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## G = S3×D23order 276 = 22·3·23

### Direct product of S3 and D23

Aliases: S3×D23, D69⋊C2, C231D6, C31D46, C69⋊C22, (S3×C23)⋊C2, (C3×D23)⋊C2, SmallGroup(276,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C69 — S3×D23
 Chief series C1 — C23 — C69 — C3×D23 — S3×D23
 Lower central C69 — S3×D23
 Upper central C1

Generators and relations for S3×D23
G = < a,b,c,d | a3=b2=c23=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

3C2
23C2
69C2
69C22
23C6
23S3
3C46
3D23
23D6
3D46

Smallest permutation representation of S3×D23
On 69 points
Generators in S69
(1 24 63)(2 25 64)(3 26 65)(4 27 66)(5 28 67)(6 29 68)(7 30 69)(8 31 47)(9 32 48)(10 33 49)(11 34 50)(12 35 51)(13 36 52)(14 37 53)(15 38 54)(16 39 55)(17 40 56)(18 41 57)(19 42 58)(20 43 59)(21 44 60)(22 45 61)(23 46 62)
(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)
(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 67 68 69)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 46)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)(47 55)(48 54)(49 53)(50 52)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)

G:=sub<Sym(69)| (1,24,63)(2,25,64)(3,26,65)(4,27,66)(5,28,67)(6,29,68)(7,30,69)(8,31,47)(9,32,48)(10,33,49)(11,34,50)(12,35,51)(13,36,52)(14,37,53)(15,38,54)(16,39,55)(17,40,56)(18,41,57)(19,42,58)(20,43,59)(21,44,60)(22,45,61)(23,46,62), (24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62), (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)(47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(47,55)(48,54)(49,53)(50,52)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)>;

G:=Group( (1,24,63)(2,25,64)(3,26,65)(4,27,66)(5,28,67)(6,29,68)(7,30,69)(8,31,47)(9,32,48)(10,33,49)(11,34,50)(12,35,51)(13,36,52)(14,37,53)(15,38,54)(16,39,55)(17,40,56)(18,41,57)(19,42,58)(20,43,59)(21,44,60)(22,45,61)(23,46,62), (24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62), (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)(47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(47,55)(48,54)(49,53)(50,52)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63) );

G=PermutationGroup([(1,24,63),(2,25,64),(3,26,65),(4,27,66),(5,28,67),(6,29,68),(7,30,69),(8,31,47),(9,32,48),(10,33,49),(11,34,50),(12,35,51),(13,36,52),(14,37,53),(15,38,54),(16,39,55),(17,40,56),(18,41,57),(19,42,58),(20,43,59),(21,44,60),(22,45,61),(23,46,62)], [(24,63),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,47),(32,48),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62)], [(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),(47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13),(24,46),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36),(47,55),(48,54),(49,53),(50,52),(56,69),(57,68),(58,67),(59,66),(60,65),(61,64),(62,63)])

39 conjugacy classes

 class 1 2A 2B 2C 3 6 23A ··· 23K 46A ··· 46K 69A ··· 69K order 1 2 2 2 3 6 23 ··· 23 46 ··· 46 69 ··· 69 size 1 3 23 69 2 46 2 ··· 2 6 ··· 6 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 S3 D6 D23 D46 S3×D23 kernel S3×D23 S3×C23 C3×D23 D69 D23 C23 S3 C3 C1 # reps 1 1 1 1 1 1 11 11 11

Matrix representation of S3×D23 in GL4(𝔽139) generated by

 1 0 0 0 0 1 0 0 0 0 1 123 0 0 61 137
,
 1 0 0 0 0 1 0 0 0 0 138 16 0 0 0 1
,
 29 1 0 0 97 128 0 0 0 0 1 0 0 0 0 1
,
 104 134 0 0 78 35 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(139))| [1,0,0,0,0,1,0,0,0,0,1,61,0,0,123,137],[1,0,0,0,0,1,0,0,0,0,138,0,0,0,16,1],[29,97,0,0,1,128,0,0,0,0,1,0,0,0,0,1],[104,78,0,0,134,35,0,0,0,0,1,0,0,0,0,1] >;

S3×D23 in GAP, Magma, Sage, TeX

S_3\times D_{23}
% in TeX

G:=Group("S3xD23");
// GroupNames label

G:=SmallGroup(276,5);
// by ID

G=gap.SmallGroup(276,5);
# by ID

G:=PCGroup([4,-2,-2,-3,-23,54,4227]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^23=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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