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## G = S3×C4○D12order 288 = 25·32

### Direct product of S3 and C4○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — S3×C4○D12
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C4×S32 — S3×C4○D12
 Lower central C32 — C3×C6 — S3×C4○D12
 Upper central C1 — C4 — C2×C4

Generators and relations for S3×C4○D12
G = < a,b,c,d,e | a3=b2=c4=e2=1, d6=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d5 >

Subgroups: 1242 in 348 conjugacy classes, 112 normal (60 characteristic)
C1, C2, C2 [×8], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×12], S3 [×2], S3 [×9], C6 [×2], C6 [×9], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C32, Dic3 [×2], Dic3 [×2], Dic3 [×6], C12 [×4], C12 [×6], D6 [×2], D6 [×2], D6 [×16], C2×C6 [×2], C2×C6 [×7], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×2], C3×S3 [×3], C3⋊S3 [×2], C3×C6, C3×C6, Dic6, Dic6 [×7], C4×S3 [×4], C4×S3 [×2], C4×S3 [×14], D12, D12 [×7], C2×Dic3, C2×Dic3 [×4], C3⋊D4 [×2], C3⋊D4 [×14], C2×C12 [×2], C2×C12 [×8], C3×D4 [×3], C3×Q8, C22×S3, C22×S3 [×4], C22×C6, C2×C4○D4, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], S32 [×4], S3×C6 [×2], S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3 [×2], C62, C2×Dic6, S3×C2×C4, S3×C2×C4 [×4], C2×D12, C4○D12, C4○D12 [×11], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, C2×C3⋊D4 [×2], C22×C12, C3×C4○D4, S3×Dic3 [×4], C6.D6 [×2], D6⋊S3 [×2], C3⋊D12 [×4], C322Q8 [×2], C3×Dic6, S3×C12 [×4], S3×C12 [×2], C3×D12, C6×Dic3, C3×C3⋊D4 [×2], C324Q8, C4×C3⋊S3 [×2], C12⋊S3, C327D4 [×2], C6×C12, C2×S32 [×2], S3×C2×C6, C2×C4○D12, S3×C4○D4, S3×Dic6, D125S3, D6.D6 [×2], D6.6D6, C4×S32 [×2], S3×D12, D6.3D6 [×2], S3×C3⋊D4 [×2], S3×C2×C12, C3×C4○D12, C12.59D6, S3×C4○D12
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C4○D4 [×2], C24, C22×S3 [×14], C2×C4○D4, S32, C4○D12 [×2], S3×C23 [×2], C2×S32 [×3], C2×C4○D12, S3×C4○D4, C22×S32, S3×C4○D12

Smallest permutation representation of S3×C4○D12
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 48)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 25)(24 26)
(1 34 7 28)(2 35 8 29)(3 36 9 30)(4 25 10 31)(5 26 11 32)(6 27 12 33)(13 47 19 41)(14 48 20 42)(15 37 21 43)(16 38 22 44)(17 39 23 45)(18 40 24 46)
(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)(2 40)(3 39)(4 38)(5 37)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 36)(24 35)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 12A ··· 12F 12G ··· 12K 12L 12M 12N 12O 12P 12Q order 1 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 12 ··· 12 12 ··· 12 12 12 12 12 12 12 size 1 1 2 3 3 6 6 6 18 18 2 2 4 1 1 2 3 3 6 6 6 18 18 2 2 2 2 4 4 4 4 6 6 6 6 12 12 2 ··· 2 4 ··· 4 6 6 6 6 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D6 D6 D6 C4○D4 C4○D12 S32 C2×S32 C2×S32 S3×C4○D4 S3×C4○D12 kernel S3×C4○D12 S3×Dic6 D12⋊5S3 D6.D6 D6.6D6 C4×S32 S3×D12 D6.3D6 S3×C3⋊D4 S3×C2×C12 C3×C4○D12 C12.59D6 S3×C2×C4 C4○D12 Dic6 C4×S3 D12 C2×Dic3 C3⋊D4 C2×C12 C22×S3 C3×S3 S3 C2×C4 C4 C22 C3 C1 # reps 1 1 1 2 1 2 1 2 2 1 1 1 1 1 1 6 1 1 2 2 1 4 8 1 2 1 2 4

Matrix representation of S3×C4○D12 in GL6(𝔽13)

 0 1 0 0 0 0 12 12 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 1
,
 1 0 0 0 0 0 12 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 8 0 0 0 0 0 0 0 1 1 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 5 0 0 0 0 0 0 0 12 12 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1] >;

S3×C4○D12 in GAP, Magma, Sage, TeX

S_3\times C_4\circ D_{12}
% in TeX

G:=Group("S3xC4oD12");
// GroupNames label

G:=SmallGroup(288,953);
// by ID

G=gap.SmallGroup(288,953);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,346,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^4=e^2=1,d^6=c^2,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,c*e=e*c,e*d*e=c^2*d^5>;
// generators/relations

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