Aliases: C23.16S4, SL2(𝔽3)⋊2D4, C22⋊2GL2(𝔽3), Q8⋊(C3⋊D4), Q8⋊Dic3⋊3C2, (C2×Q8).15D6, (C22×Q8)⋊2S3, C22.39(C2×S4), C2.9(A4⋊D4), (C2×GL2(𝔽3))⋊5C2, C2.5(C2×GL2(𝔽3)), C2.7(Q8.D6), (C22×SL2(𝔽3))⋊4C2, (C2×SL2(𝔽3)).15C22, SmallGroup(192,980)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C2×SL2(𝔽3) — C23.16S4 |
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×GL2(𝔽3) — C23.16S4 |
| SL2(𝔽3) — C2×SL2(𝔽3) — C23.16S4 |
Generators and relations for C23.16S4
G = < a,b,c,d,e,f,g | a2=b2=c2=f3=g2=1, d2=e2=c, gag=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=ce=ec, cf=fc, cg=gc, fdf-1=cde, gdg=de, fef-1=d, gfg=f-1 >
Subgroups: 429 in 95 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C8, C2×C4, D4, Q8, Q8, C23, C23, Dic3, D6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, SL2(𝔽3), C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C22×Q8, GL2(𝔽3), C2×SL2(𝔽3), C2×SL2(𝔽3), C2×C3⋊D4, Q8⋊D4, Q8⋊Dic3, C2×GL2(𝔽3), C22×SL2(𝔽3), C23.16S4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊D4, S4, GL2(𝔽3), C2×S4, C2×GL2(𝔽3), Q8.D6, A4⋊D4, C23.16S4
Character table of C23.16S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 24 | 8 | 6 | 6 | 12 | 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 | 0 | -1 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ7 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | 2 | -2 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | √-3 | √-3 | -1 | -√-3 | -√-3 | 1 | 1 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ9 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | -√-3 | -√-3 | -1 | √-3 | √-3 | 1 | 1 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ10 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from GL2(𝔽3) |
| ρ11 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -√-2 | √-2 | -√-2 | √-2 | complex lifted from GL2(𝔽3) |
| ρ12 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | √-2 | -√-2 | -√-2 | √-2 | complex lifted from GL2(𝔽3) |
| ρ13 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | √-2 | -√-2 | √-2 | -√-2 | complex lifted from GL2(𝔽3) |
| ρ14 | 3 | 3 | 3 | 3 | -3 | -3 | -1 | 0 | -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 | 1 | 0 | -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 | 1 | 0 | -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 | -1 | 0 | -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 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from GL2(𝔽3) |
| ρ19 | 4 | -4 | 4 | -4 | 4 | -4 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from GL2(𝔽3) |
| ρ20 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.D6, Schur index 2 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 1 | 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 | 0 | 1 | 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 | 0 | -2 | 2 | 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 22)(2 23)(3 24)(4 21)(5 19)(6 20)(7 17)(8 18)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)
(1 14)(2 15)(3 16)(4 13)(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 31 7 29)(6 30 8 32)(13 19 15 17)(14 18 16 20)(21 27 23 25)(22 26 24 28)
(2 10 11)(4 12 9)(5 31 8)(6 7 29)(13 20 17)(15 18 19)(21 28 25)(23 26 27)
(1 3)(2 11)(4 9)(5 23)(6 28)(7 21)(8 26)(13 17)(14 16)(15 19)(22 32)(24 30)(25 29)(27 31)
G:=sub<Sym(32)| (1,22)(2,23)(3,24)(4,21)(5,19)(6,20)(7,17)(8,18)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32), (1,14)(2,15)(3,16)(4,13)(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)(5,31,7,29)(6,30,8,32)(13,19,15,17)(14,18,16,20)(21,27,23,25)(22,26,24,28), (2,10,11)(4,12,9)(5,31,8)(6,7,29)(13,20,17)(15,18,19)(21,28,25)(23,26,27), (1,3)(2,11)(4,9)(5,23)(6,28)(7,21)(8,26)(13,17)(14,16)(15,19)(22,32)(24,30)(25,29)(27,31)>;
G:=Group( (1,22)(2,23)(3,24)(4,21)(5,19)(6,20)(7,17)(8,18)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32), (1,14)(2,15)(3,16)(4,13)(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)(5,31,7,29)(6,30,8,32)(13,19,15,17)(14,18,16,20)(21,27,23,25)(22,26,24,28), (2,10,11)(4,12,9)(5,31,8)(6,7,29)(13,20,17)(15,18,19)(21,28,25)(23,26,27), (1,3)(2,11)(4,9)(5,23)(6,28)(7,21)(8,26)(13,17)(14,16)(15,19)(22,32)(24,30)(25,29)(27,31) );
G=PermutationGroup([[(1,22),(2,23),(3,24),(4,21),(5,19),(6,20),(7,17),(8,18),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32)], [(1,14),(2,15),(3,16),(4,13),(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),(5,31,7,29),(6,30,8,32),(13,19,15,17),(14,18,16,20),(21,27,23,25),(22,26,24,28)], [(2,10,11),(4,12,9),(5,31,8),(6,7,29),(13,20,17),(15,18,19),(21,28,25),(23,26,27)], [(1,3),(2,11),(4,9),(5,23),(6,28),(7,21),(8,26),(13,17),(14,16),(15,19),(22,32),(24,30),(25,29),(27,31)]])
Matrix representation of C23.16S4 ►in GL4(𝔽73) generated by
| 72 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 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 | 32 | 52 |
| 0 | 0 | 21 | 41 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 53 | 33 |
| 0 | 0 | 52 | 20 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 20 | 41 |
| 0 | 0 | 20 | 52 |
| 1 | 0 | 0 | 0 |
| 37 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(73))| [72,0,0,0,4,1,0,0,0,0,72,0,0,0,0,72],[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],[1,0,0,0,0,1,0,0,0,0,32,21,0,0,52,41],[1,0,0,0,0,1,0,0,0,0,53,52,0,0,33,20],[1,0,0,0,0,1,0,0,0,0,20,20,0,0,41,52],[1,37,0,0,0,72,0,0,0,0,72,1,0,0,0,1] >;
C23.16S4 in GAP, Magma, Sage, TeX
C_2^3._{16}S_4 % in TeX
G:=Group("C2^3.16S4"); // GroupNames label
G:=SmallGroup(192,980);
// by ID
G=gap.SmallGroup(192,980);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,85,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^3=g^2=1,d^2=e^2=c,g*a*g=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=c*e=e*c,c*f=f*c,c*g=g*c,f*d*f^-1=c*d*e,g*d*g=d*e,f*e*f^-1=d,g*f*g=f^-1>;
// generators/relations
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