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## G = C2×Q8⋊A4order 192 = 26·3

### Direct product of C2 and Q8⋊A4

Aliases: C2×Q8⋊A4, C24.11A4, C233SL2(𝔽3), (C2×Q8)⋊3A4, Q81(C2×A4), (Q8×C23)⋊3C3, (C22×Q8)⋊8C6, C23.22(C2×A4), C22⋊(C2×SL2(𝔽3)), C22.4(C22⋊A4), C2.2(C2×C22⋊A4), SmallGroup(192,1506)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C22×Q8 — C2×Q8⋊A4
 Chief series C1 — C2 — Q8 — C22×Q8 — Q8⋊A4 — C2×Q8⋊A4
 Lower central C22×Q8 — C2×Q8⋊A4
 Upper central C1 — C22

Generators and relations for C2×Q8⋊A4
G = < a,b,c,d,e,f | a2=b4=d2=e2=f3=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, fbf-1=c, cd=dc, ce=ec, fcf-1=bc, fdf-1=de=ed, fef-1=d >

Subgroups: 559 in 175 conjugacy classes, 22 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C2×C4, Q8, Q8, C23, C23, C23, A4, C2×C6, C22×C4, C2×Q8, C2×Q8, C24, SL2(𝔽3), C2×A4, C23×C4, C22×Q8, C22×Q8, C2×SL2(𝔽3), C22×A4, Q8×C23, Q8⋊A4, C2×Q8⋊A4
Quotients: C1, C2, C3, C6, A4, SL2(𝔽3), C2×A4, C2×SL2(𝔽3), C22⋊A4, Q8⋊A4, C2×C22⋊A4, C2×Q8⋊A4

Character table of C2×Q8⋊A4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F size 1 1 1 1 3 3 3 3 16 16 6 6 6 6 6 6 6 6 16 16 16 16 16 16 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 ζ3 ζ32 -1 1 1 1 1 -1 -1 -1 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ4 1 1 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ5 1 -1 1 -1 -1 1 1 -1 ζ32 ζ3 -1 1 1 1 1 -1 -1 -1 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ6 1 1 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ7 2 2 -2 -2 -2 2 -2 2 -1 -1 0 0 0 0 0 0 0 0 1 1 -1 1 -1 1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ8 2 -2 -2 2 2 2 -2 -2 -1 -1 0 0 0 0 0 0 0 0 1 1 1 -1 1 -1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ9 2 -2 -2 2 2 2 -2 -2 ζ6 ζ65 0 0 0 0 0 0 0 0 ζ3 ζ32 ζ3 ζ65 ζ32 ζ6 complex lifted from SL2(𝔽3) ρ10 2 -2 -2 2 2 2 -2 -2 ζ65 ζ6 0 0 0 0 0 0 0 0 ζ32 ζ3 ζ32 ζ6 ζ3 ζ65 complex lifted from SL2(𝔽3) ρ11 2 2 -2 -2 -2 2 -2 2 ζ6 ζ65 0 0 0 0 0 0 0 0 ζ3 ζ32 ζ65 ζ3 ζ6 ζ32 complex lifted from SL2(𝔽3) ρ12 2 2 -2 -2 -2 2 -2 2 ζ65 ζ6 0 0 0 0 0 0 0 0 ζ32 ζ3 ζ6 ζ32 ζ65 ζ3 complex lifted from SL2(𝔽3) ρ13 3 3 3 3 -1 -1 -1 -1 0 0 -1 -1 3 -1 -1 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ14 3 -3 3 -3 1 -1 -1 1 0 0 1 -1 -1 3 -1 1 -3 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ15 3 3 3 3 -1 -1 -1 -1 0 0 -1 -1 -1 3 -1 -1 3 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ16 3 -3 3 -3 1 -1 -1 1 0 0 1 -1 3 -1 -1 -3 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ17 3 -3 3 -3 -3 3 3 -3 0 0 1 -1 -1 -1 -1 1 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ18 3 -3 3 -3 1 -1 -1 1 0 0 -3 -1 -1 -1 3 1 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ19 3 -3 3 -3 1 -1 -1 1 0 0 1 3 -1 -1 -1 1 1 -3 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ20 3 3 3 3 -1 -1 -1 -1 0 0 3 -1 -1 -1 3 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ21 3 3 3 3 3 3 3 3 0 0 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ22 3 3 3 3 -1 -1 -1 -1 0 0 -1 3 -1 -1 -1 -1 -1 3 0 0 0 0 0 0 orthogonal lifted from A4 ρ23 6 -6 -6 6 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8⋊A4, Schur index 2 ρ24 6 6 -6 -6 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8⋊A4, Schur index 2

Smallest permutation representation of C2×Q8⋊A4
On 48 points
Generators in S48
(1 27)(2 28)(3 25)(4 26)(5 31)(6 32)(7 29)(8 30)(9 35)(10 36)(11 33)(12 34)(13 39)(14 40)(15 37)(16 38)(17 43)(18 44)(19 41)(20 42)(21 47)(22 48)(23 45)(24 46)
(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 7 3 5)(2 6 4 8)(9 13 11 15)(10 16 12 14)(17 24 19 22)(18 23 20 21)(25 31 27 29)(26 30 28 32)(33 37 35 39)(34 40 36 38)(41 48 43 46)(42 47 44 45)
(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)
(1 19 13)(2 21 11)(3 17 15)(4 23 9)(5 18 12)(6 22 14)(7 20 10)(8 24 16)(25 43 37)(26 45 35)(27 41 39)(28 47 33)(29 42 36)(30 46 38)(31 44 34)(32 48 40)

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

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

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

Matrix representation of C2×Q8⋊A4 in GL7(𝔽13)

 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 10 9 0 0 0 0 0 9 3 0 0 0 0 0 0 0 10 9 0 0 0 0 0 9 3 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 1 0 0 0 0 12 1 0
,
 0 1 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 12 0 0 0 0 1 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 12 0 0 0 0 0 1 12 0
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 12 0 0 0 0 1 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 9 3 0 0 0 0 0 0 0 9 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0

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

C2×Q8⋊A4 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes A_4
% in TeX

G:=Group("C2xQ8:A4");
// GroupNames label

G:=SmallGroup(192,1506);
// by ID

G=gap.SmallGroup(192,1506);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,135,262,851,172,1524,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^4=d^2=e^2=f^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,f*b*f^-1=c,c*d=d*c,c*e=e*c,f*c*f^-1=b*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

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