direct product, non-abelian, soluble
Aliases: C2×U2(𝔽3), C4.A4⋊3C4, C4.32(C2×S4), (C2×C4).25S4, C4.7(A4⋊C4), C4○D4.13D6, C4○D4.2Dic3, Q8.2(C2×Dic3), (C2×Q8).2Dic3, C4.A4.13C22, SL2(𝔽3)⋊4(C2×C4), (C2×SL2(𝔽3))⋊4C4, C22.10(A4⋊C4), C2.7(C2×A4⋊C4), (C2×C4.A4).5C2, (C2×C4○D4).1S3, SmallGroup(192,981)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — C2×U2(𝔽3) |
Generators and relations for C2×U2(𝔽3)
G = < a,b,c,d,e,f | a2=b4=e3=1, c2=d2=b2, f2=b, 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-1=cd, ede-1=c, fdf-1=b2d, fef-1=e-1 >
Subgroups: 275 in 81 conjugacy classes, 21 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C2×C6, C42, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, SL2(𝔽3), C2×C12, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C2×SL2(𝔽3), C4.A4, C2×C4≀C2, U2(𝔽3), C2×C4.A4, C2×U2(𝔽3)
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C2×Dic3, S4, A4⋊C4, C2×S4, U2(𝔽3), C2×A4⋊C4, C2×U2(𝔽3)
►On 48 points
Generators in S48
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)
(1 16 5 12)(2 15 6 11)(3 10 7 14)(4 9 8 13)(17 19 21 23)(18 31 22 27)(20 25 24 29)(26 32 30 28)(33 39 37 35)(34 43 38 47)(36 45 40 41)(42 44 46 48)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 28 21 32)(18 25 22 29)(19 30 23 26)(20 27 24 31)(33 44 37 48)(34 41 38 45)(35 46 39 42)(36 43 40 47)
(1 38 17)(2 18 39)(3 40 19)(4 20 33)(5 34 21)(6 22 35)(7 36 23)(8 24 37)(9 27 46)(10 47 28)(11 29 48)(12 41 30)(13 31 42)(14 43 32)(15 25 44)(16 45 26)
(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)
G:=sub<Sym(48)| (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48), (1,16,5,12)(2,15,6,11)(3,10,7,14)(4,9,8,13)(17,19,21,23)(18,31,22,27)(20,25,24,29)(26,32,30,28)(33,39,37,35)(34,43,38,47)(36,45,40,41)(42,44,46,48), (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12)(17,28,21,32)(18,25,22,29)(19,30,23,26)(20,27,24,31)(33,44,37,48)(34,41,38,45)(35,46,39,42)(36,43,40,47), (1,38,17)(2,18,39)(3,40,19)(4,20,33)(5,34,21)(6,22,35)(7,36,23)(8,24,37)(9,27,46)(10,47,28)(11,29,48)(12,41,30)(13,31,42)(14,43,32)(15,25,44)(16,45,26), (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)>;
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48), (1,16,5,12)(2,15,6,11)(3,10,7,14)(4,9,8,13)(17,19,21,23)(18,31,22,27)(20,25,24,29)(26,32,30,28)(33,39,37,35)(34,43,38,47)(36,45,40,41)(42,44,46,48), (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12)(17,28,21,32)(18,25,22,29)(19,30,23,26)(20,27,24,31)(33,44,37,48)(34,41,38,45)(35,46,39,42)(36,43,40,47), (1,38,17)(2,18,39)(3,40,19)(4,20,33)(5,34,21)(6,22,35)(7,36,23)(8,24,37)(9,27,46)(10,47,28)(11,29,48)(12,41,30)(13,31,42)(14,43,32)(15,25,44)(16,45,26), (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) );
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,41),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48)], [(1,16,5,12),(2,15,6,11),(3,10,7,14),(4,9,8,13),(17,19,21,23),(18,31,22,27),(20,25,24,29),(26,32,30,28),(33,39,37,35),(34,43,38,47),(36,45,40,41),(42,44,46,48)], [(1,3,5,7),(2,8,6,4),(9,11,13,15),(10,16,14,12),(17,28,21,32),(18,25,22,29),(19,30,23,26),(20,27,24,31),(33,44,37,48),(34,41,38,45),(35,46,39,42),(36,43,40,47)], [(1,38,17),(2,18,39),(3,40,19),(4,20,33),(5,34,21),(6,22,35),(7,36,23),(8,24,37),(9,27,46),(10,47,28),(11,29,48),(12,41,30),(13,31,42),(14,43,32),(15,25,44),(16,45,26)], [(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)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 8 | 1 | 1 | 1 | 1 | 6 | ··· | 6 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 |
| type | + | + | + | + | - | - | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | Dic3 | D6 | U2(𝔽3) | S4 | A4⋊C4 | C2×S4 | A4⋊C4 | U2(𝔽3) |
| kernel | C2×U2(𝔽3) | U2(𝔽3) | C2×C4.A4 | C2×SL2(𝔽3) | C4.A4 | C2×C4○D4 | C2×Q8 | C4○D4 | C4○D4 | C2 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 |
Matrix representation of C2×U2(𝔽3) ►in 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 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 71 |
| 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 46 | 27 |
| 0 | 4 | 0 | 0 |
| 18 | 72 | 0 | 0 |
| 0 | 0 | 72 | 28 |
| 0 | 0 | 13 | 0 |
| 72 | 69 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 2 |
| 0 | 0 | 59 | 1 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,46,0,0,0,0,46],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,71,72],[1,0,0,0,0,1,0,0,0,0,46,46,0,0,0,27],[0,18,0,0,4,72,0,0,0,0,72,13,0,0,28,0],[72,0,0,0,69,1,0,0,0,0,72,59,0,0,2,1] >;
C2×U2(𝔽3) in GAP, Magma, Sage, TeX
C_2\times {\rm U}_2({\mathbb F}_3) % in TeX
G:=Group("C2xU(2,3)"); // GroupNames label
G:=SmallGroup(192,981);
// by ID
G=gap.SmallGroup(192,981);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,28,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^4=e^3=1,c^2=d^2=b^2,f^2=b,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^-1=c*d,e*d*e^-1=c,f*d*f^-1=b^2*d,f*e*f^-1=e^-1>;
// generators/relations