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## G = S3×C4⋊C4order 96 = 25·3

### Direct product of S3 and C4⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×C4⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — S3×C4⋊C4
 Lower central C3 — C6 — S3×C4⋊C4
 Upper central C1 — C22 — C4⋊C4

Generators and relations for S3×C4⋊C4
G = < a,b,c,d | a3=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 186 in 92 conjugacy classes, 49 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×6], S3 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C4⋊C4, C4⋊C4 [×3], C22×C4 [×3], C4×S3 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C2×C4⋊C4, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, S3×C2×C4 [×2], S3×C4⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4×S3 [×2], C22×S3, C2×C4⋊C4, S3×C2×C4, S3×D4, S3×Q8, S3×C4⋊C4

Character table of S3×C4⋊C4

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 12A 12B 12C 12D 12E 12F size 1 1 1 1 3 3 3 3 2 2 2 2 2 2 2 6 6 6 6 6 6 2 2 2 4 4 4 4 4 4 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 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 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 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 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 -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 -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 -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 -1 1 -1 1 linear of order 2 ρ9 1 1 -1 -1 -1 1 -1 1 1 -1 -i i i -i 1 -i -i 1 -1 i i -1 -1 1 -i i i 1 -i -1 linear of order 4 ρ10 1 1 -1 -1 1 -1 1 -1 1 1 -i i -i i -1 i -i 1 -1 -i i -1 -1 1 i -i i -1 -i 1 linear of order 4 ρ11 1 1 -1 -1 1 -1 1 -1 1 -1 -i i i -i 1 i i -1 1 -i -i -1 -1 1 -i i i 1 -i -1 linear of order 4 ρ12 1 1 -1 -1 -1 1 -1 1 1 1 -i i -i i -1 -i i -1 1 i -i -1 -1 1 i -i i -1 -i 1 linear of order 4 ρ13 1 1 -1 -1 -1 1 -1 1 1 -1 i -i -i i 1 i i 1 -1 -i -i -1 -1 1 i -i -i 1 i -1 linear of order 4 ρ14 1 1 -1 -1 1 -1 1 -1 1 1 i -i i -i -1 -i i 1 -1 i -i -1 -1 1 -i i -i -1 i 1 linear of order 4 ρ15 1 1 -1 -1 1 -1 1 -1 1 -1 i -i -i i 1 -i -i -1 1 i i -1 -1 1 i -i -i 1 i -1 linear of order 4 ρ16 1 1 -1 -1 -1 1 -1 1 1 1 i -i i -i -1 i -i -1 1 -i i -1 -1 1 -i i -i -1 i 1 linear of order 4 ρ17 2 -2 2 -2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 0 0 0 0 -1 -2 2 2 -2 -2 -2 0 0 0 0 0 0 -1 -1 -1 1 1 -1 1 -1 1 orthogonal lifted from D6 ρ19 2 2 2 2 0 0 0 0 -1 -2 -2 -2 2 2 -2 0 0 0 0 0 0 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ20 2 -2 2 -2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 2 2 0 0 0 0 -1 2 -2 -2 -2 -2 2 0 0 0 0 0 0 -1 -1 -1 1 1 1 -1 1 -1 orthogonal lifted from D6 ρ22 2 2 2 2 0 0 0 0 -1 2 2 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ23 2 -2 -2 2 -2 2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ24 2 -2 -2 2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ25 2 2 -2 -2 0 0 0 0 -1 2 2i -2i 2i -2i -2 0 0 0 0 0 0 1 1 -1 i -i i 1 -i -1 complex lifted from C4×S3 ρ26 2 2 -2 -2 0 0 0 0 -1 2 -2i 2i -2i 2i -2 0 0 0 0 0 0 1 1 -1 -i i -i 1 i -1 complex lifted from C4×S3 ρ27 2 2 -2 -2 0 0 0 0 -1 -2 -2i 2i 2i -2i 2 0 0 0 0 0 0 1 1 -1 i -i -i -1 i 1 complex lifted from C4×S3 ρ28 2 2 -2 -2 0 0 0 0 -1 -2 2i -2i -2i 2i 2 0 0 0 0 0 0 1 1 -1 -i i i -1 -i 1 complex lifted from C4×S3 ρ29 4 -4 4 -4 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ30 4 -4 -4 4 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 symplectic lifted from S3×Q8, Schur index 2

Smallest permutation representation of S3×C4⋊C4
On 48 points
Generators in S48
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 29 34)(6 30 35)(7 31 36)(8 32 33)(9 25 42)(10 26 43)(11 27 44)(12 28 41)(21 37 46)(22 38 47)(23 39 48)(24 40 45)
(1 3)(2 4)(5 7)(6 8)(9 27)(10 28)(11 25)(12 26)(13 20)(14 17)(15 18)(16 19)(21 39)(22 40)(23 37)(24 38)(29 36)(30 33)(31 34)(32 35)(41 43)(42 44)(45 47)(46 48)
(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)
(1 47 43 8)(2 46 44 7)(3 45 41 6)(4 48 42 5)(9 29 15 23)(10 32 16 22)(11 31 13 21)(12 30 14 24)(17 38 26 33)(18 37 27 36)(19 40 28 35)(20 39 25 34)

G:=sub<Sym(48)| (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,29,34)(6,30,35)(7,31,36)(8,32,33)(9,25,42)(10,26,43)(11,27,44)(12,28,41)(21,37,46)(22,38,47)(23,39,48)(24,40,45), (1,3)(2,4)(5,7)(6,8)(9,27)(10,28)(11,25)(12,26)(13,20)(14,17)(15,18)(16,19)(21,39)(22,40)(23,37)(24,38)(29,36)(30,33)(31,34)(32,35)(41,43)(42,44)(45,47)(46,48), (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), (1,47,43,8)(2,46,44,7)(3,45,41,6)(4,48,42,5)(9,29,15,23)(10,32,16,22)(11,31,13,21)(12,30,14,24)(17,38,26,33)(18,37,27,36)(19,40,28,35)(20,39,25,34)>;

G:=Group( (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,29,34)(6,30,35)(7,31,36)(8,32,33)(9,25,42)(10,26,43)(11,27,44)(12,28,41)(21,37,46)(22,38,47)(23,39,48)(24,40,45), (1,3)(2,4)(5,7)(6,8)(9,27)(10,28)(11,25)(12,26)(13,20)(14,17)(15,18)(16,19)(21,39)(22,40)(23,37)(24,38)(29,36)(30,33)(31,34)(32,35)(41,43)(42,44)(45,47)(46,48), (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), (1,47,43,8)(2,46,44,7)(3,45,41,6)(4,48,42,5)(9,29,15,23)(10,32,16,22)(11,31,13,21)(12,30,14,24)(17,38,26,33)(18,37,27,36)(19,40,28,35)(20,39,25,34) );

G=PermutationGroup([(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,29,34),(6,30,35),(7,31,36),(8,32,33),(9,25,42),(10,26,43),(11,27,44),(12,28,41),(21,37,46),(22,38,47),(23,39,48),(24,40,45)], [(1,3),(2,4),(5,7),(6,8),(9,27),(10,28),(11,25),(12,26),(13,20),(14,17),(15,18),(16,19),(21,39),(22,40),(23,37),(24,38),(29,36),(30,33),(31,34),(32,35),(41,43),(42,44),(45,47),(46,48)], [(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)], [(1,47,43,8),(2,46,44,7),(3,45,41,6),(4,48,42,5),(9,29,15,23),(10,32,16,22),(11,31,13,21),(12,30,14,24),(17,38,26,33),(18,37,27,36),(19,40,28,35),(20,39,25,34)])

Matrix representation of S3×C4⋊C4 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 0 12 0 0 1 12
,
 12 0 0 0 0 12 0 0 0 0 0 1 0 0 1 0
,
 9 3 0 0 3 4 0 0 0 0 12 0 0 0 0 12
,
 0 1 0 0 12 0 0 0 0 0 8 0 0 0 0 8
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,12],[12,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0],[9,3,0,0,3,4,0,0,0,0,12,0,0,0,0,12],[0,12,0,0,1,0,0,0,0,0,8,0,0,0,0,8] >;

S3×C4⋊C4 in GAP, Magma, Sage, TeX

S_3\times C_4\rtimes C_4
% in TeX

G:=Group("S3xC4:C4");
// GroupNames label

G:=SmallGroup(96,98);
// by ID

G=gap.SmallGroup(96,98);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,188,50,2309]);
// Polycyclic

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

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