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G = S3×C2×C12order 144 = 24·32

Direct product of C2×C12 and S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C2×C12
G = < a,b,c,d | a2=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 200 in 116 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×6], S3 [×4], C6 [×2], C6 [×4], C6 [×7], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×6], C2×C6 [×2], C2×C6 [×7], C22×C4, C3×S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×4], C2×Dic3, C2×C12 [×2], C2×C12 [×6], C22×S3, C22×C6, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×6], C62, S3×C2×C4, C22×C12, S3×C12 [×4], C6×Dic3, C6×C12, S3×C2×C6, S3×C2×C12
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], C23, C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C3×S3, C4×S3 [×2], C2×C12 [×6], C22×S3, C22×C6, S3×C6 [×3], S3×C2×C4, C22×C12, S3×C12 [×2], S3×C2×C6, S3×C2×C12

Smallest permutation representation of S3×C2×C12
On 48 points
Generators in S48
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(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 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 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)

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

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

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

S3×C2×C12 is a maximal subgroup of
C12.77D12  C62.11C23  C62.20C23  D6⋊Dic6  C62.25C23  D66Dic6  D67Dic6  C62.49C23  C62.74C23  C62.75C23  D6⋊D12  D62D12  C127D12

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6O 6P ··· 6W 12A ··· 12H 12I ··· 12T 12U ··· 12AB order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 3 3 3 3 1 1 2 2 2 1 1 1 1 3 3 3 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 S3 D6 D6 C3×S3 C4×S3 S3×C6 S3×C6 S3×C12 kernel S3×C2×C12 S3×C12 C6×Dic3 C6×C12 S3×C2×C6 S3×C2×C4 S3×C6 C4×S3 C2×Dic3 C2×C12 C22×S3 D6 C2×C12 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 2 8 8 2 2 2 16 1 2 1 2 4 4 2 8

Matrix representation of S3×C2×C12 in GL3(𝔽13) generated by

 12 0 0 0 12 0 0 0 12
,
 4 0 0 0 2 0 0 0 2
,
 1 0 0 0 3 3 0 0 9
,
 12 0 0 0 12 0 0 11 1
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,12],[4,0,0,0,2,0,0,0,2],[1,0,0,0,3,0,0,3,9],[12,0,0,0,12,11,0,0,1] >;

S3×C2×C12 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{12}
% in TeX

G:=Group("S3xC2xC12");
// GroupNames label

G:=SmallGroup(144,159);
// by ID

G=gap.SmallGroup(144,159);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,122,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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