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## G = C6×D4⋊2S3order 288 = 25·32

### Direct product of C6 and D4⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6×D4⋊2S3
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — S3×C2×C12 — C6×D4⋊2S3
 Lower central C3 — C6 — C6×D4⋊2S3
 Upper central C1 — C2×C6 — C6×D4

Generators and relations for C6×D42S3
G = < a,b,c,d,e | a6=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 682 in 355 conjugacy classes, 178 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], C22 [×8], S3 [×2], C6 [×2], C6 [×4], C6 [×17], C2×C4, C2×C4 [×15], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, C32, Dic3 [×6], C12 [×4], C12 [×8], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×8], C2×C6 [×21], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], C3×S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×4], Dic6 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×10], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×16], C3×D4 [×8], C3×D4 [×12], C3×Q8 [×4], C22×S3, C22×C6 [×4], C22×C6 [×3], C2×C4○D4, C3×Dic3 [×6], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, C62 [×4], C62 [×4], C2×Dic6, S3×C2×C4, D42S3 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C22×C12 [×3], C6×D4 [×2], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], C3×Dic6 [×4], S3×C12 [×4], C6×Dic3, C6×Dic3 [×10], C3×C3⋊D4 [×8], C6×C12, D4×C32 [×4], S3×C2×C6, C2×C62 [×2], C2×D42S3, C6×C4○D4, C6×Dic6, S3×C2×C12, C3×D42S3 [×8], Dic3×C2×C6 [×2], C6×C3⋊D4 [×2], D4×C3×C6, C6×D42S3
Quotients: C1, C2 [×15], C3, C22 [×35], S3, C6 [×15], C23 [×15], D6 [×7], C2×C6 [×35], C4○D4 [×2], C24, C3×S3, C22×S3 [×7], C22×C6 [×15], C2×C4○D4, S3×C6 [×7], D42S3 [×2], C3×C4○D4 [×2], S3×C23, C23×C6, S3×C2×C6 [×7], C2×D42S3, C6×C4○D4, C3×D42S3 [×2], S3×C22×C6, C6×D42S3

Smallest permutation representation of C6×D42S3
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 8 23 28)(2 9 24 29)(3 10 19 30)(4 11 20 25)(5 12 21 26)(6 7 22 27)(13 31 41 46)(14 32 42 47)(15 33 37 48)(16 34 38 43)(17 35 39 44)(18 36 40 45)
(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 35)(2 36)(3 31)(4 32)(5 33)(6 34)(7 38)(8 39)(9 40)(10 41)(11 42)(12 37)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6F 6G ··· 6W 6X ··· 6AI 6AJ 6AK 6AL 6AM 12A 12B 12C 12D 12E ··· 12L 12M ··· 12R 12S ··· 12Z order 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 2 2 6 6 1 1 2 2 2 2 2 3 3 3 3 6 6 6 6 1 ··· 1 2 ··· 2 4 ··· 4 6 6 6 6 2 2 2 2 3 ··· 3 4 ··· 4 6 ··· 6

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 S3 D6 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 S3×C6 C3×C4○D4 D4⋊2S3 C3×D4⋊2S3 kernel C6×D4⋊2S3 C6×Dic6 S3×C2×C12 C3×D4⋊2S3 Dic3×C2×C6 C6×C3⋊D4 D4×C3×C6 C2×D4⋊2S3 C2×Dic6 S3×C2×C4 D4⋊2S3 C22×Dic3 C2×C3⋊D4 C6×D4 C6×D4 C2×C12 C3×D4 C22×C6 C3×C6 C2×D4 C2×C4 D4 C23 C6 C6 C2 # reps 1 1 1 8 2 2 1 2 2 2 16 4 4 2 1 1 4 2 4 2 2 8 4 8 2 4

Matrix representation of C6×D42S3 in GL4(𝔽13) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 0 12 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 12
,
 3 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 0 12 0 0 12 0 0 0 0 0 0 5 0 0 8 0
G:=sub<GL(4,GF(13))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12],[3,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,12,0,0,0,0,0,0,8,0,0,5,0] >;

C6×D42S3 in GAP, Magma, Sage, TeX

C_6\times D_4\rtimes_2S_3
% in TeX

G:=Group("C6xD4:2S3");
// GroupNames label

G:=SmallGroup(288,993);
// by ID

G=gap.SmallGroup(288,993);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,268,1571,409,9414]);
// Polycyclic

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

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