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## G = S3×C4⋊1D4order 192 = 26·3

### Direct product of S3 and C4⋊1D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — S3×C4⋊1D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×S3×D4 — S3×C4⋊1D4
 Lower central C3 — C2×C6 — S3×C4⋊1D4
 Upper central C1 — C22 — C4⋊1D4

Generators and relations for S3×C41D4
G = < a,b,c,d,e | a3=b2=c4=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1584 in 498 conjugacy classes, 131 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22×C4, C2×D4, C2×D4, C24, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C2×C42, C41D4, C41D4, C22×D4, C4×Dic3, C4×C12, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C2×C41D4, S3×C42, C4⋊D12, C123D4, C3×C41D4, C2×S3×D4, S3×C41D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C41D4, C22×D4, S3×D4, S3×C23, C2×C41D4, C2×S3×D4, S3×C41D4

Smallest permutation representation of S3×C41D4
On 48 points
Generators in S48
(1 23 34)(2 24 35)(3 21 36)(4 22 33)(5 30 42)(6 31 43)(7 32 44)(8 29 41)(9 38 25)(10 39 26)(11 40 27)(12 37 28)(13 18 46)(14 19 47)(15 20 48)(16 17 45)
(1 3)(2 4)(5 32)(6 29)(7 30)(8 31)(9 40)(10 37)(11 38)(12 39)(13 15)(14 16)(17 47)(18 48)(19 45)(20 46)(21 34)(22 35)(23 36)(24 33)(25 27)(26 28)(41 43)(42 44)
(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 41 16 26)(2 42 13 27)(3 43 14 28)(4 44 15 25)(5 18 11 24)(6 19 12 21)(7 20 9 22)(8 17 10 23)(29 45 39 34)(30 46 40 35)(31 47 37 36)(32 48 38 33)
(1 2)(3 4)(5 10)(6 9)(7 12)(8 11)(13 16)(14 15)(17 18)(19 20)(21 22)(23 24)(25 43)(26 42)(27 41)(28 44)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(45 46)(47 48)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A ··· 4F 4G ··· 4L 6A 6B 6C 6D 6E 6F 6G 12A ··· 12F order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 6 6 6 6 6 6 12 ··· 12 size 1 1 1 1 3 3 3 3 4 4 4 4 12 12 12 12 2 2 ··· 2 6 ··· 6 2 2 2 8 8 8 8 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D6 D6 S3×D4 kernel S3×C4⋊1D4 S3×C42 C4⋊D12 C12⋊3D4 C3×C4⋊1D4 C2×S3×D4 C4⋊1D4 C4×S3 C42 C2×D4 C4 # reps 1 1 1 6 1 6 1 12 1 6 6

Matrix representation of S3×C41D4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 11 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 11 0 0 0 0 1 12 0 0 0 0 0 0 12 5 0 0 0 0 10 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 2 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 3 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,12,10,0,0,0,0,5,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,2,1,0,0,0,0,0,0,1,3,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

S3×C41D4 in GAP, Magma, Sage, TeX

S_3\times C_4\rtimes_1D_4
% in TeX

G:=Group("S3xC4:1D4");
// GroupNames label

G:=SmallGroup(192,1273);
// by ID

G=gap.SmallGroup(192,1273);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,570,185,6278]);
// Polycyclic

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

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