direct product, non-abelian, soluble
Aliases: C2×SU3(𝔽2), C6.4PSU3(𝔽2), He3⋊(C2×Q8), (C2×He3)⋊Q8, He3⋊C2⋊Q8, He3⋊C4.3C22, C3.(C2×PSU3(𝔽2)), He3⋊C2.1C23, (C2×He3⋊C4).5C2, (C2×He3⋊C2).7C22, SmallGroup(432,531)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×SU3(𝔽2)
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, ebe-1=dc=cd, fbf-1=bd, ce=ec, cf=fc, ede-1=b-1, fdf-1=bc-1d-1, fef-1=e-1 >
Subgroups: 433 in 67 conjugacy classes, 23 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, Q8, C32, C12, D6, C2×C6, C2×Q8, C3×S3, C3×C6, C2×C12, C3×Q8, He3, S3×C6, C6×Q8, He3⋊C2, C2×He3, He3⋊C4, C2×He3⋊C2, SU3(𝔽2), C2×He3⋊C4, C2×SU3(𝔽2)
Quotients: C1, C2, C22, Q8, C23, C2×Q8, PSU3(𝔽2), C2×PSU3(𝔽2), SU3(𝔽2), C2×SU3(𝔽2)
►On 54 points
Generators in S54
(1 2)(3 4)(5 6)(7 37)(8 38)(9 35)(10 36)(11 33)(12 34)(13 31)(14 32)(15 20)(16 21)(17 22)(18 19)(23 54)(24 51)(25 52)(26 53)(27 43)(28 44)(29 45)(30 46)(39 49)(40 50)(41 47)(42 48)
(1 48 8)(2 42 38)(3 22 40)(4 17 50)(5 10 15)(6 36 20)(7 24 32)(9 45 28)(11 16 34)(12 33 21)(13 43 49)(14 37 51)(18 30 52)(19 46 25)(23 53 41)(26 47 54)(27 39 31)(29 44 35)
(1 4 5)(2 3 6)(7 49 18)(8 50 15)(9 47 16)(10 48 17)(11 28 26)(12 29 23)(13 30 24)(14 27 25)(19 37 39)(20 38 40)(21 35 41)(22 36 42)(31 46 51)(32 43 52)(33 44 53)(34 45 54)
(1 35 39)(2 9 49)(3 47 18)(4 41 19)(5 21 37)(6 16 7)(8 33 25)(10 29 46)(11 52 38)(12 31 17)(13 22 34)(14 50 44)(15 53 27)(20 26 43)(23 51 48)(24 42 54)(28 32 40)(30 36 45)
(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)
(1 2)(3 4)(5 6)(7 46 9 44)(8 45 10 43)(11 19 13 21)(12 22 14 20)(15 34 17 32)(16 33 18 31)(23 42 25 40)(24 41 26 39)(27 38 29 36)(28 37 30 35)(47 53 49 51)(48 52 50 54)
G:=sub<Sym(54)| (1,2)(3,4)(5,6)(7,37)(8,38)(9,35)(10,36)(11,33)(12,34)(13,31)(14,32)(15,20)(16,21)(17,22)(18,19)(23,54)(24,51)(25,52)(26,53)(27,43)(28,44)(29,45)(30,46)(39,49)(40,50)(41,47)(42,48), (1,48,8)(2,42,38)(3,22,40)(4,17,50)(5,10,15)(6,36,20)(7,24,32)(9,45,28)(11,16,34)(12,33,21)(13,43,49)(14,37,51)(18,30,52)(19,46,25)(23,53,41)(26,47,54)(27,39,31)(29,44,35), (1,4,5)(2,3,6)(7,49,18)(8,50,15)(9,47,16)(10,48,17)(11,28,26)(12,29,23)(13,30,24)(14,27,25)(19,37,39)(20,38,40)(21,35,41)(22,36,42)(31,46,51)(32,43,52)(33,44,53)(34,45,54), (1,35,39)(2,9,49)(3,47,18)(4,41,19)(5,21,37)(6,16,7)(8,33,25)(10,29,46)(11,52,38)(12,31,17)(13,22,34)(14,50,44)(15,53,27)(20,26,43)(23,51,48)(24,42,54)(28,32,40)(30,36,45), (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), (1,2)(3,4)(5,6)(7,46,9,44)(8,45,10,43)(11,19,13,21)(12,22,14,20)(15,34,17,32)(16,33,18,31)(23,42,25,40)(24,41,26,39)(27,38,29,36)(28,37,30,35)(47,53,49,51)(48,52,50,54)>;
G:=Group( (1,2)(3,4)(5,6)(7,37)(8,38)(9,35)(10,36)(11,33)(12,34)(13,31)(14,32)(15,20)(16,21)(17,22)(18,19)(23,54)(24,51)(25,52)(26,53)(27,43)(28,44)(29,45)(30,46)(39,49)(40,50)(41,47)(42,48), (1,48,8)(2,42,38)(3,22,40)(4,17,50)(5,10,15)(6,36,20)(7,24,32)(9,45,28)(11,16,34)(12,33,21)(13,43,49)(14,37,51)(18,30,52)(19,46,25)(23,53,41)(26,47,54)(27,39,31)(29,44,35), (1,4,5)(2,3,6)(7,49,18)(8,50,15)(9,47,16)(10,48,17)(11,28,26)(12,29,23)(13,30,24)(14,27,25)(19,37,39)(20,38,40)(21,35,41)(22,36,42)(31,46,51)(32,43,52)(33,44,53)(34,45,54), (1,35,39)(2,9,49)(3,47,18)(4,41,19)(5,21,37)(6,16,7)(8,33,25)(10,29,46)(11,52,38)(12,31,17)(13,22,34)(14,50,44)(15,53,27)(20,26,43)(23,51,48)(24,42,54)(28,32,40)(30,36,45), (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), (1,2)(3,4)(5,6)(7,46,9,44)(8,45,10,43)(11,19,13,21)(12,22,14,20)(15,34,17,32)(16,33,18,31)(23,42,25,40)(24,41,26,39)(27,38,29,36)(28,37,30,35)(47,53,49,51)(48,52,50,54) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,37),(8,38),(9,35),(10,36),(11,33),(12,34),(13,31),(14,32),(15,20),(16,21),(17,22),(18,19),(23,54),(24,51),(25,52),(26,53),(27,43),(28,44),(29,45),(30,46),(39,49),(40,50),(41,47),(42,48)], [(1,48,8),(2,42,38),(3,22,40),(4,17,50),(5,10,15),(6,36,20),(7,24,32),(9,45,28),(11,16,34),(12,33,21),(13,43,49),(14,37,51),(18,30,52),(19,46,25),(23,53,41),(26,47,54),(27,39,31),(29,44,35)], [(1,4,5),(2,3,6),(7,49,18),(8,50,15),(9,47,16),(10,48,17),(11,28,26),(12,29,23),(13,30,24),(14,27,25),(19,37,39),(20,38,40),(21,35,41),(22,36,42),(31,46,51),(32,43,52),(33,44,53),(34,45,54)], [(1,35,39),(2,9,49),(3,47,18),(4,41,19),(5,21,37),(6,16,7),(8,33,25),(10,29,46),(11,52,38),(12,31,17),(13,22,34),(14,50,44),(15,53,27),(20,26,43),(23,51,48),(24,42,54),(28,32,40),(30,36,45)], [(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)], [(1,2),(3,4),(5,6),(7,46,9,44),(8,45,10,43),(11,19,13,21),(12,22,14,20),(15,34,17,32),(16,33,18,31),(23,42,25,40),(24,41,26,39),(27,38,29,36),(28,37,30,35),(47,53,49,51),(48,52,50,54)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | ··· | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 24 | 18 | ··· | 18 | 1 | 1 | 9 | 9 | 9 | 9 | 24 | 18 | ··· | 18 |
32 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 6 | 6 | 8 | 8 |
| type | + | + | + | - | - | + | + | ||||
| image | C1 | C2 | C2 | Q8 | Q8 | SU3(𝔽2) | C2×SU3(𝔽2) | SU3(𝔽2) | C2×SU3(𝔽2) | PSU3(𝔽2) | C2×PSU3(𝔽2) |
| kernel | C2×SU3(𝔽2) | SU3(𝔽2) | C2×He3⋊C4 | He3⋊C2 | C2×He3 | C2 | C1 | C2 | C1 | C6 | C3 |
| # reps | 1 | 4 | 3 | 1 | 1 | 8 | 8 | 2 | 2 | 1 | 1 |
Matrix representation of C2×SU3(𝔽2) ►in GL3(𝔽7) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 5 | 3 | 3 |
| 0 | 6 | 5 |
| 6 | 2 | 3 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 5 | 5 | 4 |
| 5 | 1 | 6 |
| 3 | 5 | 1 |
| 3 | 5 | 2 |
| 4 | 6 | 5 |
| 1 | 0 | 6 |
| 0 | 6 | 1 |
| 0 | 3 | 2 |
| 3 | 6 | 3 |
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[5,0,6,3,6,2,3,5,3],[4,0,0,0,4,0,0,0,4],[5,5,3,5,1,5,4,6,1],[3,4,1,5,6,0,2,5,6],[0,0,3,6,3,6,1,2,3] >;
C2×SU3(𝔽2) in GAP, Magma, Sage, TeX
C_2\times {\rm SU}_3({\mathbb F}_2) % in TeX
G:=Group("C2xSU(3,2)"); // GroupNames label
G:=SmallGroup(432,531);
// by ID
G=gap.SmallGroup(432,531);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,141,64,3924,851,795,17477,3708,1286,537]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=1,f^2=e^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,d*b*d^-1=b*c^-1,e*b*e^-1=d*c=c*d,f*b*f^-1=b*d,c*e=e*c,c*f=f*c,e*d*e^-1=b^-1,f*d*f^-1=b*c^-1*d^-1,f*e*f^-1=e^-1>;
// generators/relations