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G = S3×D19order 228 = 22·3·19

Direct product of S3 and D19

Aliases: S3×D19, D57⋊C2, C31D38, C191D6, C57⋊C22, (S3×C19)⋊C2, (C3×D19)⋊C2, SmallGroup(228,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C57 — S3×D19
 Chief series C1 — C19 — C57 — C3×D19 — S3×D19
 Lower central C57 — S3×D19
 Upper central C1

Generators and relations for S3×D19
G = < a,b,c,d | a3=b2=c19=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

3C2
19C2
57C2
57C22
19C6
19S3
3C38
3D19
19D6
3D38

Smallest permutation representation of S3×D19
On 57 points
Generators in S57
(1 34 42)(2 35 43)(3 36 44)(4 37 45)(5 38 46)(6 20 47)(7 21 48)(8 22 49)(9 23 50)(10 24 51)(11 25 52)(12 26 53)(13 27 54)(14 28 55)(15 29 56)(16 30 57)(17 31 39)(18 32 40)(19 33 41)
(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 28)(21 27)(22 26)(23 25)(29 38)(30 37)(31 36)(32 35)(33 34)(39 44)(40 43)(41 42)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)

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

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

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

S3×D19 is a maximal quotient of   D57⋊C4  C57⋊D4  C3⋊D76  C19⋊D12  C57⋊Q8

33 conjugacy classes

 class 1 2A 2B 2C 3 6 19A ··· 19I 38A ··· 38I 57A ··· 57I order 1 2 2 2 3 6 19 ··· 19 38 ··· 38 57 ··· 57 size 1 3 19 57 2 38 2 ··· 2 6 ··· 6 4 ··· 4

33 irreducible representations

 dim 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 S3 D6 D19 D38 S3×D19 kernel S3×D19 S3×C19 C3×D19 D57 D19 C19 S3 C3 C1 # reps 1 1 1 1 1 1 9 9 9

Matrix representation of S3×D19 in GL4(𝔽229) generated by

 1 0 0 0 0 1 0 0 0 0 227 11 0 0 83 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 146 228
,
 78 1 0 0 120 210 0 0 0 0 1 0 0 0 0 1
,
 58 123 0 0 224 171 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(229))| [1,0,0,0,0,1,0,0,0,0,227,83,0,0,11,1],[1,0,0,0,0,1,0,0,0,0,1,146,0,0,0,228],[78,120,0,0,1,210,0,0,0,0,1,0,0,0,0,1],[58,224,0,0,123,171,0,0,0,0,1,0,0,0,0,1] >;

S3×D19 in GAP, Magma, Sage, TeX

S_3\times D_{19}
% in TeX

G:=Group("S3xD19");
// GroupNames label

G:=SmallGroup(228,8);
// by ID

G=gap.SmallGroup(228,8);
# by ID

G:=PCGroup([4,-2,-2,-3,-19,54,3459]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^19=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

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