Aliases: C6.5S4, C3⋊CSU2(𝔽3), SL2(𝔽3).S3, Q8.(C3⋊S3), C2.2(C3⋊S4), (C3×Q8).3S3, (C3×SL2(𝔽3)).1C2, SmallGroup(144,124)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3×SL2(𝔽3) — C6.5S4 |
| C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C6.5S4 |
| C3×SL2(𝔽3) — C6.5S4 |
Generators and relations for C6.5S4
G = < a,b,c,d,e | a6=d3=1, b2=c2=e2=a3, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a3b, dbd-1=a3bc, ebe-1=bc, dcd-1=b, ece-1=a3c, ede-1=d-1 >
Character table of C6.5S4
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 12 | |
| size | 1 | 1 | 2 | 8 | 8 | 8 | 6 | 36 | 2 | 8 | 8 | 8 | 18 | 18 | 12 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | 0 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | orthogonal lifted from S3 |
| ρ5 | 2 | 2 | -1 | 2 | -1 | -1 | 2 | 0 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | -2 | 1 | 1 | 1 | -√2 | √2 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ8 | 2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | -2 | 1 | 1 | 1 | √2 | -√2 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ9 | 3 | 3 | 3 | 0 | 0 | 0 | -1 | 1 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ10 | 3 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | 3 | 0 | 0 | 0 | 1 | 1 | -1 | orthogonal lifted from S4 |
| ρ11 | 4 | -4 | -2 | 1 | -2 | 1 | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ12 | 4 | -4 | 4 | 1 | 1 | 1 | 0 | 0 | -4 | -1 | -1 | -1 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ13 | 4 | -4 | -2 | 1 | 1 | -2 | 0 | 0 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ14 | 4 | -4 | -2 | -2 | 1 | 1 | 0 | 0 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ15 | 6 | 6 | -3 | 0 | 0 | 0 | -2 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from C3⋊S4 |
►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 22 4 19)(2 23 5 20)(3 24 6 21)(7 37 10 40)(8 38 11 41)(9 39 12 42)(13 43 16 46)(14 44 17 47)(15 45 18 48)(25 31 28 34)(26 32 29 35)(27 33 30 36)
(1 33 4 36)(2 34 5 31)(3 35 6 32)(7 14 10 17)(8 15 11 18)(9 16 12 13)(19 30 22 27)(20 25 23 28)(21 26 24 29)(37 47 40 44)(38 48 41 45)(39 43 42 46)
(7 37 47)(8 38 48)(9 39 43)(10 40 44)(11 41 45)(12 42 46)(19 36 27)(20 31 28)(21 32 29)(22 33 30)(23 34 25)(24 35 26)
(1 13 4 16)(2 18 5 15)(3 17 6 14)(7 35 10 32)(8 34 11 31)(9 33 12 36)(19 39 22 42)(20 38 23 41)(21 37 24 40)(25 45 28 48)(26 44 29 47)(27 43 30 46)
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), (1,22,4,19)(2,23,5,20)(3,24,6,21)(7,37,10,40)(8,38,11,41)(9,39,12,42)(13,43,16,46)(14,44,17,47)(15,45,18,48)(25,31,28,34)(26,32,29,35)(27,33,30,36), (1,33,4,36)(2,34,5,31)(3,35,6,32)(7,14,10,17)(8,15,11,18)(9,16,12,13)(19,30,22,27)(20,25,23,28)(21,26,24,29)(37,47,40,44)(38,48,41,45)(39,43,42,46), (7,37,47)(8,38,48)(9,39,43)(10,40,44)(11,41,45)(12,42,46)(19,36,27)(20,31,28)(21,32,29)(22,33,30)(23,34,25)(24,35,26), (1,13,4,16)(2,18,5,15)(3,17,6,14)(7,35,10,32)(8,34,11,31)(9,33,12,36)(19,39,22,42)(20,38,23,41)(21,37,24,40)(25,45,28,48)(26,44,29,47)(27,43,30,46)>;
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), (1,22,4,19)(2,23,5,20)(3,24,6,21)(7,37,10,40)(8,38,11,41)(9,39,12,42)(13,43,16,46)(14,44,17,47)(15,45,18,48)(25,31,28,34)(26,32,29,35)(27,33,30,36), (1,33,4,36)(2,34,5,31)(3,35,6,32)(7,14,10,17)(8,15,11,18)(9,16,12,13)(19,30,22,27)(20,25,23,28)(21,26,24,29)(37,47,40,44)(38,48,41,45)(39,43,42,46), (7,37,47)(8,38,48)(9,39,43)(10,40,44)(11,41,45)(12,42,46)(19,36,27)(20,31,28)(21,32,29)(22,33,30)(23,34,25)(24,35,26), (1,13,4,16)(2,18,5,15)(3,17,6,14)(7,35,10,32)(8,34,11,31)(9,33,12,36)(19,39,22,42)(20,38,23,41)(21,37,24,40)(25,45,28,48)(26,44,29,47)(27,43,30,46) );
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)], [(1,22,4,19),(2,23,5,20),(3,24,6,21),(7,37,10,40),(8,38,11,41),(9,39,12,42),(13,43,16,46),(14,44,17,47),(15,45,18,48),(25,31,28,34),(26,32,29,35),(27,33,30,36)], [(1,33,4,36),(2,34,5,31),(3,35,6,32),(7,14,10,17),(8,15,11,18),(9,16,12,13),(19,30,22,27),(20,25,23,28),(21,26,24,29),(37,47,40,44),(38,48,41,45),(39,43,42,46)], [(7,37,47),(8,38,48),(9,39,43),(10,40,44),(11,41,45),(12,42,46),(19,36,27),(20,31,28),(21,32,29),(22,33,30),(23,34,25),(24,35,26)], [(1,13,4,16),(2,18,5,15),(3,17,6,14),(7,35,10,32),(8,34,11,31),(9,33,12,36),(19,39,22,42),(20,38,23,41),(21,37,24,40),(25,45,28,48),(26,44,29,47),(27,43,30,46)]])
C6.5S4 is a maximal subgroup of
CSU2(𝔽3)⋊S3 Dic3.4S4 S3×CSU2(𝔽3) D6.S4 SL2(𝔽3).D6 C12.6S4 C12.14S4 C32⋊CSU2(𝔽3) C18.5S4 C32⋊4CSU2(𝔽3)
C6.5S4 is a maximal quotient of
C6.GL2(𝔽3) C18.5S4 C32.3CSU2(𝔽3) C32⋊2CSU2(𝔽3) C32⋊4CSU2(𝔽3)
Matrix representation of C6.5S4 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 1 |
| 4 | 1 | 4 | 0 |
| 4 | 1 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 4 | 0 | 0 | 4 |
| 3 | 0 | 4 | 0 |
| 4 | 1 | 4 | 0 |
| 0 | 1 | 0 | 4 |
| 4 | 0 | 4 | 1 |
| 0 | 0 | 4 | 2 |
| 0 | 0 | 4 | 1 |
| 1 | 0 | 1 | 4 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 2 |
| 0 | 1 | 4 | 1 |
| 2 | 0 | 0 | 2 |
| 0 | 3 | 2 | 3 |
| 0 | 0 | 2 | 1 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(5))| [1,4,4,4,0,1,1,0,0,4,0,0,1,0,4,0],[1,4,3,4,0,0,0,1,1,0,4,4,0,4,0,0],[0,4,0,0,1,0,0,0,0,4,4,4,4,1,2,1],[1,0,0,0,0,0,0,1,1,0,4,4,4,1,2,1],[2,0,0,0,0,3,0,0,0,2,2,0,2,3,1,3] >;
C6.5S4 in GAP, Magma, Sage, TeX
C_6._5S_4
% in TeX
G:=Group("C6.5S4"); // GroupNames label
G:=SmallGroup(144,124);
// by ID
G=gap.SmallGroup(144,124);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,2,-2,432,49,218,867,1305,117,544,820,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=d^3=1,b^2=c^2=e^2=a^3,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c^-1=a^3*b,d*b*d^-1=a^3*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^3*c,e*d*e^-1=d^-1>;
// generators/relations
Export
Subgroup lattice of C6.5S4 in TeX
Character table of C6.5S4 in TeX