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## G = C2×A4⋊Q8order 192 = 26·3

### Direct product of C2 and A4⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — C2×A4⋊Q8
 Chief series C1 — C22 — A4 — C2×A4 — A4⋊C4 — C2×A4⋊C4 — C2×A4⋊Q8
 Lower central A4 — C2×A4 — C2×A4⋊Q8
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×A4⋊Q8
G = < a,b,c,d,e,f | a2=b2=c2=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd-1=fbf-1=bc=cb, be=eb, dcd-1=b, ce=ec, cf=fc, de=ed, fdf-1=d-1, fef-1=e-1 >

Subgroups: 534 in 165 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, Q8, C23, C23, C23, Dic3, C12, A4, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C2×A4, C2×A4, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, A4⋊C4, C4×A4, C2×Dic6, C22×A4, C2×C22⋊Q8, A4⋊Q8, C2×A4⋊C4, C2×C4×A4, C2×A4⋊Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, S4, C22×S3, C2×Dic6, C2×S4, A4⋊Q8, C22×S4, C2×A4⋊Q8

Character table of C2×A4⋊Q8

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 12A 12B 12C 12D size 1 1 1 1 3 3 3 3 8 2 2 6 6 12 12 12 12 12 12 12 12 8 8 8 8 8 8 8 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 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 -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 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 -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 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 -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 -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 1 1 -1 -1 linear of order 2 ρ9 2 2 2 2 2 2 2 2 -1 -2 -2 -2 -2 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 2 -2 -2 -2 -2 2 2 -1 2 -2 2 -2 0 0 0 0 0 0 0 0 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ11 2 2 2 2 2 2 2 2 -1 2 2 2 2 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 -2 -2 -2 -2 2 2 -1 -2 2 -2 2 0 0 0 0 0 0 0 0 -1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ13 2 -2 -2 2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 2 -2 -2 2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ15 2 -2 2 -2 -2 2 -2 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 √3 -√3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ16 2 -2 2 -2 -2 2 -2 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 -√3 √3 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ17 2 -2 -2 2 2 -2 -2 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 -√3 √3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ18 2 -2 -2 2 2 -2 -2 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 √3 -√3 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ19 3 3 -3 -3 1 1 -1 -1 0 3 -3 -1 1 1 1 -1 1 -1 -1 1 -1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ20 3 3 -3 -3 1 1 -1 -1 0 -3 3 1 -1 1 -1 1 1 -1 1 -1 -1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ21 3 3 3 3 -1 -1 -1 -1 0 -3 -3 1 1 1 1 -1 -1 1 1 -1 -1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ22 3 3 3 3 -1 -1 -1 -1 0 3 3 -1 -1 1 -1 1 -1 1 -1 1 -1 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ23 3 3 -3 -3 1 1 -1 -1 0 3 -3 -1 1 -1 -1 1 -1 1 1 -1 1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ24 3 3 -3 -3 1 1 -1 -1 0 -3 3 1 -1 -1 1 -1 -1 1 -1 1 1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ25 3 3 3 3 -1 -1 -1 -1 0 -3 -3 1 1 -1 -1 1 1 -1 -1 1 1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ26 3 3 3 3 -1 -1 -1 -1 0 3 3 -1 -1 -1 1 -1 1 -1 1 -1 1 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ27 6 -6 6 -6 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from A4⋊Q8, Schur index 2 ρ28 6 -6 -6 6 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from A4⋊Q8, Schur index 2

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

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

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

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

Matrix representation of C2×A4⋊Q8 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 12 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 12 0 0 0 1 0 12
,
 0 8 0 0 0 8 12 0 0 0 0 0 1 0 11 0 0 0 0 12 0 0 0 1 12
,
 10 4 0 0 0 4 3 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 8 12 0 0 0 0 5 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 12 0

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,12,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0,0,0,12,0,0,0,0,0,12],[0,8,0,0,0,8,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,11,12,12],[10,4,0,0,0,4,3,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[8,0,0,0,0,12,5,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0] >;

C2×A4⋊Q8 in GAP, Magma, Sage, TeX

C_2\times A_4\rtimes Q_8
% in TeX

G:=Group("C2xA4:Q8");
// GroupNames label

G:=SmallGroup(192,1468);
// by ID

G=gap.SmallGroup(192,1468);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,56,254,58,1124,4037,285,2358,475]);
// Polycyclic

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

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