Copied to
clipboard

## 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, C2, C3, C3, C22, C22, S3, C6, C6, C23, C23, C32, D6, C2×C6, C2×C6, C24, C3×S3, C3×C6, C22×S3, C22×C6, C22×C6, S3×C6, C62, S3×C23, C23×C6, S3×C2×C6, C2×C62, S3×C22×C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C24, C3×S3, C22×S3, C22×C6, S3×C6, S3×C23, C23×C6, S3×C2×C6, 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

׿
×
𝔽