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G = S3×C22order 132 = 22·3·11

Direct product of C22 and S3

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3×C22, C6⋊C22, C663C2, C334C22, C3⋊(C2×C22), SmallGroup(132,8)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C22
C1C3C33S3×C11 — S3×C22
C3 — S3×C22
C1C22

Generators and relations for S3×C22
 G = < a,b,c | a22=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C22
3C22
3C22
3C2×C22

Smallest permutation representation of S3×C22
On 66 points
Generators in S66
(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 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66)
(1 39 58)(2 40 59)(3 41 60)(4 42 61)(5 43 62)(6 44 63)(7 23 64)(8 24 65)(9 25 66)(10 26 45)(11 27 46)(12 28 47)(13 29 48)(14 30 49)(15 31 50)(16 32 51)(17 33 52)(18 34 53)(19 35 54)(20 36 55)(21 37 56)(22 38 57)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 45)(38 46)(39 47)(40 48)(41 49)(42 50)(43 51)(44 52)

G:=sub<Sym(66)| (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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66), (1,39,58)(2,40,59)(3,41,60)(4,42,61)(5,43,62)(6,44,63)(7,23,64)(8,24,65)(9,25,66)(10,26,45)(11,27,46)(12,28,47)(13,29,48)(14,30,49)(15,31,50)(16,32,51)(17,33,52)(18,34,53)(19,35,54)(20,36,55)(21,37,56)(22,38,57), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,45)(38,46)(39,47)(40,48)(41,49)(42,50)(43,51)(44,52)>;

G:=Group( (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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66), (1,39,58)(2,40,59)(3,41,60)(4,42,61)(5,43,62)(6,44,63)(7,23,64)(8,24,65)(9,25,66)(10,26,45)(11,27,46)(12,28,47)(13,29,48)(14,30,49)(15,31,50)(16,32,51)(17,33,52)(18,34,53)(19,35,54)(20,36,55)(21,37,56)(22,38,57), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,45)(38,46)(39,47)(40,48)(41,49)(42,50)(43,51)(44,52) );

G=PermutationGroup([(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),(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66)], [(1,39,58),(2,40,59),(3,41,60),(4,42,61),(5,43,62),(6,44,63),(7,23,64),(8,24,65),(9,25,66),(10,26,45),(11,27,46),(12,28,47),(13,29,48),(14,30,49),(15,31,50),(16,32,51),(17,33,52),(18,34,53),(19,35,54),(20,36,55),(21,37,56),(22,38,57)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,45),(38,46),(39,47),(40,48),(41,49),(42,50),(43,51),(44,52)])

S3×C22 is a maximal subgroup of   C33⋊D4  C11⋊D12

66 conjugacy classes

class 1 2A2B2C 3  6 11A···11J22A···22J22K···22AD33A···33J66A···66J
order12223611···1122···2222···2233···3366···66
size1133221···11···13···32···22···2

66 irreducible representations

dim1111112222
type+++++
imageC1C2C2C11C22C22S3D6S3×C11S3×C22
kernelS3×C22S3×C11C66D6S3C6C22C11C2C1
# reps121102010111010

Matrix representation of S3×C22 in GL2(𝔽23) generated by

70
07
,
2210
160
,
2210
01
G:=sub<GL(2,GF(23))| [7,0,0,7],[22,16,10,0],[22,0,10,1] >;

S3×C22 in GAP, Magma, Sage, TeX

S_3\times C_{22}
% in TeX

G:=Group("S3xC22");
// GroupNames label

G:=SmallGroup(132,8);
// by ID

G=gap.SmallGroup(132,8);
# by ID

G:=PCGroup([4,-2,-2,-11,-3,1411]);
// Polycyclic

G:=Group<a,b,c|a^22=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of S3×C22 in TeX

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