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## G = (C2×C4).S4order 192 = 26·3

### 15th non-split extension by C2×C4 of S4 acting via S4/A4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — (C2×C4).S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — Q8⋊Dic3 — (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=b2, f2=a, 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: 291 in 79 conjugacy classes, 21 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C2×C6, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, SL2(𝔽3), C2×Dic3, C2×C12, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C4⋊Dic3, C2×SL2(𝔽3), C4.A4, C23.36D4, Q8⋊Dic3, C2×C4.A4, (C2×C4).S4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C2×Dic3, S4, A4⋊C4, C2×S4, C4.S4, C4.3S4, C2×A4⋊C4, (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 i -i -i i -1 1 -1 i -i -i i -1 -1 1 1 linear of order 4 ρ6 1 -1 -1 1 -1 1 1 1 -1 1 -1 i i -i -i -1 1 -1 i -i i -i 1 1 -1 -1 linear of order 4 ρ7 1 -1 -1 1 1 -1 1 -1 1 1 -1 -i i i -i -1 1 -1 -i i i -i -1 -1 1 1 linear of order 4 ρ8 1 -1 -1 1 -1 1 1 1 -1 1 -1 -i -i i i -1 1 -1 -i i -i i 1 1 -1 -1 linear of order 4 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 symplectic lifted from Dic3, Schur index 2 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 symplectic lifted from Dic3, Schur index 2 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 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 C2×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 S4 ρ17 3 -3 -3 3 1 -1 0 3 -3 -1 1 i i -i -i 0 0 0 -i i -i i 0 0 0 0 complex lifted from A4⋊C4 ρ18 3 -3 -3 3 -1 1 0 -3 3 -1 1 -i i i -i 0 0 0 i -i -i i 0 0 0 0 complex lifted from A4⋊C4 ρ19 3 -3 -3 3 -1 1 0 -3 3 -1 1 i -i -i i 0 0 0 -i i i -i 0 0 0 0 complex lifted from A4⋊C4 ρ20 3 -3 -3 3 1 -1 0 3 -3 -1 1 -i -i i i 0 0 0 i -i i -i 0 0 0 0 complex lifted from A4⋊C4 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 orthogonal lifted from C4.3S4 ρ22 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 orthogonal lifted from C4.3S4 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 orthogonal lifted from C4.3S4 ρ24 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 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 7)(2 8)(3 5)(4 6)(9 21)(10 22)(11 23)(12 24)(13 36)(14 33)(15 34)(16 35)(17 46)(18 47)(19 48)(20 45)(25 29)(26 30)(27 31)(28 32)(37 41)(38 42)(39 43)(40 44)(49 56)(50 53)(51 54)(52 55)(57 61)(58 62)(59 63)(60 64)
(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 9 3 11)(2 10 4 12)(5 23 7 21)(6 24 8 22)(13 25 15 27)(14 26 16 28)(17 52 19 50)(18 49 20 51)(29 34 31 36)(30 35 32 33)(37 58 39 60)(38 59 40 57)(41 62 43 64)(42 63 44 61)(45 54 47 56)(46 55 48 53)
(1 15 3 13)(2 16 4 14)(5 36 7 34)(6 33 8 35)(9 25 11 27)(10 26 12 28)(17 44 19 42)(18 41 20 43)(21 29 23 31)(22 30 24 32)(37 45 39 47)(38 46 40 48)(49 64 51 62)(50 61 52 63)(53 57 55 59)(54 58 56 60)
(9 15 27)(10 16 28)(11 13 25)(12 14 26)(17 52 63)(18 49 64)(19 50 61)(20 51 62)(21 34 31)(22 35 32)(23 36 29)(24 33 30)(45 54 58)(46 55 59)(47 56 60)(48 53 57)
(1 42 7 38)(2 41 8 37)(3 44 5 40)(4 43 6 39)(9 50 21 53)(10 49 22 56)(11 52 23 55)(12 51 24 54)(13 17 36 46)(14 20 33 45)(15 19 34 48)(16 18 35 47)(25 63 29 59)(26 62 30 58)(27 61 31 57)(28 64 32 60)

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

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

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

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 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72
,
 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 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
,
 0 72 1 0 0 0 0 0 72 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 72 0 0 0 0 0 0 0 72 0 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 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 27 0 0 0 0 0 27 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 43 69 4 69 0 0 0 69 4 43 69 0 0 0 4 43 69 69 0 0 0 69 69 69 30

G:=sub<GL(7,GF(73))| [72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,7,66,7,0,0,0,66,0,7,7,0,0,0,7,66,0,7,0,0,0,66,66,66,0],[0,0,1,0,0,0,0,72,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,72,72,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0],[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],[0,27,0,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,43,69,4,69,0,0,0,69,4,43,69,0,0,0,4,43,69,69,0,0,0,69,69,69,30] >;

(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(192,985);
// by ID

G=gap.SmallGroup(192,985);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,28,1373,680,451,1684,655,172,1013,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^4=e^3=1,c^2=d^2=b^2,f^2=a,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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