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G = S3×D23order 276 = 22·3·23

Direct product of S3 and D23

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C69 — S3×D23
C1C23C69C3×D23 — S3×D23
C69 — S3×D23
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 2A2B2C 3  6 23A···23K46A···46K69A···69K
order12223623···2346···4669···69
size1323692462···26···64···4

39 irreducible representations

dim111122224
type+++++++++
imageC1C2C2C2S3D6D23D46S3×D23
kernelS3×D23S3×C23C3×D23D69D23C23S3C3C1
# reps111111111111

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

1000
0100
001123
0061137
,
1000
0100
0013816
0001
,
29100
9712800
0010
0001
,
10413400
783500
0010
0001
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

Export

Subgroup lattice of S3×D23 in TeX

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