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## G = C4×S3×D4order 192 = 26·3

### Direct product of C4, S3 and D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C4×S3×D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — C4×S3×D4
 Lower central C3 — C6 — C4×S3×D4
 Upper central C1 — C2×C4 — C4×D4

Generators and relations for C4×S3×D4
G = < a,b,c,d,e | a4=b3=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1032 in 426 conjugacy classes, 169 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C23×C4, C22×D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, S3×C2×C4, C2×D12, S3×D4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C2×C4×D4, S3×C42, C4×D12, S3×C22⋊C4, Dic34D4, S3×C4⋊C4, Dic35D4, C4×C3⋊D4, D4×Dic3, D4×C12, S3×C22×C4, C2×S3×D4, C4×S3×D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C24, C4×S3, C22×S3, C4×D4, C23×C4, C22×D4, C2×C4○D4, S3×C2×C4, S3×D4, S3×C23, C2×C4×D4, S3×C22×C4, C2×S3×D4, S3×C4○D4, C4×S3×D4

Smallest permutation representation of C4×S3×D4
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 25 22)(2 26 23)(3 27 24)(4 28 21)(5 29 19)(6 30 20)(7 31 17)(8 32 18)(9 43 37)(10 44 38)(11 41 39)(12 42 40)(13 34 48)(14 35 45)(15 36 46)(16 33 47)
(1 3)(2 4)(5 7)(6 8)(9 41)(10 42)(11 43)(12 44)(13 36)(14 33)(15 34)(16 35)(17 29)(18 30)(19 31)(20 32)(21 26)(22 27)(23 28)(24 25)(37 39)(38 40)(45 47)(46 48)
(1 45 39 6)(2 46 40 7)(3 47 37 8)(4 48 38 5)(9 32 27 16)(10 29 28 13)(11 30 25 14)(12 31 26 15)(17 23 36 42)(18 24 33 43)(19 21 34 44)(20 22 35 41)
(5 48)(6 45)(7 46)(8 47)(13 29)(14 30)(15 31)(16 32)(17 36)(18 33)(19 34)(20 35)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 D6 D6 D6 C4○D4 C4×S3 S3×D4 S3×C4○D4 kernel C4×S3×D4 S3×C42 C4×D12 S3×C22⋊C4 Dic3⋊4D4 S3×C4⋊C4 Dic3⋊5D4 C4×C3⋊D4 D4×Dic3 D4×C12 S3×C22×C4 C2×S3×D4 S3×D4 C4×D4 C4×S3 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D6 D4 C4 C2 # reps 1 1 1 2 2 1 1 2 1 1 2 1 16 1 4 1 2 1 2 1 4 8 2 2

Matrix representation of C4×S3×D4 in GL4(𝔽13) generated by

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

C4×S3×D4 in GAP, Magma, Sage, TeX

C_4\times S_3\times D_4
% in TeX

G:=Group("C4xS3xD4");
// GroupNames label

G:=SmallGroup(192,1103);
// by ID

G=gap.SmallGroup(192,1103);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,80,6278]);
// Polycyclic

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

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