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G = C2xQ8.D6order 192 = 26·3

Direct product of C2 and Q8.D6

direct product, non-abelian, soluble

Aliases: C2xQ8.D6, C23.18S4, GL2(F3):1C22, CSU2(F3):1C22, SL2(F3).2C23, (C2xQ8):3D6, (C22xQ8):4S3, C22.27(C2xS4), C2.10(C22xS4), Q8.2(C22xS3), (C2xGL2(F3)):1C2, (C2xCSU2(F3)):4C2, (C22xSL2(F3)):6C2, (C2xSL2(F3)):5C22, SmallGroup(192,1476)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(F3) — C2xQ8.D6
C1C2Q8SL2(F3)GL2(F3)C2xGL2(F3) — C2xQ8.D6
SL2(F3) — C2xQ8.D6
C1C22C23

Generators and relations for C2xQ8.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, C2, C3, C4, C22, C22, C22, S3, C6, C8, C2xC4, D4, Q8, Q8, C23, C23, Dic3, D6, C2xC6, C2xC8, M4(2), SD16, Q16, C22xC4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, SL2(F3), C2xDic3, C3:D4, C22xS3, C22xC6, C2xM4(2), C2xSD16, C2xQ16, C8.C22, C22xQ8, C2xC4oD4, CSU2(F3), GL2(F3), C2xSL2(F3), C2xSL2(F3), C2xC3:D4, C2xC8.C22, C2xCSU2(F3), C2xGL2(F3), Q8.D6, C22xSL2(F3), C2xQ8.D6
Quotients: C1, C2, C22, S3, C23, D6, S4, C22xS3, C2xS4, Q8.D6, C22xS4, C2xQ8.D6

Character table of C2xQ8.D6

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F6A6B6C6D6E6F6G8A8B8C8D
 size 1111221212866661212888888812121212
ρ111111111111111111111111111    trivial
ρ21-1-11-11-111-1-111-11-111-11-1-11-1-11    linear of order 2
ρ31111-1-11111-1-11-1-11-1-111-1-111-1-1    linear of order 2
ρ41-1-111-1-111-11-111-1-1-1-1-11111-11-1    linear of order 2
ρ51111-1-1-1-111-1-11111-1-111-1-1-1-111    linear of order 2
ρ61-1-111-11-11-11-11-11-1-1-1-1111-11-11    linear of order 2
ρ7111111-1-111111-1-11111111-1-1-1-1    linear of order 2
ρ81-1-11-111-11-1-1111-1-111-11-1-1-111-1    linear of order 2
ρ92222-2-200-12-2-2200-111-1-1110000    orthogonal lifted from D6
ρ102-2-222-200-1-22-22001111-1-1-10000    orthogonal lifted from D6
ρ1122222200-1222200-1-1-1-1-1-1-10000    orthogonal lifted from S3
ρ122-2-22-2200-1-2-222001-1-11-1110000    orthogonal lifted from D6
ρ133333-3-3110-111-1-1-10000000-1-111    orthogonal lifted from C2xS4
ρ143-3-333-31-101-11-1-1100000001-11-1    orthogonal lifted from C2xS4
ρ153-3-33-331-1011-1-11-100000001-1-11    orthogonal lifted from C2xS4
ρ16333333110-1-1-1-1110000000-1-1-1-1    orthogonal lifted from S4
ρ173-3-33-33-11011-1-1-110000000-111-1    orthogonal lifted from C2xS4
ρ18333333-1-10-1-1-1-1-1-100000001111    orthogonal lifted from S4
ρ193333-3-3-1-10-111-111000000011-1-1    orthogonal lifted from C2xS4
ρ203-3-333-3-1101-11-11-10000000-11-11    orthogonal lifted from C2xS4
ρ214-44-40000-2000000-20022000000    symplectic lifted from Q8.D6, Schur index 2
ρ2244-4-40000-2000000200-22000000    symplectic lifted from Q8.D6, Schur index 2
ρ234-44-4000010000001--3-3-1-1-3--30000    complex lifted from Q8.D6
ρ244-44-4000010000001-3--3-1-1--3-30000    complex lifted from Q8.D6
ρ2544-4-400001000000-1-3--31-1-3--30000    complex lifted from Q8.D6
ρ2644-4-400001000000-1--3-31-1--3-30000    complex lifted from Q8.D6

Smallest permutation representation of C2xQ8.D6
On 32 points
Generators in S32
(1 6)(2 5)(3 4)(7 8)(9 19)(10 20)(11 15)(12 16)(13 17)(14 18)(21 24)(22 25)(23 26)(27 30)(28 31)(29 32)
(1 12 5 19)(2 9 6 16)(3 26 8 29)(4 23 7 32)(10 15 17 14)(11 13 18 20)(21 28 30 25)(22 24 31 27)
(1 10 5 17)(2 13 6 20)(3 24 8 27)(4 21 7 30)(9 11 16 18)(12 14 19 15)(22 29 31 26)(23 25 32 28)
(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 8 5 3)(2 4 6 7)(9 32 16 23)(10 22 17 31)(11 30 18 21)(12 26 19 29)(13 28 20 25)(14 24 15 27)

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

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

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

Matrix representation of C2xQ8.D6 in GL7(F73)

72000000
07200000
00720000
0001000
0000100
0000010
0000001
,
0010000
7272720000
1000000
00096500
000656400
0000001
00000720
,
0100000
1000000
7272720000
00007200
0001000
000685648
00012589
,
1000000
0010000
7272720000
00072000
00096500
0001010
0006946564
,
72000000
1110000
00720000
000720710
0001611618
0001010
000648072

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] >;

C2xQ8.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

Export

Character table of C2xQ8.D6 in TeX

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