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## G = C2×C4.S4order 192 = 26·3

### Direct product of C2 and C4.S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C2×C4.S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — CSU2(𝔽3) — C2×CSU2(𝔽3) — C2×C4.S4
 Lower central SL2(𝔽3) — C2×C4.S4
 Upper central C1 — C22 — C2×C4

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

Subgroups: 459 in 141 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C2×C6, C2×C8, M4(2), SD16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, SL2(𝔽3), Dic6, C2×Dic3, C2×C12, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, CSU2(𝔽3), C2×SL2(𝔽3), C4.A4, C2×Dic6, C2×C8.C22, C2×CSU2(𝔽3), C4.S4, C2×C4.A4, C2×C4.S4
Quotients: C1, C2, C22, S3, C23, D6, S4, C22×S3, C2×S4, C4.S4, C22×S4, C2×C4.S4

Character table of C2×C4.S4

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

Smallest permutation representation of C2×C4.S4
On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 23)(6 24)(7 21)(8 22)(9 52)(10 49)(11 50)(12 51)(13 31)(14 32)(15 29)(16 30)(17 34)(18 35)(19 36)(20 33)(37 55)(38 56)(39 53)(40 54)(41 59)(42 60)(43 57)(44 58)(45 63)(46 64)(47 61)(48 62)
(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)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 13 3 15)(2 14 4 16)(5 20 7 18)(6 17 8 19)(9 63 11 61)(10 64 12 62)(21 35 23 33)(22 36 24 34)(25 31 27 29)(26 32 28 30)(37 41 39 43)(38 42 40 44)(45 50 47 52)(46 51 48 49)(53 57 55 59)(54 58 56 60)
(1 17 3 19)(2 18 4 20)(5 14 7 16)(6 15 8 13)(9 56 11 54)(10 53 12 55)(21 30 23 32)(22 31 24 29)(25 34 27 36)(26 35 28 33)(37 49 39 51)(38 50 40 52)(41 48 43 46)(42 45 44 47)(57 64 59 62)(58 61 60 63)
(5 14 18)(6 15 19)(7 16 20)(8 13 17)(9 63 58)(10 64 59)(11 61 60)(12 62 57)(21 30 33)(22 31 34)(23 32 35)(24 29 36)(41 49 46)(42 50 47)(43 51 48)(44 52 45)
(1 54 3 56)(2 53 4 55)(5 59 7 57)(6 58 8 60)(9 17 11 19)(10 20 12 18)(13 61 15 63)(14 64 16 62)(21 43 23 41)(22 42 24 44)(25 40 27 38)(26 39 28 37)(29 45 31 47)(30 48 32 46)(33 51 35 49)(34 50 36 52)

G:=sub<Sym(64)| (1,25)(2,26)(3,27)(4,28)(5,23)(6,24)(7,21)(8,22)(9,52)(10,49)(11,50)(12,51)(13,31)(14,32)(15,29)(16,30)(17,34)(18,35)(19,36)(20,33)(37,55)(38,56)(39,53)(40,54)(41,59)(42,60)(43,57)(44,58)(45,63)(46,64)(47,61)(48,62), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,13,3,15)(2,14,4,16)(5,20,7,18)(6,17,8,19)(9,63,11,61)(10,64,12,62)(21,35,23,33)(22,36,24,34)(25,31,27,29)(26,32,28,30)(37,41,39,43)(38,42,40,44)(45,50,47,52)(46,51,48,49)(53,57,55,59)(54,58,56,60), (1,17,3,19)(2,18,4,20)(5,14,7,16)(6,15,8,13)(9,56,11,54)(10,53,12,55)(21,30,23,32)(22,31,24,29)(25,34,27,36)(26,35,28,33)(37,49,39,51)(38,50,40,52)(41,48,43,46)(42,45,44,47)(57,64,59,62)(58,61,60,63), (5,14,18)(6,15,19)(7,16,20)(8,13,17)(9,63,58)(10,64,59)(11,61,60)(12,62,57)(21,30,33)(22,31,34)(23,32,35)(24,29,36)(41,49,46)(42,50,47)(43,51,48)(44,52,45), (1,54,3,56)(2,53,4,55)(5,59,7,57)(6,58,8,60)(9,17,11,19)(10,20,12,18)(13,61,15,63)(14,64,16,62)(21,43,23,41)(22,42,24,44)(25,40,27,38)(26,39,28,37)(29,45,31,47)(30,48,32,46)(33,51,35,49)(34,50,36,52)>;

G:=Group( (1,25)(2,26)(3,27)(4,28)(5,23)(6,24)(7,21)(8,22)(9,52)(10,49)(11,50)(12,51)(13,31)(14,32)(15,29)(16,30)(17,34)(18,35)(19,36)(20,33)(37,55)(38,56)(39,53)(40,54)(41,59)(42,60)(43,57)(44,58)(45,63)(46,64)(47,61)(48,62), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,13,3,15)(2,14,4,16)(5,20,7,18)(6,17,8,19)(9,63,11,61)(10,64,12,62)(21,35,23,33)(22,36,24,34)(25,31,27,29)(26,32,28,30)(37,41,39,43)(38,42,40,44)(45,50,47,52)(46,51,48,49)(53,57,55,59)(54,58,56,60), (1,17,3,19)(2,18,4,20)(5,14,7,16)(6,15,8,13)(9,56,11,54)(10,53,12,55)(21,30,23,32)(22,31,24,29)(25,34,27,36)(26,35,28,33)(37,49,39,51)(38,50,40,52)(41,48,43,46)(42,45,44,47)(57,64,59,62)(58,61,60,63), (5,14,18)(6,15,19)(7,16,20)(8,13,17)(9,63,58)(10,64,59)(11,61,60)(12,62,57)(21,30,33)(22,31,34)(23,32,35)(24,29,36)(41,49,46)(42,50,47)(43,51,48)(44,52,45), (1,54,3,56)(2,53,4,55)(5,59,7,57)(6,58,8,60)(9,17,11,19)(10,20,12,18)(13,61,15,63)(14,64,16,62)(21,43,23,41)(22,42,24,44)(25,40,27,38)(26,39,28,37)(29,45,31,47)(30,48,32,46)(33,51,35,49)(34,50,36,52) );

G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,23),(6,24),(7,21),(8,22),(9,52),(10,49),(11,50),(12,51),(13,31),(14,32),(15,29),(16,30),(17,34),(18,35),(19,36),(20,33),(37,55),(38,56),(39,53),(40,54),(41,59),(42,60),(43,57),(44,58),(45,63),(46,64),(47,61),(48,62)], [(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),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,13,3,15),(2,14,4,16),(5,20,7,18),(6,17,8,19),(9,63,11,61),(10,64,12,62),(21,35,23,33),(22,36,24,34),(25,31,27,29),(26,32,28,30),(37,41,39,43),(38,42,40,44),(45,50,47,52),(46,51,48,49),(53,57,55,59),(54,58,56,60)], [(1,17,3,19),(2,18,4,20),(5,14,7,16),(6,15,8,13),(9,56,11,54),(10,53,12,55),(21,30,23,32),(22,31,24,29),(25,34,27,36),(26,35,28,33),(37,49,39,51),(38,50,40,52),(41,48,43,46),(42,45,44,47),(57,64,59,62),(58,61,60,63)], [(5,14,18),(6,15,19),(7,16,20),(8,13,17),(9,63,58),(10,64,59),(11,61,60),(12,62,57),(21,30,33),(22,31,34),(23,32,35),(24,29,36),(41,49,46),(42,50,47),(43,51,48),(44,52,45)], [(1,54,3,56),(2,53,4,55),(5,59,7,57),(6,58,8,60),(9,17,11,19),(10,20,12,18),(13,61,15,63),(14,64,16,62),(21,43,23,41),(22,42,24,44),(25,40,27,38),(26,39,28,37),(29,45,31,47),(30,48,32,46),(33,51,35,49),(34,50,36,52)]])

Matrix representation of C2×C4.S4 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 0 7 66 7 0 0 0 66 0 7 7 0 0 0 7 66 0 7 0 0 0 66 66 66 0
,
 72 0 0 0 0 0 0 72 0 1 0 0 0 0 72 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 72 0 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 0 0 1 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0
,
 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 1 0 0
,
 0 72 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 46 0 0 0 0 0 0 0 0 46 0 0 0 0 0 46 0 0 0 0 0 0 0 0 27

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],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,66,7,66,0,0,0,7,0,66,66,0,0,0,66,7,0,66,0,0,0,7,7,7,0],[72,72,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,0],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,72,72,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0],[0,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,0,0,0,0,0,27] >;

C2×C4.S4 in GAP, Magma, Sage, TeX

C_2\times C_4.S_4
% in TeX

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

G:=SmallGroup(192,1479);
// by ID

G=gap.SmallGroup(192,1479);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,2102,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic

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

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