9,36,29),(10,30,37),(11,38,31),(12,32,39),(13,40,33),(14,34,41)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,14),(7,13),(15,40,22,33),(16,39,23,32),(17,38,24,31),(18,37,25,30),(19,36,26,29),(20,35,27,42),(21,34,28,41)]])
Matrix representation of C32⋊Dic7 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 297 | 0 | 336 | 336 |
| 1 | 269 | 0 | 0 |
| 5 | 335 | 0 | 0 |
| 0 | 8 | 0 | 1 |
| 297 | 8 | 336 | 336 |
| 8 | 0 | 0 | 0 |
| 40 | 329 | 0 | 0 |
| 232 | 0 | 295 | 0 |
| 227 | 0 | 42 | 42 |
| 329 | 0 | 269 | 0 |
| 0 | 0 | 336 | 1 |
| 105 | 0 | 8 | 0 |
| 110 | 336 | 8 | 0 |
G:=sub<GL(4,GF(337))| [1,0,0,297,0,1,0,0,0,0,0,336,0,0,1,336],[1,5,0,297,269,335,8,8,0,0,0,336,0,0,1,336],[8,40,232,227,0,329,0,0,0,0,295,42,0,0,0,42],[329,0,105,110,0,0,0,336,269,336,8,8,0,1,0,0] >;
C32⋊Dic7 in GAP, Magma, Sage, TeX
C_3^2\rtimes {\rm Dic}_7 % in TeX
G:=Group("C3^2:Dic7"); // GroupNames label
G:=SmallGroup(252,32);
// by ID
G=gap.SmallGroup(252,32);
# by ID
G:=PCGroup([5,-2,-2,-3,3,-7,10,302,67,323,248,5404]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^14=1,d^2=c^7,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of C32⋊Dic7 in TeX
Character table of C32⋊Dic7 in TeX