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G = C2×Q8.D6order 192 = 26·3

Direct product of C2 and Q8.D6

direct product, non-abelian, soluble

Aliases: C2×Q8.D6, C23.18S4, GL2(𝔽3)⋊1C22, CSU2(𝔽3)⋊1C22, SL2(𝔽3).2C23, (C2×Q8)⋊3D6, (C22×Q8)⋊4S3, C22.27(C2×S4), C2.10(C22×S4), Q8.2(C22×S3), (C2×GL2(𝔽3))⋊1C2, (C2×CSU2(𝔽3))⋊4C2, (C22×SL2(𝔽3))⋊6C2, (C2×SL2(𝔽3))⋊5C22, SmallGroup(192,1476)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — C2×Q8.D6
C1C2Q8SL2(𝔽3)GL2(𝔽3)C2×GL2(𝔽3) — C2×Q8.D6
SL2(𝔽3) — C2×Q8.D6
C1C22C23

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 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 C2×S4
ρ143-3-333-31-101-11-1-1100000001-11-1    orthogonal lifted from C2×S4
ρ153-3-33-331-1011-1-11-100000001-1-11    orthogonal lifted from C2×S4
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 C2×S4
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 C2×S4
ρ203-3-333-3-1101-11-11-10000000-11-11    orthogonal lifted from C2×S4
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 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)

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] >;

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

Export

Character table of C2×Q8.D6 in TeX

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