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G = S3×C22×C6order 144 = 24·32

Direct product of C22×C6 and S3

direct product, metabelian, supersoluble, monomial, A-group

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

C1C3 — S3×C22×C6
C1C3C32C3×S3S3×C6S3×C2×C6 — S3×C22×C6
C3 — S3×C22×C6
C1C22×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···2G2H···2O3A3B3C3D3E6A···6N6O···6AI6AJ···6AY
order12···22···2333336···66···66···6
size11···13···3112221···12···23···3

72 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6S3D6C3×S3S3×C6
kernelS3×C22×C6S3×C2×C6C2×C62S3×C23C22×S3C22×C6C22×C6C2×C6C23C22
# reps1141228217214

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

1000
0600
0060
0006
,
6000
0100
0060
0006
,
4000
0200
0030
0003
,
1000
0100
0020
0004
,
6000
0600
0006
0060
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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