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## G = C6×D4.S3order 288 = 25·32

### Direct product of C6 and D4.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C6×D4.S3
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×Dic6 — C6×Dic6 — C6×D4.S3
 Lower central C3 — C6 — C12 — C6×D4.S3
 Upper central C1 — C2×C6 — C2×C12 — C6×D4

Generators and relations for C6×D4.S3
G = < a,b,c,d,e | a6=b4=c2=d3=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 362 in 163 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C6 [×2], C6 [×4], C6 [×11], C8 [×2], C2×C4, C2×C4, D4 [×2], D4, Q8 [×3], C23, C32, Dic3 [×2], C12 [×4], C12 [×4], C2×C6 [×2], C2×C6 [×17], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×C6, C3×C6 [×2], C3×C6 [×2], C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6, C2×Dic3, C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C3×D4 [×6], C3×Q8 [×3], C22×C6 [×4], C2×SD16, C3×Dic3 [×2], C3×C12 [×2], C62, C62 [×4], C2×C3⋊C8, D4.S3 [×4], C2×C24, C3×SD16 [×4], C2×Dic6, C6×D4 [×2], C6×D4, C6×Q8, C3×C3⋊C8 [×2], C3×Dic6 [×2], C3×Dic6, C6×Dic3, C6×C12, D4×C32 [×2], D4×C32, C2×C62, C2×D4.S3, C6×SD16, C6×C3⋊C8, C3×D4.S3 [×4], C6×Dic6, D4×C3×C6, C6×D4.S3
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], SD16 [×2], C2×D4, C3×S3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C22×C6, C2×SD16, S3×C6 [×3], D4.S3 [×2], C3×SD16 [×2], C2×C3⋊D4, C6×D4, C3×C3⋊D4 [×2], S3×C2×C6, C2×D4.S3, C6×SD16, C3×D4.S3 [×2], C6×C3⋊D4, C6×D4.S3

Smallest permutation representation of C6×D4.S3
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 29 19 7)(2 30 20 8)(3 25 21 9)(4 26 22 10)(5 27 23 11)(6 28 24 12)(13 35 40 47)(14 36 41 48)(15 31 42 43)(16 32 37 44)(17 33 38 45)(18 34 39 46)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 16)(14 17)(15 18)(19 26)(20 27)(21 28)(22 29)(23 30)(24 25)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(37 40)(38 41)(39 42)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 46 19 34)(2 47 20 35)(3 48 21 36)(4 43 22 31)(5 44 23 32)(6 45 24 33)(7 18 29 39)(8 13 30 40)(9 14 25 41)(10 15 26 42)(11 16 27 37)(12 17 28 38)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6O 6P ··· 6AE 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 24A ··· 24H order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 4 4 1 1 2 2 2 2 2 12 12 1 ··· 1 2 ··· 2 4 ··· 4 6 6 6 6 2 2 2 2 4 ··· 4 12 12 12 12 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D4 D4 D6 D6 SD16 C3×S3 C3⋊D4 C3×D4 C3⋊D4 C3×D4 S3×C6 S3×C6 C3×SD16 C3×C3⋊D4 C3×C3⋊D4 D4.S3 C3×D4.S3 kernel C6×D4.S3 C6×C3⋊C8 C3×D4.S3 C6×Dic6 D4×C3×C6 C2×D4.S3 C2×C3⋊C8 D4.S3 C2×Dic6 C6×D4 C6×D4 C3×C12 C62 C2×C12 C3×D4 C3×C6 C2×D4 C12 C12 C2×C6 C2×C6 C2×C4 D4 C6 C4 C22 C6 C2 # reps 1 1 4 1 1 2 2 8 2 2 1 1 1 1 2 4 2 2 2 2 2 2 4 8 4 4 2 4

Matrix representation of C6×D4.S3 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 64 0 0 0 0 0 0 64
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 55 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1
,
 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 8 0 0 0 0 0 0 64
,
 62 15 0 0 0 0 65 11 0 0 0 0 0 0 67 6 0 0 0 0 6 6 0 0 0 0 0 0 0 51 0 0 0 0 63 0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,55,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,64],[62,65,0,0,0,0,15,11,0,0,0,0,0,0,67,6,0,0,0,0,6,6,0,0,0,0,0,0,0,63,0,0,0,0,51,0] >;

C6×D4.S3 in GAP, Magma, Sage, TeX

C_6\times D_4.S_3
% in TeX

G:=Group("C6xD4.S3");
// GroupNames label

G:=SmallGroup(288,704);
// by ID

G=gap.SmallGroup(288,704);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,590,2524,648,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^4=c^2=d^3=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

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