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## G = C2×C33⋊Q8order 432 = 24·33

### Direct product of C2 and C33⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊S3 — C2×C33⋊Q8
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C33⋊C4 — C33⋊Q8 — C2×C33⋊Q8
 Lower central C33 — C3×C3⋊S3 — C2×C33⋊Q8
 Upper central C1 — C2

Generators and relations for C2×C33⋊Q8
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, bd=db, ebe-1=c, fbf-1=bc-1, cd=dc, ece-1=b-1, fcf-1=b-1c-1, de=ed, fdf-1=d-1, fef-1=e-1 >

Subgroups: 688 in 90 conjugacy classes, 31 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, Q8, C32, C32, Dic3, C12, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C33, C32⋊C4, C32⋊C4, S3×C6, C2×C3⋊S3, C2×Dic6, C3×C3⋊S3, C32×C6, PSU3(𝔽2), C2×C32⋊C4, C2×C32⋊C4, C3×C32⋊C4, C33⋊C4, C6×C3⋊S3, C2×PSU3(𝔽2), C33⋊Q8, C6×C32⋊C4, C2×C33⋊C4, C2×C33⋊Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, C22×S3, C2×Dic6, PSU3(𝔽2), C2×PSU3(𝔽2), C33⋊Q8, C2×C33⋊Q8

Character table of C2×C33⋊Q8

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D size 1 1 9 9 2 8 8 8 18 18 54 54 54 54 2 8 8 8 18 18 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 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 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 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 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 2 2 -1 -1 2 -1 -2 -2 0 0 0 0 -1 2 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 -2 -2 2 -1 -1 2 -1 -2 2 0 0 0 0 1 -2 1 1 1 -1 1 -1 1 -1 orthogonal lifted from D6 ρ11 2 -2 -2 2 -1 -1 2 -1 2 -2 0 0 0 0 1 -2 1 1 1 -1 -1 1 -1 1 orthogonal lifted from D6 ρ12 2 2 2 2 -1 -1 2 -1 2 2 0 0 0 0 -1 2 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 -2 2 -2 2 2 2 2 0 0 0 0 0 0 -2 -2 -2 -2 2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 2 -2 -2 2 2 2 2 0 0 0 0 0 0 2 2 2 2 -2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ15 2 2 -2 -2 -1 -1 2 -1 0 0 0 0 0 0 -1 2 -1 -1 1 1 -√3 -√3 √3 √3 symplectic lifted from Dic6, Schur index 2 ρ16 2 2 -2 -2 -1 -1 2 -1 0 0 0 0 0 0 -1 2 -1 -1 1 1 √3 √3 -√3 -√3 symplectic lifted from Dic6, Schur index 2 ρ17 2 -2 2 -2 -1 -1 2 -1 0 0 0 0 0 0 1 -2 1 1 -1 1 -√3 √3 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ18 2 -2 2 -2 -1 -1 2 -1 0 0 0 0 0 0 1 -2 1 1 -1 1 √3 -√3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ19 8 -8 0 0 8 -1 -1 -1 0 0 0 0 0 0 -8 1 1 1 0 0 0 0 0 0 orthogonal lifted from C2×PSU3(𝔽2) ρ20 8 8 0 0 8 -1 -1 -1 0 0 0 0 0 0 8 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from PSU3(𝔽2) ρ21 8 8 0 0 -4 1-3√-3/2 -1 1+3√-3/2 0 0 0 0 0 0 -4 -1 1-3√-3/2 1+3√-3/2 0 0 0 0 0 0 complex lifted from C33⋊Q8 ρ22 8 -8 0 0 -4 1-3√-3/2 -1 1+3√-3/2 0 0 0 0 0 0 4 1 -1+3√-3/2 -1-3√-3/2 0 0 0 0 0 0 complex faithful ρ23 8 -8 0 0 -4 1+3√-3/2 -1 1-3√-3/2 0 0 0 0 0 0 4 1 -1-3√-3/2 -1+3√-3/2 0 0 0 0 0 0 complex faithful ρ24 8 8 0 0 -4 1+3√-3/2 -1 1-3√-3/2 0 0 0 0 0 0 -4 -1 1+3√-3/2 1-3√-3/2 0 0 0 0 0 0 complex lifted from C33⋊Q8

Smallest permutation representation of C2×C33⋊Q8
On 48 points
Generators in S48
(1 11)(2 12)(3 9)(4 10)(5 24)(6 21)(7 22)(8 23)(13 41)(14 42)(15 43)(16 44)(17 26)(18 27)(19 28)(20 25)(29 45)(30 46)(31 47)(32 48)(33 37)(34 38)(35 39)(36 40)
(1 19 6)(3 8 17)(9 23 26)(11 28 21)(13 38 47)(14 48 39)(15 45 40)(16 37 46)(29 36 43)(30 44 33)(31 41 34)(32 35 42)
(2 7 20)(4 18 5)(10 27 24)(12 22 25)(13 47 38)(14 48 39)(15 40 45)(16 37 46)(29 43 36)(30 44 33)(31 34 41)(32 35 42)
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 23 26)(10 24 27)(11 21 28)(12 22 25)(13 47 38)(14 48 39)(15 45 40)(16 46 37)(29 36 43)(30 33 44)(31 34 41)(32 35 42)
(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)
(1 40 3 38)(2 39 4 37)(5 46 7 48)(6 45 8 47)(9 34 11 36)(10 33 12 35)(13 19 15 17)(14 18 16 20)(21 29 23 31)(22 32 24 30)(25 42 27 44)(26 41 28 43)

G:=sub<Sym(48)| (1,11)(2,12)(3,9)(4,10)(5,24)(6,21)(7,22)(8,23)(13,41)(14,42)(15,43)(16,44)(17,26)(18,27)(19,28)(20,25)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40), (1,19,6)(3,8,17)(9,23,26)(11,28,21)(13,38,47)(14,48,39)(15,45,40)(16,37,46)(29,36,43)(30,44,33)(31,41,34)(32,35,42), (2,7,20)(4,18,5)(10,27,24)(12,22,25)(13,47,38)(14,48,39)(15,40,45)(16,37,46)(29,43,36)(30,44,33)(31,34,41)(32,35,42), (1,6,19)(2,7,20)(3,8,17)(4,5,18)(9,23,26)(10,24,27)(11,21,28)(12,22,25)(13,47,38)(14,48,39)(15,45,40)(16,46,37)(29,36,43)(30,33,44)(31,34,41)(32,35,42), (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), (1,40,3,38)(2,39,4,37)(5,46,7,48)(6,45,8,47)(9,34,11,36)(10,33,12,35)(13,19,15,17)(14,18,16,20)(21,29,23,31)(22,32,24,30)(25,42,27,44)(26,41,28,43)>;

G:=Group( (1,11)(2,12)(3,9)(4,10)(5,24)(6,21)(7,22)(8,23)(13,41)(14,42)(15,43)(16,44)(17,26)(18,27)(19,28)(20,25)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40), (1,19,6)(3,8,17)(9,23,26)(11,28,21)(13,38,47)(14,48,39)(15,45,40)(16,37,46)(29,36,43)(30,44,33)(31,41,34)(32,35,42), (2,7,20)(4,18,5)(10,27,24)(12,22,25)(13,47,38)(14,48,39)(15,40,45)(16,37,46)(29,43,36)(30,44,33)(31,34,41)(32,35,42), (1,6,19)(2,7,20)(3,8,17)(4,5,18)(9,23,26)(10,24,27)(11,21,28)(12,22,25)(13,47,38)(14,48,39)(15,45,40)(16,46,37)(29,36,43)(30,33,44)(31,34,41)(32,35,42), (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), (1,40,3,38)(2,39,4,37)(5,46,7,48)(6,45,8,47)(9,34,11,36)(10,33,12,35)(13,19,15,17)(14,18,16,20)(21,29,23,31)(22,32,24,30)(25,42,27,44)(26,41,28,43) );

G=PermutationGroup([[(1,11),(2,12),(3,9),(4,10),(5,24),(6,21),(7,22),(8,23),(13,41),(14,42),(15,43),(16,44),(17,26),(18,27),(19,28),(20,25),(29,45),(30,46),(31,47),(32,48),(33,37),(34,38),(35,39),(36,40)], [(1,19,6),(3,8,17),(9,23,26),(11,28,21),(13,38,47),(14,48,39),(15,45,40),(16,37,46),(29,36,43),(30,44,33),(31,41,34),(32,35,42)], [(2,7,20),(4,18,5),(10,27,24),(12,22,25),(13,47,38),(14,48,39),(15,40,45),(16,37,46),(29,43,36),(30,44,33),(31,34,41),(32,35,42)], [(1,6,19),(2,7,20),(3,8,17),(4,5,18),(9,23,26),(10,24,27),(11,21,28),(12,22,25),(13,47,38),(14,48,39),(15,45,40),(16,46,37),(29,36,43),(30,33,44),(31,34,41),(32,35,42)], [(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)], [(1,40,3,38),(2,39,4,37),(5,46,7,48),(6,45,8,47),(9,34,11,36),(10,33,12,35),(13,19,15,17),(14,18,16,20),(21,29,23,31),(22,32,24,30),(25,42,27,44),(26,41,28,43)]])

Matrix representation of C2×C33⋊Q8 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 9 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 12 0 3 3 0 9 4 3
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 9 0 9 9 4 0 9 0 4 3
,
 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 1 1 9 9 0 0 0 9
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 10 10 12 12 1 1 12 8 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 7 7 7 7 0 0 0 12
,
 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 3 3 1 1 12 12 1 5 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 7 7 0 0 6 6 0 12

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[9,0,0,0,0,0,0,12,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,0,0,1,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,9,0,9,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,9,0,1,0,0,0,0,0,9,0,0,9,0,0,0,0,4,0,0,0,3,0,0,0,0,0,0,0,0,9,0,0,9,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,1,0,3,0,0,0,0,0,1,0,0,3,0,0,0,0,9,0,0,0,3,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9],[0,0,0,1,10,0,0,7,0,0,1,0,10,0,0,7,1,0,0,0,12,0,0,7,0,1,0,0,12,0,0,7,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,8,0,0,12],[0,0,0,3,0,12,0,7,0,0,0,3,12,0,0,7,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,0,12,0,0,12,0,0,0,6,0,12,0,12,0,0,0,6,0,0,12,1,0,0,0,0,0,0,0,5,0,0,0,12] >;

C2×C33⋊Q8 in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes Q_8
% in TeX

G:=Group("C2xC3^3:Q8");
// GroupNames label

G:=SmallGroup(432,758);
// by ID

G=gap.SmallGroup(432,758);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,141,64,2804,1691,165,2693,348,530,14118]);
// 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,b*d=d*b,e*b*e^-1=c,f*b*f^-1=b*c^-1,c*d=d*c,e*c*e^-1=b^-1,f*c*f^-1=b^-1*c^-1,d*e=e*d,f*d*f^-1=d^-1,f*e*f^-1=e^-1>;
// generators/relations

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