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## G = S3×C22×C6order 144 = 24·32

### Direct product of C22×C6 and S3

Aliases: S3×C22×C6, C322C24, C629C22, C6⋊(C22×C6), C3⋊(C23×C6), (C22×C6)⋊5C6, (C2×C62)⋊4C2, (C3×C6)⋊2C23, (C2×C6)⋊6(C2×C6), SmallGroup(144,195)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C22×C6
 Chief series C1 — C3 — C32 — C3×S3 — S3×C6 — S3×C2×C6 — S3×C22×C6
 Lower central C3 — S3×C22×C6
 Upper central C1 — C22×C6

Generators and relations for S3×C22×C6
G = < a,b,c,d,e | a2=b2=c6=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 504 in 284 conjugacy classes, 166 normal (10 characteristic)
C1, C2 [×7], C2 [×8], C3 [×2], C3, C22 [×7], C22 [×28], S3 [×8], C6 [×14], C6 [×15], C23, C23 [×14], C32, D6 [×28], C2×C6 [×14], C2×C6 [×35], C24, C3×S3 [×8], C3×C6 [×7], C22×S3 [×14], C22×C6 [×2], C22×C6 [×15], S3×C6 [×28], C62 [×7], S3×C23, C23×C6, S3×C2×C6 [×14], C2×C62, S3×C22×C6
Quotients: C1, C2 [×15], C3, C22 [×35], S3, C6 [×15], C23 [×15], D6 [×7], C2×C6 [×35], C24, C3×S3, C22×S3 [×7], C22×C6 [×15], S3×C6 [×7], S3×C23, C23×C6, S3×C2×C6 [×7], S3×C22×C6

Smallest permutation representation of S3×C22×C6
On 48 points
Generators in S48
(1 14)(2 15)(3 16)(4 17)(5 18)(6 13)(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)
(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 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 34)(8 35)(9 36)(10 31)(11 32)(12 33)(13 40)(14 41)(15 42)(16 37)(17 38)(18 39)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)

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

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

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

S3×C22×C6 is a maximal subgroup of   C624D4  C625D4

72 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 3C 3D 3E 6A ··· 6N 6O ··· 6AI 6AJ ··· 6AY order 1 2 ··· 2 2 ··· 2 3 3 3 3 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 ··· 1 3 ··· 3 1 1 2 2 2 1 ··· 1 2 ··· 2 3 ··· 3

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 kernel S3×C22×C6 S3×C2×C6 C2×C62 S3×C23 C22×S3 C22×C6 C22×C6 C2×C6 C23 C22 # reps 1 14 1 2 28 2 1 7 2 14

Matrix representation of S3×C22×C6 in GL4(𝔽7) generated by

 1 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6
,
 6 0 0 0 0 1 0 0 0 0 6 0 0 0 0 6
,
 4 0 0 0 0 2 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 2 0 0 0 0 4
,
 6 0 0 0 0 6 0 0 0 0 0 6 0 0 6 0
G:=sub<GL(4,GF(7))| [1,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[6,0,0,0,0,1,0,0,0,0,6,0,0,0,0,6],[4,0,0,0,0,2,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,4],[6,0,0,0,0,6,0,0,0,0,0,6,0,0,6,0] >;

S3×C22×C6 in GAP, Magma, Sage, TeX

S_3\times C_2^2\times C_6
% in TeX

G:=Group("S3xC2^2xC6");
// GroupNames label

G:=SmallGroup(144,195);
// by ID

G=gap.SmallGroup(144,195);
# by ID

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

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

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