Aliases: C2.U2(𝔽3), SL2(𝔽3)⋊C8, C4.2GL2(𝔽3), C4.2CSU2(𝔽3), Q8⋊(C3⋊C8), (C2×C4).20S4, (C4×Q8).1S3, C2.2(A4⋊C8), C2.(Q8⋊Dic3), (C2×Q8).1Dic3, C22.7(A4⋊C4), (C2×SL2(𝔽3)).2C4, (C4×SL2(𝔽3)).7C2, SmallGroup(192,183)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C4×SL2(𝔽3) — C2.U2(𝔽3) |
| 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=ab2, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=ab2c, ece-1=ab2cd, fcf-1=cd, ede-1=c, fdf-1=ab2d, fef-1=e-1 >
Smallest permutation representation of C2.U2(𝔽3)
►On 64 points
Generators in S64
(1 56)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(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)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(1 20 52 36)(2 13 53 29)(3 22 54 38)(4 15 55 31)(5 24 56 40)(6 9 49 25)(7 18 50 34)(8 11 51 27)(10 62 26 42)(12 64 28 44)(14 58 30 46)(16 60 32 48)(17 41 33 61)(19 43 35 63)(21 45 37 57)(23 47 39 59)
(1 64 52 44)(2 45 53 57)(3 58 54 46)(4 47 55 59)(5 60 56 48)(6 41 49 61)(7 62 50 42)(8 43 51 63)(9 17 25 33)(10 34 26 18)(11 19 27 35)(12 36 28 20)(13 21 29 37)(14 38 30 22)(15 23 31 39)(16 40 32 24)
(9 41 33)(10 34 42)(11 43 35)(12 36 44)(13 45 37)(14 38 46)(15 47 39)(16 40 48)(17 25 61)(18 62 26)(19 27 63)(20 64 28)(21 29 57)(22 58 30)(23 31 59)(24 60 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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
G:=sub<Sym(64)| (1,56)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,20,52,36)(2,13,53,29)(3,22,54,38)(4,15,55,31)(5,24,56,40)(6,9,49,25)(7,18,50,34)(8,11,51,27)(10,62,26,42)(12,64,28,44)(14,58,30,46)(16,60,32,48)(17,41,33,61)(19,43,35,63)(21,45,37,57)(23,47,39,59), (1,64,52,44)(2,45,53,57)(3,58,54,46)(4,47,55,59)(5,60,56,48)(6,41,49,61)(7,62,50,42)(8,43,51,63)(9,17,25,33)(10,34,26,18)(11,19,27,35)(12,36,28,20)(13,21,29,37)(14,38,30,22)(15,23,31,39)(16,40,32,24), (9,41,33)(10,34,42)(11,43,35)(12,36,44)(13,45,37)(14,38,46)(15,47,39)(16,40,48)(17,25,61)(18,62,26)(19,27,63)(20,64,28)(21,29,57)(22,58,30)(23,31,59)(24,60,32), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)>;
G:=Group( (1,56)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,20,52,36)(2,13,53,29)(3,22,54,38)(4,15,55,31)(5,24,56,40)(6,9,49,25)(7,18,50,34)(8,11,51,27)(10,62,26,42)(12,64,28,44)(14,58,30,46)(16,60,32,48)(17,41,33,61)(19,43,35,63)(21,45,37,57)(23,47,39,59), (1,64,52,44)(2,45,53,57)(3,58,54,46)(4,47,55,59)(5,60,56,48)(6,41,49,61)(7,62,50,42)(8,43,51,63)(9,17,25,33)(10,34,26,18)(11,19,27,35)(12,36,28,20)(13,21,29,37)(14,38,30,22)(15,23,31,39)(16,40,32,24), (9,41,33)(10,34,42)(11,43,35)(12,36,44)(13,45,37)(14,38,46)(15,47,39)(16,40,48)(17,25,61)(18,62,26)(19,27,63)(20,64,28)(21,29,57)(22,58,30)(23,31,59)(24,60,32), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64) );
G=PermutationGroup([[(1,56),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,29),(10,30),(11,31),(12,32),(13,25),(14,26),(15,27),(16,28),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(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),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(1,20,52,36),(2,13,53,29),(3,22,54,38),(4,15,55,31),(5,24,56,40),(6,9,49,25),(7,18,50,34),(8,11,51,27),(10,62,26,42),(12,64,28,44),(14,58,30,46),(16,60,32,48),(17,41,33,61),(19,43,35,63),(21,45,37,57),(23,47,39,59)], [(1,64,52,44),(2,45,53,57),(3,58,54,46),(4,47,55,59),(5,60,56,48),(6,41,49,61),(7,62,50,42),(8,43,51,63),(9,17,25,33),(10,34,26,18),(11,19,27,35),(12,36,28,20),(13,21,29,37),(14,38,30,22),(15,23,31,39),(16,40,32,24)], [(9,41,33),(10,34,42),(11,43,35),(12,36,44),(13,45,37),(14,38,46),(15,47,39),(16,40,48),(17,25,61),(18,62,26),(19,27,63),(20,64,28),(21,29,57),(22,58,30),(23,31,59),(24,60,32)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 |
| type | + | + | + | - | - | + | - | + | ||||||||
| image | C1 | C2 | C4 | C8 | S3 | Dic3 | C3⋊C8 | CSU2(𝔽3) | GL2(𝔽3) | U2(𝔽3) | S4 | A4⋊C4 | A4⋊C8 | CSU2(𝔽3) | GL2(𝔽3) | U2(𝔽3) |
| kernel | C2.U2(𝔽3) | C4×SL2(𝔽3) | C2×SL2(𝔽3) | SL2(𝔽3) | C4×Q8 | C2×Q8 | Q8 | C4 | C4 | C2 | C2×C4 | C22 | C2 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 1 | 1 | 2 |
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 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 58 | 43 |
| 0 | 0 | 27 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 42 | 46 |
| 0 | 0 | 14 | 31 |
| 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 72 |
| 0 | 10 | 0 | 0 |
| 10 | 0 | 0 | 0 |
| 0 | 0 | 62 | 36 |
| 0 | 0 | 25 | 11 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[27,0,0,0,0,27,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,58,27,0,0,43,15],[1,0,0,0,0,1,0,0,0,0,42,14,0,0,46,31],[0,1,0,0,72,72,0,0,0,0,0,1,0,0,72,72],[0,10,0,0,10,0,0,0,0,0,62,25,0,0,36,11] >;
C2.U2(𝔽3) in GAP, Magma, Sage, TeX
C_2.{\rm U}_2({\mathbb F}_3) % in TeX
G:=Group("C2.U(2,3)"); // GroupNames label
G:=SmallGroup(192,183);
// by ID
G=gap.SmallGroup(192,183);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,14,36,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^4=e^3=1,c^2=d^2=a*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=a*b^2*c,e*c*e^-1=a*b^2*c*d,f*c*f^-1=c*d,e*d*e^-1=c,f*d*f^-1=a*b^2*d,f*e*f^-1=e^-1>;
// generators/relations
Export