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## G = S3×C42order 96 = 25·3

### Direct product of C42 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C42
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — S3×C42
 Lower central C3 — S3×C42
 Upper central C1 — C42

Generators and relations for S3×C42
G = < a,b,c,d | a4=b4=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 186 in 108 conjugacy classes, 69 normal (8 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, Dic3, C12, D6, C2×C6, C42, C42, C22×C4, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×C42, C4×Dic3, C4×C12, S3×C2×C4, S3×C42
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C42, C22×C4, C4×S3, C22×S3, C2×C42, S3×C2×C4, S3×C42

Smallest permutation representation of S3×C42
On 48 points
Generators in S48
(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 15 29 21)(2 16 30 22)(3 13 31 23)(4 14 32 24)(5 25 43 35)(6 26 44 36)(7 27 41 33)(8 28 42 34)(9 47 17 39)(10 48 18 40)(11 45 19 37)(12 46 20 38)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 45 27)(14 46 28)(15 47 25)(16 48 26)(17 43 29)(18 44 30)(19 41 31)(20 42 32)(21 39 35)(22 40 36)(23 37 33)(24 38 34)
(1 31)(2 32)(3 29)(4 30)(5 19)(6 20)(7 17)(8 18)(9 41)(10 42)(11 43)(12 44)(13 21)(14 22)(15 23)(16 24)(25 37)(26 38)(27 39)(28 40)(33 47)(34 48)(35 45)(36 46)

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

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

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

S3×C42 is a maximal subgroup of
C42.282D6  C42.182D6  Dic35M4(2)  C42.200D6  C42.202D6  C12⋊M4(2)  C42.188D6  C42.93D6  C42.228D6  C42.229D6  C42.232D6  C42.131D6  C42.233D6  C42.234D6  C42.236D6  C42.237D6  C42.189D6  C42.238D6  C42.240D6  C42.241D6
S3×C42 is a maximal quotient of
Dic3.5C42  Dic3⋊C42  D6⋊C42  D6.C42  Dic35M4(2)  D6.4C42

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A ··· 4L 4M ··· 4X 6A 6B 6C 12A ··· 12L order 1 2 2 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 6 6 12 ··· 12 size 1 1 1 1 3 3 3 3 2 1 ··· 1 3 ··· 3 2 2 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 S3 D6 C4×S3 kernel S3×C42 C4×Dic3 C4×C12 S3×C2×C4 C4×S3 C42 C2×C4 C4 # reps 1 3 1 3 24 1 3 12

Matrix representation of S3×C42 in GL3(𝔽13) generated by

 8 0 0 0 1 0 0 0 1
,
 1 0 0 0 5 0 0 0 5
,
 1 0 0 0 12 12 0 1 0
,
 1 0 0 0 12 0 0 1 1
G:=sub<GL(3,GF(13))| [8,0,0,0,1,0,0,0,1],[1,0,0,0,5,0,0,0,5],[1,0,0,0,12,1,0,12,0],[1,0,0,0,12,1,0,0,1] >;

S3×C42 in GAP, Magma, Sage, TeX

S_3\times C_4^2
% in TeX

G:=Group("S3xC4^2");
// GroupNames label

G:=SmallGroup(96,78);
// by ID

G=gap.SmallGroup(96,78);
# by ID

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

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

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