direct product, non-abelian, soluble
Aliases: C4×GL2(𝔽3), Q8⋊(C4×S3), C2.5(C4×S4), (C4×Q8)⋊1S3, (C2×C4).22S4, (C2×Q8).8D6, Q8⋊Dic3⋊8C2, C22.11(C2×S4), C2.2(C4.6S4), (C4×SL2(𝔽3))⋊7C2, SL2(𝔽3)⋊1(C2×C4), C2.1(C2×GL2(𝔽3)), (C2×GL2(𝔽3)).3C2, (C2×SL2(𝔽3)).8C22, SmallGroup(192,951)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×GL2(𝔽3) — C4×GL2(𝔽3) |
| SL2(𝔽3) — C4×GL2(𝔽3) |
Generators and relations for C4×GL2(𝔽3)
G = < a,b,c,d,e | a4=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece=b-1, dbd-1=bc, ebe=b2c, dcd-1=b, ede=d-1 >
Subgroups: 343 in 88 conjugacy classes, 19 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, SL2(𝔽3), C4×S3, C2×Dic3, C2×C12, C22×S3, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, GL2(𝔽3), C2×SL2(𝔽3), S3×C2×C4, C4×SD16, Q8⋊Dic3, C4×SL2(𝔽3), C2×GL2(𝔽3), C4×GL2(𝔽3)
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, C4×S3, S4, GL2(𝔽3), C2×S4, C4×S4, C2×GL2(𝔽3), C4.6S4, C4×GL2(𝔽3)
►On 32 points
(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 29 12 22)(2 30 9 23)(3 31 10 24)(4 32 11 21)(5 25 17 13)(6 26 18 14)(7 27 19 15)(8 28 20 16)
(1 17 12 5)(2 18 9 6)(3 19 10 7)(4 20 11 8)(13 29 25 22)(14 30 26 23)(15 31 27 24)(16 32 28 21)
(5 13 22)(6 14 23)(7 15 24)(8 16 21)(17 25 29)(18 26 30)(19 27 31)(20 28 32)
(5 29)(6 30)(7 31)(8 32)(13 25)(14 26)(15 27)(16 28)(17 22)(18 23)(19 24)(20 21)
G:=sub<Sym(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), (1,29,12,22)(2,30,9,23)(3,31,10,24)(4,32,11,21)(5,25,17,13)(6,26,18,14)(7,27,19,15)(8,28,20,16), (1,17,12,5)(2,18,9,6)(3,19,10,7)(4,20,11,8)(13,29,25,22)(14,30,26,23)(15,31,27,24)(16,32,28,21), (5,13,22)(6,14,23)(7,15,24)(8,16,21)(17,25,29)(18,26,30)(19,27,31)(20,28,32), (5,29)(6,30)(7,31)(8,32)(13,25)(14,26)(15,27)(16,28)(17,22)(18,23)(19,24)(20,21)>;
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), (1,29,12,22)(2,30,9,23)(3,31,10,24)(4,32,11,21)(5,25,17,13)(6,26,18,14)(7,27,19,15)(8,28,20,16), (1,17,12,5)(2,18,9,6)(3,19,10,7)(4,20,11,8)(13,29,25,22)(14,30,26,23)(15,31,27,24)(16,32,28,21), (5,13,22)(6,14,23)(7,15,24)(8,16,21)(17,25,29)(18,26,30)(19,27,31)(20,28,32), (5,29)(6,30)(7,31)(8,32)(13,25)(14,26)(15,27)(16,28)(17,22)(18,23)(19,24)(20,21) );
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)], [(1,29,12,22),(2,30,9,23),(3,31,10,24),(4,32,11,21),(5,25,17,13),(6,26,18,14),(7,27,19,15),(8,28,20,16)], [(1,17,12,5),(2,18,9,6),(3,19,10,7),(4,20,11,8),(13,29,25,22),(14,30,26,23),(15,31,27,24),(16,32,28,21)], [(5,13,22),(6,14,23),(7,15,24),(8,16,21),(17,25,29),(18,26,30),(19,27,31),(20,28,32)], [(5,29),(6,30),(7,31),(8,32),(13,25),(14,26),(15,27),(16,28),(17,22),(18,23),(19,24),(20,21)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | ··· | 8H | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 8 | 1 | 1 | 1 | 1 | 6 | 6 | 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 | 2 | 3 | 3 | 3 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | C4×S3 | GL2(𝔽3) | C4.6S4 | S4 | C2×S4 | C4×S4 | GL2(𝔽3) | C4.6S4 |
| kernel | C4×GL2(𝔽3) | Q8⋊Dic3 | C4×SL2(𝔽3) | C2×GL2(𝔽3) | GL2(𝔽3) | C4×Q8 | C2×Q8 | Q8 | C4 | C2 | C2×C4 | C22 | C2 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 4 | 2 | 2 |
Matrix representation of C4×GL2(𝔽3) ►in GL4(𝔽73) generated by
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 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 | 53 | 40 |
| 0 | 0 | 21 | 20 |
| 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 72 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [27,0,0,0,0,27,0,0,0,0,27,0,0,0,0,27],[1,0,0,0,0,1,0,0,0,0,53,33,0,0,52,20],[1,0,0,0,0,1,0,0,0,0,53,21,0,0,40,20],[0,1,0,0,72,72,0,0,0,0,0,1,0,0,72,72],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C4×GL2(𝔽3) in GAP, Magma, Sage, TeX
C_4\times {\rm GL}_2({\mathbb F}_3) % in TeX
G:=Group("C4xGL(2,3)"); // GroupNames label
G:=SmallGroup(192,951);
// by ID
G=gap.SmallGroup(192,951);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,36,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*c*e=b^-1,d*b*d^-1=b*c,e*b*e=b^2*c,d*c*d^-1=b,e*d*e=d^-1>;
// generators/relations