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## G = C3×S3×D11order 396 = 22·32·11

### Direct product of C3, S3 and D11

Aliases: C3×S3×D11, D33⋊C6, C334D6, C323D22, C33⋊(C2×C6), (S3×C11)⋊C6, C111(S3×C6), (C3×D11)⋊C6, C31(C6×D11), (S3×C33)⋊2C2, (C3×D33)⋊1C2, (C3×C33)⋊1C22, (C32×D11)⋊1C2, SmallGroup(396,19)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C3×S3×D11
 Chief series C1 — C11 — C33 — C3×C33 — C32×D11 — C3×S3×D11
 Lower central C33 — C3×S3×D11
 Upper central C1 — C3

Generators and relations for C3×S3×D11
G = < a,b,c,d,e | a3=b3=c2=d11=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Smallest permutation representation of C3×S3×D11
On 66 points
Generators in S66
(1 21 32)(2 22 33)(3 12 23)(4 13 24)(5 14 25)(6 15 26)(7 16 27)(8 17 28)(9 18 29)(10 19 30)(11 20 31)(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 21 32)(2 22 33)(3 12 23)(4 13 24)(5 14 25)(6 15 26)(7 16 27)(8 17 28)(9 18 29)(10 19 30)(11 20 31)(34 56 45)(35 57 46)(36 58 47)(37 59 48)(38 60 49)(39 61 50)(40 62 51)(41 63 52)(42 64 53)(43 65 54)(44 66 55)
(1 43)(2 44)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 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 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 6A 6B 6C 6D 6E 6F 6G 6H 6I 11A ··· 11E 22A ··· 22E 33A ··· 33J 33K ··· 33Y 66A ··· 66J order 1 2 2 2 3 3 3 3 3 6 6 6 6 6 6 6 6 6 11 ··· 11 22 ··· 22 33 ··· 33 33 ··· 33 66 ··· 66 size 1 3 11 33 1 1 2 2 2 3 3 11 11 22 22 22 33 33 2 ··· 2 6 ··· 6 2 ··· 2 4 ··· 4 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D6 C3×S3 D11 S3×C6 D22 C3×D11 C6×D11 S3×D11 C3×S3×D11 kernel C3×S3×D11 C32×D11 S3×C33 C3×D33 S3×D11 S3×C11 C3×D11 D33 C3×D11 C33 D11 C3×S3 C11 C32 S3 C3 C3 C1 # reps 1 1 1 1 2 2 2 2 1 1 2 5 2 5 10 10 5 10

Matrix representation of C3×S3×D11 in GL4(𝔽67) generated by

 37 0 0 0 0 37 0 0 0 0 1 0 0 0 0 1
,
 37 0 0 0 0 29 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 48 1 0 0 55 57
,
 1 0 0 0 0 1 0 0 0 0 57 66 0 0 32 10
G:=sub<GL(4,GF(67))| [37,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[37,0,0,0,0,29,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,48,55,0,0,1,57],[1,0,0,0,0,1,0,0,0,0,57,32,0,0,66,10] >;

C3×S3×D11 in GAP, Magma, Sage, TeX

C_3\times S_3\times D_{11}
% in TeX

G:=Group("C3xS3xD11");
// GroupNames label

G:=SmallGroup(396,19);
// by ID

G=gap.SmallGroup(396,19);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-11,248,9004]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^11=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,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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