Aliases: C23.14S4, SL2(𝔽3).6D4, C22⋊2CSU2(𝔽3), Q8⋊Dic3⋊1C2, (C2×Q8).13D6, C22.38(C2×S4), Q8.1(C3⋊D4), (C22×Q8).3S3, C2.8(A4⋊D4), (C2×CSU2(𝔽3))⋊2C2, C2.6(Q8.D6), C2.5(C2×CSU2(𝔽3)), (C22×SL2(𝔽3)).3C2, (C2×SL2(𝔽3)).13C22, SmallGroup(192,978)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C2×SL2(𝔽3) — C23.14S4 |
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×CSU2(𝔽3) — C23.14S4 |
| SL2(𝔽3) — C2×SL2(𝔽3) — C23.14S4 |
Generators and relations for C23.14S4
G = < a,b,c,d,e,f,g | a2=b2=c2=f3=1, d2=e2=g2=c, gag-1=ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, bg=gb, ede-1=cd=dc, geg-1=ce=ec, cf=fc, cg=gc, fdf-1=cde, gdg-1=de, fef-1=d, gfg-1=f-1 >
Subgroups: 309 in 80 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C8, C2×C4, Q8, Q8, C23, Dic3, C2×C6, C22⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C2×Q8, C2×Q8, SL2(𝔽3), C2×Dic3, C22×C6, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C6.D4, CSU2(𝔽3), C2×SL2(𝔽3), C2×SL2(𝔽3), C22⋊Q16, Q8⋊Dic3, C2×CSU2(𝔽3), C22×SL2(𝔽3), C23.14S4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊D4, S4, CSU2(𝔽3), C2×S4, C2×CSU2(𝔽3), Q8.D6, A4⋊D4, C23.14S4
Character table of C23.14S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 6 | 6 | 12 | 24 | 24 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 |
| ρ4 | 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 |
| ρ5 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | 2 | -2 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ8 | 2 | -2 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -√2 | √2 | -√2 | √2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ9 | 2 | -2 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | √2 | -√2 | √2 | -√2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ10 | 2 | -2 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -√2 | √2 | √2 | -√2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ11 | 2 | -2 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | √2 | -√2 | -√2 | √2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | √-3 | √-3 | -1 | -√-3 | -√-3 | 1 | 1 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -√-3 | -√-3 | -1 | √-3 | √-3 | 1 | 1 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ14 | 3 | 3 | 3 | 3 | -3 | -3 | 0 | -1 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×S4 |
| ρ15 | 3 | 3 | 3 | 3 | 3 | 3 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ16 | 3 | 3 | 3 | 3 | -3 | -3 | 0 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×S4 |
| ρ17 | 3 | 3 | 3 | 3 | 3 | 3 | 0 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ18 | 4 | -4 | 4 | -4 | -4 | 4 | 1 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ19 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.D6, Schur index 2 |
| ρ20 | 4 | -4 | 4 | -4 | 4 | -4 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | -√-3 | √-3 | -1 | -√-3 | √-3 | -1 | 1 | 0 | 0 | 0 | 0 | complex lifted from Q8.D6 |
| ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | √-3 | -√-3 | -1 | √-3 | -√-3 | -1 | 1 | 0 | 0 | 0 | 0 | complex lifted from Q8.D6 |
| ρ23 | 6 | 6 | -6 | -6 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4⋊D4 |
►On 32 points
Generators in S32
(1 3)(2 4)(5 25)(6 26)(7 27)(8 28)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 31)(22 32)(23 29)(24 30)
(1 16)(2 13)(3 14)(4 15)(5 27)(6 28)(7 25)(8 26)(9 17)(10 18)(11 19)(12 20)(21 29)(22 30)(23 31)(24 32)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(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)
(1 10 3 12)(2 9 4 11)(5 30 7 32)(6 29 8 31)(13 17 15 19)(14 20 16 18)(21 26 23 28)(22 25 24 27)
(2 10 11)(4 12 9)(5 30 8)(6 7 32)(13 18 19)(15 20 17)(22 26 27)(24 28 25)
(1 21 3 23)(2 25 4 27)(5 13 7 15)(6 20 8 18)(9 22 11 24)(10 28 12 26)(14 31 16 29)(17 30 19 32)
G:=sub<Sym(32)| (1,3)(2,4)(5,25)(6,26)(7,27)(8,28)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,31)(22,32)(23,29)(24,30), (1,16)(2,13)(3,14)(4,15)(5,27)(6,28)(7,25)(8,26)(9,17)(10,18)(11,19)(12,20)(21,29)(22,30)(23,31)(24,32), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (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), (1,10,3,12)(2,9,4,11)(5,30,7,32)(6,29,8,31)(13,17,15,19)(14,20,16,18)(21,26,23,28)(22,25,24,27), (2,10,11)(4,12,9)(5,30,8)(6,7,32)(13,18,19)(15,20,17)(22,26,27)(24,28,25), (1,21,3,23)(2,25,4,27)(5,13,7,15)(6,20,8,18)(9,22,11,24)(10,28,12,26)(14,31,16,29)(17,30,19,32)>;
G:=Group( (1,3)(2,4)(5,25)(6,26)(7,27)(8,28)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,31)(22,32)(23,29)(24,30), (1,16)(2,13)(3,14)(4,15)(5,27)(6,28)(7,25)(8,26)(9,17)(10,18)(11,19)(12,20)(21,29)(22,30)(23,31)(24,32), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (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), (1,10,3,12)(2,9,4,11)(5,30,7,32)(6,29,8,31)(13,17,15,19)(14,20,16,18)(21,26,23,28)(22,25,24,27), (2,10,11)(4,12,9)(5,30,8)(6,7,32)(13,18,19)(15,20,17)(22,26,27)(24,28,25), (1,21,3,23)(2,25,4,27)(5,13,7,15)(6,20,8,18)(9,22,11,24)(10,28,12,26)(14,31,16,29)(17,30,19,32) );
G=PermutationGroup([[(1,3),(2,4),(5,25),(6,26),(7,27),(8,28),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,31),(22,32),(23,29),(24,30)], [(1,16),(2,13),(3,14),(4,15),(5,27),(6,28),(7,25),(8,26),(9,17),(10,18),(11,19),(12,20),(21,29),(22,30),(23,31),(24,32)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(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)], [(1,10,3,12),(2,9,4,11),(5,30,7,32),(6,29,8,31),(13,17,15,19),(14,20,16,18),(21,26,23,28),(22,25,24,27)], [(2,10,11),(4,12,9),(5,30,8),(6,7,32),(13,18,19),(15,20,17),(22,26,27),(24,28,25)], [(1,21,3,23),(2,25,4,27),(5,13,7,15),(6,20,8,18),(9,22,11,24),(10,28,12,26),(14,31,16,29),(17,30,19,32)]])
Matrix representation of C23.14S4 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 65 | 64 |
| 0 | 0 | 64 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 1 | 64 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 71 | 34 |
| 0 | 0 | 2 | 2 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,65,64,0,0,64,8],[1,0,0,0,0,1,0,0,0,0,8,1,0,0,0,64],[0,1,0,0,1,0,0,0,0,0,71,2,0,0,34,2] >;
C23.14S4 in GAP, Magma, Sage, TeX
C_2^3._{14}S_4 % in TeX
G:=Group("C2^3.14S4"); // GroupNames label
G:=SmallGroup(192,978);
// by ID
G=gap.SmallGroup(192,978);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,85,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^3=1,d^2=e^2=g^2=c,g*a*g^-1=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,e*d*e^-1=c*d=d*c,g*e*g^-1=c*e=e*c,c*f=f*c,c*g=g*c,f*d*f^-1=c*d*e,g*d*g^-1=d*e,f*e*f^-1=d,g*f*g^-1=f^-1>;
// generators/relations
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