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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, C3, C3, C4, C4, C22, C22, S3, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C2×C4○D4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, S3×C6, C2×C3⋊S3, C62, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C4○D12, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C2×C3⋊D4, C22×C12, C3×C4○D4, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C322Q8, C3×Dic6, S3×C12, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, C2×S32, S3×C2×C6, C2×C4○D12, S3×C4○D4, S3×Dic6, D125S3, D6.D6, D6.6D6, C4×S32, S3×D12, D6.3D6, S3×C3⋊D4, S3×C2×C12, C3×C4○D12, C12.59D6, S3×C4○D12
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, S32, C4○D12, S3×C23, C2×S32, 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 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 25)(11 26)(12 27)(13 44)(14 45)(15 46)(16 47)(17 48)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)
(1 14 7 20)(2 15 8 21)(3 16 9 22)(4 17 10 23)(5 18 11 24)(6 19 12 13)(25 42 31 48)(26 43 32 37)(27 44 33 38)(28 45 34 39)(29 46 35 40)(30 47 36 41)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 39)(14 38)(15 37)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)

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

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

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

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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