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G = S3xD33order 396 = 22·32·11

Direct product of S3 and D33

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

Aliases: S3xD33, C33:2D6, C3:1D66, C32:1D22, C11:1S32, (S3xC11):S3, (C3xS3):D11, C3:D33:2C2, C3:1(S3xD11), (S3xC33):1C2, (C3xD33):2C2, (C3xC33):3C22, SmallGroup(396,22)

Series: Derived Chief Lower central Upper central

C1C3xC33 — S3xD33
C1C11C33C3xC33C3xD33 — S3xD33
C3xC33 — S3xD33
C1

Generators and relations for S3xD33
 G = < a,b,c,d | a3=b2=c33=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 580 in 44 conjugacy classes, 16 normal (all characteristic)
Quotients: C1, C2, C22, S3, D6, D11, S32, D22, D33, S3xD11, D66, S3xD33
3C2
33C2
99C2
2C3
99C22
3C6
11S3
33S3
33C6
33S3
66S3
3C22
3D11
9D11
2C33
33D6
33D6
11C3:S3
11C3xS3
9D22
3D33
3C3xD11
3D33
3C66
6D33
11S32
3D66
3S3xD11

Smallest permutation representation of S3xD33
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++++++++++++++
imageC1C2C2C2S3S3D6D11D22D33D66S32S3xD11S3xD33
kernelS3xD33S3xC33C3xD33C3:D33S3xC11D33C33C3xS3C32S3C3C11C3C1
# reps11111125510101510

Matrix representation of S3xD33 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] >;

S3xD33 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 S3xD33 in TeX

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