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## G = C2×He3⋊C8order 432 = 24·33

### Direct product of C2 and He3⋊C8

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

 Derived series C1 — C3 — He3 — C2×He3⋊C8
 Chief series C1 — C3 — He3 — He3⋊C2 — He3⋊C4 — He3⋊C8 — C2×He3⋊C8
 Lower central He3 — C2×He3⋊C8
 Upper central C1 — C2

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 >

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

Smallest permutation representation of C2×He3⋊C8
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

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