direct product, non-abelian, soluble
Aliases: C2×He3⋊C8, C6.2F9, He3⋊(C2×C8), C3.(C2×F9), (C2×He3)⋊C8, He3⋊C4.C4, He3⋊C2⋊C8, He3⋊C4.2C22, (C2×He3⋊C2).C4, (C2×He3⋊C4).3C2, He3⋊C2.1(C2×C4), SmallGroup(432,529)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C2×He3⋊C8 |
Generators and relations for C2×He3⋊C8
G = < a,b,c,d,e | a2=b3=c3=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=d, cd=dc, ece-1=c-1, ede-1=bc-1d >
9C2
9C2
12C3
9C4
9C4
9C22
9C6
9C6
12S3
12S3
12C6
4C32
27C8
27C8
9C12
9C12
12D6
12C3×S3
12C3×S3
27C2×C8
12S3×C6
Character table of C2×He3⋊C8
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 9 | 9 | 2 | 24 | 9 | 9 | 9 | 9 | 2 | 18 | 18 | 24 | 27 | 27 | 27 | 27 | 27 | 27 | 27 | 27 | 18 | 18 | 18 | 18 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | i | -i | -i | i | i | -i | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -i | -i | -i | i | i | i | i | -i | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | i | -i | -i | i | i | -i | -i | i | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | i | i | i | -i | -i | -i | -i | i | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ9 | 1 | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | 1 | -1 | -1 | 1 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | i | -i | i | -i | linear of order 8 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | 1 | -i | i | -i | i | -1 | 1 | -1 | -1 | ζ8 | ζ83 | ζ87 | ζ8 | ζ85 | ζ87 | ζ83 | ζ85 | -i | -i | i | i | linear of order 8 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | 1 | -1 | -1 | 1 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | -i | i | -i | i | linear of order 8 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | 1 | i | -i | i | -i | -1 | 1 | -1 | -1 | ζ87 | ζ85 | ζ8 | ζ87 | ζ83 | ζ8 | ζ85 | ζ83 | i | i | -i | -i | linear of order 8 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | 1 | -i | i | -i | i | -1 | 1 | -1 | -1 | ζ85 | ζ87 | ζ83 | ζ85 | ζ8 | ζ83 | ζ87 | ζ8 | -i | -i | i | i | linear of order 8 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | 1 | i | -i | i | -i | -1 | 1 | -1 | -1 | ζ83 | ζ8 | ζ85 | ζ83 | ζ87 | ζ85 | ζ8 | ζ87 | i | i | -i | -i | linear of order 8 |
| ρ15 | 1 | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | 1 | -1 | -1 | 1 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | -i | i | -i | i | linear of order 8 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | 1 | -1 | -1 | 1 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | i | -i | i | -i | linear of order 8 |
| ρ17 | 6 | -6 | 2 | -2 | -3 | 0 | -2 | 2 | 2 | -2 | 3 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | orthogonal faithful |
| ρ18 | 6 | 6 | -2 | -2 | -3 | 0 | 2 | 2 | 2 | 2 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from He3⋊C8 |
| ρ19 | 6 | 6 | -2 | -2 | -3 | 0 | -2 | -2 | -2 | -2 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | symplectic lifted from He3⋊C8, Schur index 2 |
| ρ20 | 6 | -6 | 2 | -2 | -3 | 0 | 2 | -2 | -2 | 2 | 3 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | symplectic faithful, Schur index 2 |
| ρ21 | 6 | 6 | 2 | 2 | -3 | 0 | -2i | -2i | 2i | 2i | -3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | -i | i | complex lifted from He3⋊C8 |
| ρ22 | 6 | -6 | -2 | 2 | -3 | 0 | 2i | -2i | 2i | -2i | 3 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | -i | i | i | complex faithful |
| ρ23 | 6 | -6 | -2 | 2 | -3 | 0 | -2i | 2i | -2i | 2i | 3 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | i | -i | -i | complex faithful |
| ρ24 | 6 | 6 | 2 | 2 | -3 | 0 | 2i | 2i | -2i | -2i | -3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | i | -i | complex lifted from He3⋊C8 |
| ρ25 | 8 | -8 | 0 | 0 | 8 | -1 | 0 | 0 | 0 | 0 | -8 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×F9 |
| ρ26 | 8 | 8 | 0 | 0 | 8 | -1 | 0 | 0 | 0 | 0 | 8 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from F9 |
►On 54 points
Generators in S54
(1 2)(3 6)(4 5)(7 54)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 36)(16 37)(17 38)(18 31)(19 32)(20 33)(21 34)(22 35)(23 43)(24 44)(25 45)(26 46)(27 39)(28 40)(29 41)(30 42)
(1 34 52)(2 21 13)(3 48 45)(4 41 38)(5 29 17)(6 9 25)(7 10 28)(8 27 22)(11 16 18)(12 14 15)(19 30 20)(23 24 26)(31 50 37)(32 42 33)(35 47 39)(36 51 53)(40 54 49)(43 44 46)
(1 3 4)(2 6 5)(7 27 19)(8 20 28)(9 29 21)(10 22 30)(11 23 15)(12 16 24)(13 25 17)(14 18 26)(31 46 53)(32 54 39)(33 40 47)(34 48 41)(35 42 49)(36 50 43)(37 44 51)(38 52 45)
(1 33 51)(2 20 12)(3 40 37)(4 47 44)(5 8 24)(6 28 16)(7 26 21)(9 27 14)(10 15 17)(11 13 22)(18 29 19)(23 25 30)(31 41 32)(34 54 46)(35 50 52)(36 38 49)(39 53 48)(42 43 45)
(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)
G:=sub<Sym(54)| (1,2)(3,6)(4,5)(7,54)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,36)(16,37)(17,38)(18,31)(19,32)(20,33)(21,34)(22,35)(23,43)(24,44)(25,45)(26,46)(27,39)(28,40)(29,41)(30,42), (1,34,52)(2,21,13)(3,48,45)(4,41,38)(5,29,17)(6,9,25)(7,10,28)(8,27,22)(11,16,18)(12,14,15)(19,30,20)(23,24,26)(31,50,37)(32,42,33)(35,47,39)(36,51,53)(40,54,49)(43,44,46), (1,3,4)(2,6,5)(7,27,19)(8,20,28)(9,29,21)(10,22,30)(11,23,15)(12,16,24)(13,25,17)(14,18,26)(31,46,53)(32,54,39)(33,40,47)(34,48,41)(35,42,49)(36,50,43)(37,44,51)(38,52,45), (1,33,51)(2,20,12)(3,40,37)(4,47,44)(5,8,24)(6,28,16)(7,26,21)(9,27,14)(10,15,17)(11,13,22)(18,29,19)(23,25,30)(31,41,32)(34,54,46)(35,50,52)(36,38,49)(39,53,48)(42,43,45), (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)>;
G:=Group( (1,2)(3,6)(4,5)(7,54)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,36)(16,37)(17,38)(18,31)(19,32)(20,33)(21,34)(22,35)(23,43)(24,44)(25,45)(26,46)(27,39)(28,40)(29,41)(30,42), (1,34,52)(2,21,13)(3,48,45)(4,41,38)(5,29,17)(6,9,25)(7,10,28)(8,27,22)(11,16,18)(12,14,15)(19,30,20)(23,24,26)(31,50,37)(32,42,33)(35,47,39)(36,51,53)(40,54,49)(43,44,46), (1,3,4)(2,6,5)(7,27,19)(8,20,28)(9,29,21)(10,22,30)(11,23,15)(12,16,24)(13,25,17)(14,18,26)(31,46,53)(32,54,39)(33,40,47)(34,48,41)(35,42,49)(36,50,43)(37,44,51)(38,52,45), (1,33,51)(2,20,12)(3,40,37)(4,47,44)(5,8,24)(6,28,16)(7,26,21)(9,27,14)(10,15,17)(11,13,22)(18,29,19)(23,25,30)(31,41,32)(34,54,46)(35,50,52)(36,38,49)(39,53,48)(42,43,45), (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) );
G=PermutationGroup([[(1,2),(3,6),(4,5),(7,54),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,53),(15,36),(16,37),(17,38),(18,31),(19,32),(20,33),(21,34),(22,35),(23,43),(24,44),(25,45),(26,46),(27,39),(28,40),(29,41),(30,42)], [(1,34,52),(2,21,13),(3,48,45),(4,41,38),(5,29,17),(6,9,25),(7,10,28),(8,27,22),(11,16,18),(12,14,15),(19,30,20),(23,24,26),(31,50,37),(32,42,33),(35,47,39),(36,51,53),(40,54,49),(43,44,46)], [(1,3,4),(2,6,5),(7,27,19),(8,20,28),(9,29,21),(10,22,30),(11,23,15),(12,16,24),(13,25,17),(14,18,26),(31,46,53),(32,54,39),(33,40,47),(34,48,41),(35,42,49),(36,50,43),(37,44,51),(38,52,45)], [(1,33,51),(2,20,12),(3,40,37),(4,47,44),(5,8,24),(6,28,16),(7,26,21),(9,27,14),(10,15,17),(11,13,22),(18,29,19),(23,25,30),(31,41,32),(34,54,46),(35,50,52),(36,38,49),(39,53,48),(42,43,45)], [(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)]])
Matrix representation of C2×He3⋊C8 ►in GL6(ℤ)
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| -1 | -1 | 0 | -2 | -2 | -1 |
| -1 | -1 | 0 | -1 | -1 | -2 |
| -1 | 0 | 0 | 0 | 1 | -1 |
| 1 | 0 | 1 | 1 | 2 | 1 |
| 0 | 1 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 | 1 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 1 | -1 | 0 | 0 |
| -1 | 0 | -1 | 0 | -1 | -1 |
| 0 | 1 | 0 | 1 | 1 | 0 |
| 1 | -1 | 1 | 0 | -1 | 1 |
| 0 | -1 | -1 | 0 | -2 | -1 |
| -1 | 0 | -1 | -1 | -2 | -1 |
| -2 | 0 | -1 | -1 | -1 | -2 |
| 1 | 1 | 1 | 1 | 2 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | -1 | 0 | 0 |
| 1 | 1 | 1 | 0 | 1 | 2 |
| 1 | 1 | 0 | 1 | 2 | 1 |
| 0 | -1 | 0 | 0 | -1 | 0 |
| -1 | 0 | -1 | 0 | 0 | -1 |
G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,-1,-1,1,0,1,-1,-1,0,0,1,0,0,0,0,1,0,0,-2,-1,0,1,0,1,-2,-1,1,2,0,0,-1,-2,-1,1,1,1],[0,1,0,0,-1,0,-1,-1,0,0,0,1,0,0,0,1,-1,0,0,0,-1,-1,0,1,0,0,0,0,-1,1,0,0,0,0,-1,0],[1,0,-1,-2,1,0,-1,-1,0,0,1,0,1,-1,-1,-1,1,0,0,0,-1,-1,1,0,-1,-2,-2,-1,2,1,1,-1,-1,-2,2,0],[0,0,1,1,0,-1,0,0,1,1,-1,0,1,1,1,0,0,-1,0,-1,0,1,0,0,0,0,1,2,-1,0,0,0,2,1,0,-1] >;
C2×He3⋊C8 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3\rtimes C_8 % in TeX
G:=Group("C2xHe3:C8"); // GroupNames label
G:=SmallGroup(432,529);
// by ID
G=gap.SmallGroup(432,529);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,58,1684,998,795,4709,4387,2042,915,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=d,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=b*c^-1*d>;
// generators/relations
Export
Subgroup lattice of C2×He3⋊C8 in TeX
Character table of C2×He3⋊C8 in TeX