direct product, non-abelian, soluble
Aliases: C2×C4.6S4, GL2(𝔽3)⋊6C22, CSU2(𝔽3)⋊6C22, SL2(𝔽3).4C23, C4○D4⋊2D6, C4.35(C2×S4), (C2×C4).27S4, C4.A4⋊4C22, (C2×Q8).22D6, C2.14(C22×S4), C22.29(C2×S4), Q8.4(C22×S3), (C2×GL2(𝔽3))⋊7C2, (C2×CSU2(𝔽3))⋊7C2, (C2×SL2(𝔽3)).22C22, (C2×C4.A4)⋊6C2, (C2×C4○D4)⋊2S3, SmallGroup(192,1480)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — C2×C4.6S4 |
Subgroups: 539 in 153 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×4], C22, C22 [×8], S3 [×4], C6 [×3], C8 [×4], C2×C4, C2×C4 [×10], D4 [×10], Q8, Q8 [×4], C23 [×2], Dic3 [×2], C12 [×2], D6 [×6], C2×C6, C2×C8 [×6], D8 [×4], SD16 [×8], Q16 [×4], C22×C4 [×2], C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×2], C4○D4 [×8], SL2(𝔽3), C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4, C2×C4○D4, CSU2(𝔽3) [×2], GL2(𝔽3) [×2], C2×SL2(𝔽3), C4.A4 [×2], S3×C2×C4, C2×C4○D8, C2×CSU2(𝔽3), C2×GL2(𝔽3), C4.6S4 [×4], C2×C4.A4, C2×C4.6S4
Quotients:
C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], S4, C22×S3, C2×S4 [×3], C4.6S4 [×2], C22×S4, C2×C4.6S4
Generators and relations
G = < a,b,c,d,e,f | a2=b4=e3=f2=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=b2c, ece-1=b2cd, fcf=cd, ede-1=c, fdf=b2d, fef=e-1 >
►On 32 points
(1 27)(2 28)(3 25)(4 26)(5 23)(6 24)(7 21)(8 22)(9 17)(10 18)(11 19)(12 20)(13 31)(14 32)(15 29)(16 30)
(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 13 3 15)(2 14 4 16)(5 18 7 20)(6 19 8 17)(9 24 11 22)(10 21 12 23)(25 29 27 31)(26 30 28 32)
(1 17 3 19)(2 18 4 20)(5 16 7 14)(6 13 8 15)(9 25 11 27)(10 26 12 28)(21 32 23 30)(22 29 24 31)
(5 16 20)(6 13 17)(7 14 18)(8 15 19)(9 24 31)(10 21 32)(11 22 29)(12 23 30)
(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)(17 19)(18 20)(21 30)(22 31)(23 32)(24 29)
G:=sub<Sym(32)| (1,27)(2,28)(3,25)(4,26)(5,23)(6,24)(7,21)(8,22)(9,17)(10,18)(11,19)(12,20)(13,31)(14,32)(15,29)(16,30), (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), (1,13,3,15)(2,14,4,16)(5,18,7,20)(6,19,8,17)(9,24,11,22)(10,21,12,23)(25,29,27,31)(26,30,28,32), (1,17,3,19)(2,18,4,20)(5,16,7,14)(6,13,8,15)(9,25,11,27)(10,26,12,28)(21,32,23,30)(22,29,24,31), (5,16,20)(6,13,17)(7,14,18)(8,15,19)(9,24,31)(10,21,32)(11,22,29)(12,23,30), (5,14)(6,15)(7,16)(8,13)(9,11)(10,12)(17,19)(18,20)(21,30)(22,31)(23,32)(24,29)>;
G:=Group( (1,27)(2,28)(3,25)(4,26)(5,23)(6,24)(7,21)(8,22)(9,17)(10,18)(11,19)(12,20)(13,31)(14,32)(15,29)(16,30), (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), (1,13,3,15)(2,14,4,16)(5,18,7,20)(6,19,8,17)(9,24,11,22)(10,21,12,23)(25,29,27,31)(26,30,28,32), (1,17,3,19)(2,18,4,20)(5,16,7,14)(6,13,8,15)(9,25,11,27)(10,26,12,28)(21,32,23,30)(22,29,24,31), (5,16,20)(6,13,17)(7,14,18)(8,15,19)(9,24,31)(10,21,32)(11,22,29)(12,23,30), (5,14)(6,15)(7,16)(8,13)(9,11)(10,12)(17,19)(18,20)(21,30)(22,31)(23,32)(24,29) );
G=PermutationGroup([(1,27),(2,28),(3,25),(4,26),(5,23),(6,24),(7,21),(8,22),(9,17),(10,18),(11,19),(12,20),(13,31),(14,32),(15,29),(16,30)], [(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)], [(1,13,3,15),(2,14,4,16),(5,18,7,20),(6,19,8,17),(9,24,11,22),(10,21,12,23),(25,29,27,31),(26,30,28,32)], [(1,17,3,19),(2,18,4,20),(5,16,7,14),(6,13,8,15),(9,25,11,27),(10,26,12,28),(21,32,23,30),(22,29,24,31)], [(5,16,20),(6,13,17),(7,14,18),(8,15,19),(9,24,31),(10,21,32),(11,22,29),(12,23,30)], [(5,14),(6,15),(7,16),(8,13),(9,11),(10,12),(17,19),(18,20),(21,30),(22,31),(23,32),(24,29)])
Matrix representation ►G ⊆ GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 20 | 33 |
| 0 | 0 | 52 | 53 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 53 | 52 |
| 0 | 0 | 33 | 20 |
| 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 20 | 20 |
| 0 | 0 | 41 | 52 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,27,0,0,0,0,27],[1,0,0,0,0,1,0,0,0,0,20,52,0,0,33,53],[1,0,0,0,0,1,0,0,0,0,53,33,0,0,52,20],[0,1,0,0,72,72,0,0,0,0,20,41,0,0,20,52],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,72,72] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 8 | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 8 | 8 | 8 | 6 | ··· | 6 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4.6S4 | S4 | C2×S4 | C2×S4 | C4.6S4 |
| kernel | C2×C4.6S4 | C2×CSU2(𝔽3) | C2×GL2(𝔽3) | C4.6S4 | C2×C4.A4 | C2×C4○D4 | C2×Q8 | C4○D4 | C2 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 8 | 2 | 4 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2\times C_4._6S_4
% in TeX
G:=Group("C2xC4.6S4"); // GroupNames label
G:=SmallGroup(192,1480);
// by ID
G=gap.SmallGroup(192,1480);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^4=e^3=f^2=1,c^2=d^2=b^2,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,d*c*d^-1=b^2*c,e*c*e^-1=b^2*c*d,f*c*f=c*d,e*d*e^-1=c,f*d*f=b^2*d,f*e*f=e^-1>;
// generators/relations