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G = S3×D33  order 396 = 22·32·11

Direct product of S3 and D33

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

Aliases: S3×D33, C33⋊2D6, C3⋊1D66, C32⋊1D22, C11⋊1S32, (S3×C11)⋊S3, (C3×S3)⋊D11, C3⋊D33⋊2C2, C3⋊1(S3×D11), (S3×C33)⋊1C2, (C3×D33)⋊2C2, (C3×C33)⋊3C22, SmallGroup(396,22)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C33 — S3×D33
Chief series
C1 — C11 — C33 — C3×C33 — C3×D33 — S3×D33
Lower central
C3×C33 — S3×D33
Upper central
C1

Generators and relations for S3×D33
 G = < a,b,c,d | a3=b2=c33=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

C1
3C2
33C2
99C2
C3
C3
2C3
C11
99C22
S3
3C6
11S3
33S3
33C6
33S3
66S3
C32
3C22
3D11
9D11
C33
C33
2C33
33D6
33D6
C3×S3
11C3⋊S3
11C3×S3
9D22
D33
S3×C11
3D33
3C3×D11
3D33
3C66
6D33
C3×C33
11S32
3D66
3S3×D11
C3×D33
C3⋊D33
S3×C33
S3×D33

Smallest permutation representation of S3×D33
►On 66 points
Generators in S66
(1 23 12)(2 24 13)(3 25 14)(4 26 15)(5 27 16)(6 28 17)(7 29 18)(8 30 19)(9 31 20)(10 32 21)(11 33 22)(34 45 56)(35 46 57)(36 47 58)(37 48 59)(38 49 60)(39 50 61)(40 51 62)(41 52 63)(42 53 64)(43 54 65)(44 55 66)

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

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

54 conjugacy classes

class 1 2A2B2C3A3B3C6A6B11A···11E22A···22E33A···33J33K···33Y66A···66J
order12223336611···1122···2233···3333···3366···66
size1333992246662···26···62···24···46···6

54 irreducible representations

dim11112222222444
type++++++++++++++
imageC1C2C2C2S3S3D6D11D22D33D66S32S3×D11S3×D33
kernelS3×D33S3×C33C3×D33C3⋊D33S3×C11D33C33C3×S3C32S3C3C11C3C1
# reps11111125510101510

Matrix representation of S3×D33 ►in GL6(𝔽67)

100000
010000
001000
000100
0000661
0000660
,
100000
010000
0066000
0006600
000001
000010
,
28410000
2670000
0006600
0016600
000010
000001
,
3530000
58640000
0066100
000100
000010
000001

G:=sub<GL(6,GF(67))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,66,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,0,0,0,0,0,0,66,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[28,26,0,0,0,0,41,7,0,0,0,0,0,0,0,1,0,0,0,0,66,66,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,58,0,0,0,0,53,64,0,0,0,0,0,0,66,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

S3×D33 in GAP, Magma, Sage, TeX 

S_3\times D_{33}
 
% in TeX
 
G:=Group("S3xD33");
 
// GroupNames label
 
G:=SmallGroup(396,22);
 
// by ID
 
G=gap.SmallGroup(396,22);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-3,-11,67,483,9004]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^2=c^33=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of S3×D33 in TeX

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