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G = D33:S3  order 396 = 22·32·11

The semidirect product of D33 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, A-group

Aliases: D33:S3, C33:3D6, C32:2D22, C11:2S32, C3:S3:D11, C3:3(S3xD11), (C3xD33):3C2, (C3xC33):4C22, (C11xC3:S3):2C2, SmallGroup(396,23)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC33 — D33:S3
Chief series
C1 — C11 — C33 — C3xC33 — C3xD33 — D33:S3
Lower central
C3xC33 — D33:S3
Upper central
C1

Generators and relations for D33:S3
 G = < a,b,c,d | a33=b2=c3=d2=1, bab=a-1, ac=ca, dad=a23, bc=cb, dbd=a22b, dcd=c-1 >

Subgroups: 440 in 44 conjugacy classes, 15 normal (7 characteristic)
Quotients: C1, C2, C22, S3, D6, D11, S32, D22, S3xD11, D33:S3
C1
9C2
33C2
33C2
C3
C3
2C3
C11
99C22
3S3
3S3
6S3
11S3
11S3
33C6
33C6
C32
3D11
3D11
9C22
C33
C33
2C33
33D6
33D6
C3:S3
11C3xS3
11C3xS3
9D22
D33
D33
3C3xD11
3S3xC11
3S3xC11
3C3xD11
6S3xC11
C3xC33
11S32
3S3xD11
3S3xD11
C3xD33
C3xD33
C11xC3:S3
D33:S3

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

(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)

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

39 conjugacy classes

class 1 2A2B2C3A3B3C6A6B11A···11E22A···22E33A···33T
order12223336611···1122···2233···33
size19333322466662···218···184···4

39 irreducible representations

dim1112222444
type+++++++++
imageC1C2C2S3D6D11D22S32S3xD11D33:S3
kernelD33:S3C3xD33C11xC3:S3D33C33C3:S3C32C11C3C1
# reps121225511010

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

900000
31150000
001000
000100
0000066
0000166
,
6660000
35610000
001000
000100
0000166
0000066
,
100000
010000
0066100
0066000
000010
000001
,
6600000
0660000
000100
001000
000001
000010

G:=sub<GL(6,GF(67))| [9,31,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,66,66],[6,35,0,0,0,0,66,61,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,66,66],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,66,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[66,0,0,0,0,0,0,66,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

D33:S3 in GAP, Magma, Sage, TeX 

D_{33}\rtimes S_3
 
% in TeX
 
G:=Group("D33:S3");
 
// GroupNames label
 
G:=SmallGroup(396,23);
 
// by ID
 
G=gap.SmallGroup(396,23);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-3,-11,122,67,248,9004]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^33=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^23,b*c=c*b,d*b*d=a^22*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D33:S3 in TeX

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