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## G = C2×Q8.D6order 192 = 26·3

### Direct product of C2 and Q8.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C2×Q8.D6
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — GL2(𝔽3) — C2×GL2(𝔽3) — C2×Q8.D6
 Lower central SL2(𝔽3) — C2×Q8.D6
 Upper central C1 — C22 — C23

Generators and relations for C2×Q8.D6
G = < a,b,c,d,e | a2=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, dbd-1=bc, dcd-1=b, ece-1=b-1c, ede-1=b2d-1 >

Subgroups: 555 in 153 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×6], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×5], C8 [×4], C2×C4 [×11], D4 [×7], Q8, Q8 [×7], C23, C23, Dic3 [×2], D6 [×4], C2×C6 [×5], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4 [×2], C2×D4 [×2], C2×Q8, C2×Q8 [×2], C2×Q8 [×4], C4○D4 [×6], SL2(𝔽3), C2×Dic3, C3⋊D4 [×4], C22×S3, C22×C6, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, CSU2(𝔽3) [×2], GL2(𝔽3) [×2], C2×SL2(𝔽3), C2×SL2(𝔽3) [×2], C2×C3⋊D4, C2×C8.C22, C2×CSU2(𝔽3), C2×GL2(𝔽3), Q8.D6 [×4], C22×SL2(𝔽3), C2×Q8.D6
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], S4, C22×S3, C2×S4 [×3], Q8.D6 [×2], C22×S4, C2×Q8.D6

Character table of C2×Q8.D6

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

Smallest permutation representation of C2×Q8.D6
On 32 points
Generators in S32
(1 4)(2 3)(5 6)(7 8)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 18)(16 19)(17 20)(27 30)(28 31)(29 32)
(1 21 3 12)(2 24 4 9)(5 31 8 20)(6 28 7 17)(10 23 25 14)(11 13 26 22)(15 30 32 19)(16 18 27 29)
(1 25 3 10)(2 22 4 13)(5 29 8 18)(6 32 7 15)(9 11 24 26)(12 14 21 23)(16 31 27 20)(17 19 28 30)
(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)
(1 7 3 6)(2 5 4 8)(9 31 24 20)(10 19 25 30)(11 29 26 18)(12 17 21 28)(13 27 22 16)(14 15 23 32)

G:=sub<Sym(32)| (1,4)(2,3)(5,6)(7,8)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,18)(16,19)(17,20)(27,30)(28,31)(29,32), (1,21,3,12)(2,24,4,9)(5,31,8,20)(6,28,7,17)(10,23,25,14)(11,13,26,22)(15,30,32,19)(16,18,27,29), (1,25,3,10)(2,22,4,13)(5,29,8,18)(6,32,7,15)(9,11,24,26)(12,14,21,23)(16,31,27,20)(17,19,28,30), (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), (1,7,3,6)(2,5,4,8)(9,31,24,20)(10,19,25,30)(11,29,26,18)(12,17,21,28)(13,27,22,16)(14,15,23,32)>;

G:=Group( (1,4)(2,3)(5,6)(7,8)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,18)(16,19)(17,20)(27,30)(28,31)(29,32), (1,21,3,12)(2,24,4,9)(5,31,8,20)(6,28,7,17)(10,23,25,14)(11,13,26,22)(15,30,32,19)(16,18,27,29), (1,25,3,10)(2,22,4,13)(5,29,8,18)(6,32,7,15)(9,11,24,26)(12,14,21,23)(16,31,27,20)(17,19,28,30), (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), (1,7,3,6)(2,5,4,8)(9,31,24,20)(10,19,25,30)(11,29,26,18)(12,17,21,28)(13,27,22,16)(14,15,23,32) );

G=PermutationGroup([(1,4),(2,3),(5,6),(7,8),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,18),(16,19),(17,20),(27,30),(28,31),(29,32)], [(1,21,3,12),(2,24,4,9),(5,31,8,20),(6,28,7,17),(10,23,25,14),(11,13,26,22),(15,30,32,19),(16,18,27,29)], [(1,25,3,10),(2,22,4,13),(5,29,8,18),(6,32,7,15),(9,11,24,26),(12,14,21,23),(16,31,27,20),(17,19,28,30)], [(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)], [(1,7,3,6),(2,5,4,8),(9,31,24,20),(10,19,25,30),(11,29,26,18),(12,17,21,28),(13,27,22,16),(14,15,23,32)])

Matrix representation of C2×Q8.D6 in GL7(𝔽73)

 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 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 1
,
 0 0 1 0 0 0 0 72 72 72 0 0 0 0 1 0 0 0 0 0 0 0 0 0 9 65 0 0 0 0 0 65 64 0 0 0 0 0 0 0 0 1 0 0 0 0 0 72 0
,
 0 1 0 0 0 0 0 1 0 0 0 0 0 0 72 72 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 0 68 5 64 8 0 0 0 12 5 8 9
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 72 72 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 9 65 0 0 0 0 0 1 0 1 0 0 0 0 69 4 65 64
,
 72 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 71 0 0 0 0 16 1 16 18 0 0 0 1 0 1 0 0 0 0 64 8 0 72

G:=sub<GL(7,GF(73))| [72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,72,1,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,9,65,0,0,0,0,0,65,64,0,0,0,0,0,0,0,0,72,0,0,0,0,0,1,0],[0,1,72,0,0,0,0,1,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,68,12,0,0,0,72,0,5,5,0,0,0,0,0,64,8,0,0,0,0,0,8,9],[1,0,72,0,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,9,1,69,0,0,0,0,65,0,4,0,0,0,0,0,1,65,0,0,0,0,0,0,64],[72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,16,1,64,0,0,0,0,1,0,8,0,0,0,71,16,1,0,0,0,0,0,18,0,72] >;

C2×Q8.D6 in GAP, Magma, Sage, TeX

C_2\times Q_8.D_6
% in TeX

G:=Group("C2xQ8.D6");
// GroupNames label

G:=SmallGroup(192,1476);
// by ID

G=gap.SmallGroup(192,1476);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,2102,451,1684,655,172,1013,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^6=1,c^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,d*b*d^-1=b*c,d*c*d^-1=b,e*c*e^-1=b^-1*c,e*d*e^-1=b^2*d^-1>;
// generators/relations

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