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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×D12 — C6×D12 — 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=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Subgroups: 490 in 179 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C22, C22 [×8], S3 [×2], C6 [×2], C6 [×4], C6 [×13], C8 [×2], C2×C4, D4 [×2], D4 [×4], C23 [×2], C32, C12 [×4], C12 [×2], D6 [×4], C2×C6 [×2], C2×C6 [×21], C2×C8, D8 [×4], C2×D4, C2×D4, C3×S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C3⋊C8 [×2], C24 [×2], D12 [×2], D12, C2×C12 [×2], C2×C12, C3×D4 [×4], C3×D4 [×9], C22×S3, C22×C6 [×5], C2×D8, C3×C12 [×2], S3×C6 [×4], C62, C62 [×4], C2×C3⋊C8, D4⋊S3 [×4], C2×C24, C3×D8 [×4], C2×D12, C6×D4 [×2], C6×D4 [×2], C3×C3⋊C8 [×2], C3×D12 [×2], C3×D12, C6×C12, D4×C32 [×2], D4×C32, S3×C2×C6, C2×C62, C2×D4⋊S3, C6×D8, C6×C3⋊C8, C3×D4⋊S3 [×4], C6×D12, D4×C3×C6, C6×D4⋊S3
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], D8 [×2], C2×D4, C3×S3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C22×C6, C2×D8, S3×C6 [×3], D4⋊S3 [×2], C3×D8 [×2], C2×C3⋊D4, C6×D4, C3×C3⋊D4 [×2], S3×C2×C6, C2×D4⋊S3, C6×D8, 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 11 19 29)(2 12 20 30)(3 7 21 25)(4 8 22 26)(5 9 23 27)(6 10 24 28)(13 47 33 41)(14 48 34 42)(15 43 35 37)(16 44 36 38)(17 45 31 39)(18 46 32 40)
(1 29)(2 30)(3 25)(4 26)(5 27)(6 28)(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(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 40)(2 41)(3 42)(4 37)(5 38)(6 39)(7 34)(8 35)(9 36)(10 31)(11 32)(12 33)(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), (1,11,19,29)(2,12,20,30)(3,7,21,25)(4,8,22,26)(5,9,23,27)(6,10,24,28)(13,47,33,41)(14,48,34,42)(15,43,35,37)(16,44,36,38)(17,45,31,39)(18,46,32,40), (1,29)(2,30)(3,25)(4,26)(5,27)(6,28)(7,21)(8,22)(9,23)(10,24)(11,19)(12,20)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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), (1,40)(2,41)(3,42)(4,37)(5,38)(6,39)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(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), (1,11,19,29)(2,12,20,30)(3,7,21,25)(4,8,22,26)(5,9,23,27)(6,10,24,28)(13,47,33,41)(14,48,34,42)(15,43,35,37)(16,44,36,38)(17,45,31,39)(18,46,32,40), (1,29)(2,30)(3,25)(4,26)(5,27)(6,28)(7,21)(8,22)(9,23)(10,24)(11,19)(12,20)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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), (1,40)(2,41)(3,42)(4,37)(5,38)(6,39)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(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)], [(1,11,19,29),(2,12,20,30),(3,7,21,25),(4,8,22,26),(5,9,23,27),(6,10,24,28),(13,47,33,41),(14,48,34,42),(15,43,35,37),(16,44,36,38),(17,45,31,39),(18,46,32,40)], [(1,29),(2,30),(3,25),(4,26),(5,27),(6,28),(7,21),(8,22),(9,23),(10,24),(11,19),(12,20),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(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)], [(1,40),(2,41),(3,42),(4,37),(5,38),(6,39),(7,34),(8,35),(9,36),(10,31),(11,32),(12,33),(13,30),(14,25),(15,26),(16,27),(17,28),(18,29),(19,46),(20,47),(21,48),(22,43),(23,44),(24,45)])

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A ··· 6F 6G ··· 6O 6P ··· 6AE 6AF 6AG 6AH 6AI 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 24A ··· 24H order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 4 4 12 12 1 1 2 2 2 2 2 1 ··· 1 2 ··· 2 4 ··· 4 12 12 12 12 6 6 6 6 2 2 2 2 4 ··· 4 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 D8 C3×S3 C3⋊D4 C3×D4 C3⋊D4 C3×D4 S3×C6 S3×C6 C3×D8 C3×C3⋊D4 C3×C3⋊D4 D4⋊S3 C3×D4⋊S3 kernel C6×D4⋊S3 C6×C3⋊C8 C3×D4⋊S3 C6×D12 D4×C3×C6 C2×D4⋊S3 C2×C3⋊C8 D4⋊S3 C2×D12 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 GL4(𝔽73) generated by

 9 0 0 0 0 9 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 8 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 0 72 0 0 72 0 0 0 0 0 57 16 0 0 16 16
G:=sub<GL(4,GF(73))| [9,0,0,0,0,9,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[8,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,57,16,0,0,16,16] >;

C6×D4⋊S3 in GAP, Magma, Sage, TeX

C_6\times D_4\rtimes S_3
% in TeX

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

G:=SmallGroup(288,702);
// by ID

G=gap.SmallGroup(288,702);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,2524,648,102,9414]);
// 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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations

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