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## G = C2×C4.3S4order 192 = 26·3

### Direct product of C2 and C4.3S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C2×C4.3S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — GL2(𝔽3) — C2×GL2(𝔽3) — C2×C4.3S4
 Lower central SL2(𝔽3) — C2×C4.3S4
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C4.3S4
G = < a,b,c,d,e,f | a2=b4=e3=f2=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, dcd-1=b2c, ece-1=b2cd, fcf=cd, ede-1=c, fdf=b2d, fef=e-1 >

Subgroups: 763 in 169 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, D6, C2×C6, C2×C8, M4(2), D8, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, SL2(𝔽3), D12, C2×C12, C22×S3, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, GL2(𝔽3), C2×SL2(𝔽3), C4.A4, C2×D12, C2×C8⋊C22, C2×GL2(𝔽3), C4.3S4, C2×C4.A4, C2×C4.3S4
Quotients: C1, C2, C22, S3, C23, D6, S4, C22×S3, C2×S4, C4.3S4, C22×S4, C2×C4.3S4

Character table of C2×C4.3S4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D size 1 1 1 1 6 6 12 12 12 12 8 2 2 6 6 8 8 8 12 12 12 12 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 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 0 0 -1 2 -2 -2 2 1 1 -1 0 0 0 0 -1 1 -1 1 orthogonal lifted from D6 ρ10 2 -2 -2 2 2 -2 0 0 0 0 -1 -2 2 -2 2 1 1 -1 0 0 0 0 1 -1 1 -1 orthogonal lifted from D6 ρ11 2 2 2 2 2 2 0 0 0 0 -1 2 2 2 2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 2 -2 -2 0 0 0 0 -1 -2 -2 2 2 -1 -1 -1 0 0 0 0 1 1 1 1 orthogonal lifted from D6 ρ13 3 -3 -3 3 -1 1 1 -1 -1 1 0 -3 3 1 -1 0 0 0 -1 1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ14 3 -3 -3 3 -1 1 -1 1 1 -1 0 -3 3 1 -1 0 0 0 1 -1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ15 3 3 3 3 -1 -1 1 1 1 1 0 3 3 -1 -1 0 0 0 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ16 3 3 3 3 -1 -1 -1 -1 -1 -1 0 3 3 -1 -1 0 0 0 1 1 1 1 0 0 0 0 orthogonal lifted from S4 ρ17 3 3 3 3 1 1 -1 1 -1 1 0 -3 -3 -1 -1 0 0 0 1 -1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ18 3 3 3 3 1 1 1 -1 1 -1 0 -3 -3 -1 -1 0 0 0 -1 1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ19 3 -3 -3 3 1 -1 -1 -1 1 1 0 3 -3 1 -1 0 0 0 1 1 -1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ20 3 -3 -3 3 1 -1 1 1 -1 -1 0 3 -3 1 -1 0 0 0 -1 -1 1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ21 4 4 -4 -4 0 0 0 0 0 0 -2 0 0 0 0 2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from C4.3S4 ρ22 4 -4 4 -4 0 0 0 0 0 0 -2 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from C4.3S4 ρ23 4 4 -4 -4 0 0 0 0 0 0 1 0 0 0 0 -1 1 -1 0 0 0 0 √3 √3 -√3 -√3 orthogonal lifted from C4.3S4 ρ24 4 -4 4 -4 0 0 0 0 0 0 1 0 0 0 0 1 -1 -1 0 0 0 0 √3 -√3 -√3 √3 orthogonal lifted from C4.3S4 ρ25 4 -4 4 -4 0 0 0 0 0 0 1 0 0 0 0 1 -1 -1 0 0 0 0 -√3 √3 √3 -√3 orthogonal lifted from C4.3S4 ρ26 4 4 -4 -4 0 0 0 0 0 0 1 0 0 0 0 -1 1 -1 0 0 0 0 -√3 -√3 √3 √3 orthogonal lifted from C4.3S4

Smallest permutation representation of C2×C4.3S4
On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 23)(6 24)(7 21)(8 22)(9 17)(10 18)(11 19)(12 20)(13 31)(14 32)(15 29)(16 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 13 3 15)(2 14 4 16)(5 18 7 20)(6 19 8 17)(9 24 11 22)(10 21 12 23)(25 29 27 31)(26 30 28 32)
(1 17 3 19)(2 18 4 20)(5 16 7 14)(6 13 8 15)(9 25 11 27)(10 26 12 28)(21 32 23 30)(22 29 24 31)
(5 16 20)(6 13 17)(7 14 18)(8 15 19)(9 24 31)(10 21 32)(11 22 29)(12 23 30)
(1 28)(2 27)(3 26)(4 25)(5 31)(6 30)(7 29)(8 32)(9 20)(10 19)(11 18)(12 17)(13 23)(14 22)(15 21)(16 24)

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

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

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

Matrix representation of C2×C4.3S4 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
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 7 0 14 14 0 0 0 14 7 0 14 0 0 0 0 14 7 14 0 0 0 59 59 59 52
,
 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 1 0 0 0 0 0 72 0 0 0 0 0 0 1 1 1 2 0 0 0 0 72 72 72
,
 72 72 72 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 72 72 71 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 72 72 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 1 1 1 2 0 0 0 72 72 0 72
,
 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 0 0 0 66 0 59 59 0 0 0 0 59 66 59 0 0 0 59 66 0 59 0 0 0 7 7 7 21

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],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,14,0,59,0,0,0,0,7,14,59,0,0,0,14,0,7,59,0,0,0,14,14,14,52],[0,1,72,0,0,0,0,1,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,1,0,0,0,0,1,0,1,72,0,0,0,0,0,1,72,0,0,0,0,0,2,72],[72,0,0,0,0,0,0,72,0,1,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,1,0,0,0,72,0,1,0,0,0,0,72,72,0,1,0,0,0,71,0,0,1],[1,0,72,0,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,0,1,0,1,72,0,0,0,0,0,1,72,0,0,0,0,72,1,0,0,0,0,0,0,2,72],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,66,0,59,7,0,0,0,0,59,66,7,0,0,0,59,66,0,7,0,0,0,59,59,59,21] >;

C2×C4.3S4 in GAP, Magma, Sage, TeX

C_2\times C_4._3S_4
% in TeX

G:=Group("C2xC4.3S4");
// GroupNames label

G:=SmallGroup(192,1481);
// by ID

G=gap.SmallGroup(192,1481);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,2102,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic

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

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