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## G = S3×D25order 300 = 22·3·52

### Direct product of S3 and D25

Aliases: S3×D25, D75⋊C2, C31D50, C251D6, C75⋊C22, C15.D10, C5.(S3×D5), (S3×C25)⋊C2, (C3×D25)⋊C2, (C5×S3).D5, SmallGroup(300,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C75 — S3×D25
 Chief series C1 — C5 — C25 — C75 — C3×D25 — S3×D25
 Lower central C75 — S3×D25
 Upper central C1

Generators and relations for S3×D25
G = < a,b,c,d | a3=b2=c25=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

3C2
25C2
75C2
75C22
25C6
25S3
3C10
5D5
15D5
25D6
15D10
5D15
3C50
3D25
3D50

Smallest permutation representation of S3×D25
On 75 points
Generators in S75
(1 46 70)(2 47 71)(3 48 72)(4 49 73)(5 50 74)(6 26 75)(7 27 51)(8 28 52)(9 29 53)(10 30 54)(11 31 55)(12 32 56)(13 33 57)(14 34 58)(15 35 59)(16 36 60)(17 37 61)(18 38 62)(19 39 63)(20 40 64)(21 41 65)(22 42 66)(23 43 67)(24 44 68)(25 45 69)
(26 75)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)
(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 70 71 72 73 74 75)
(1 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 50)(42 49)(43 48)(44 47)(45 46)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)

G:=sub<Sym(75)| (1,46,70)(2,47,71)(3,48,72)(4,49,73)(5,50,74)(6,26,75)(7,27,51)(8,28,52)(9,29,53)(10,30,54)(11,31,55)(12,32,56)(13,33,57)(14,34,58)(15,35,59)(16,36,60)(17,37,61)(18,38,62)(19,39,63)(20,40,64)(21,41,65)(22,42,66)(23,43,67)(24,44,68)(25,45,69), (26,75)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74), (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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,50)(42,49)(43,48)(44,47)(45,46)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)>;

G:=Group( (1,46,70)(2,47,71)(3,48,72)(4,49,73)(5,50,74)(6,26,75)(7,27,51)(8,28,52)(9,29,53)(10,30,54)(11,31,55)(12,32,56)(13,33,57)(14,34,58)(15,35,59)(16,36,60)(17,37,61)(18,38,62)(19,39,63)(20,40,64)(21,41,65)(22,42,66)(23,43,67)(24,44,68)(25,45,69), (26,75)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74), (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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,50)(42,49)(43,48)(44,47)(45,46)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70) );

G=PermutationGroup([[(1,46,70),(2,47,71),(3,48,72),(4,49,73),(5,50,74),(6,26,75),(7,27,51),(8,28,52),(9,29,53),(10,30,54),(11,31,55),(12,32,56),(13,33,57),(14,34,58),(15,35,59),(16,36,60),(17,37,61),(18,38,62),(19,39,63),(20,40,64),(21,41,65),(22,42,66),(23,43,67),(24,44,68),(25,45,69)], [(26,75),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,73),(50,74)], [(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),(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(41,50),(42,49),(43,48),(44,47),(45,46),(51,63),(52,62),(53,61),(54,60),(55,59),(56,58),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70)]])

42 conjugacy classes

 class 1 2A 2B 2C 3 5A 5B 6 10A 10B 15A 15B 25A ··· 25J 50A ··· 50J 75A ··· 75J order 1 2 2 2 3 5 5 6 10 10 15 15 25 ··· 25 50 ··· 50 75 ··· 75 size 1 3 25 75 2 2 2 50 6 6 4 4 2 ··· 2 6 ··· 6 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 S3 D5 D6 D10 D25 D50 S3×D5 S3×D25 kernel S3×D25 S3×C25 C3×D25 D75 D25 C5×S3 C25 C15 S3 C3 C5 C1 # reps 1 1 1 1 1 2 1 2 10 10 2 10

Matrix representation of S3×D25 in GL4(𝔽151) generated by

 1 0 0 0 0 1 0 0 0 0 1 121 0 0 136 149
,
 1 0 0 0 0 1 0 0 0 0 150 30 0 0 0 1
,
 88 136 0 0 133 70 0 0 0 0 1 0 0 0 0 1
,
 136 40 0 0 85 15 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(151))| [1,0,0,0,0,1,0,0,0,0,1,136,0,0,121,149],[1,0,0,0,0,1,0,0,0,0,150,0,0,0,30,1],[88,133,0,0,136,70,0,0,0,0,1,0,0,0,0,1],[136,85,0,0,40,15,0,0,0,0,1,0,0,0,0,1] >;

S3×D25 in GAP, Magma, Sage, TeX

S_3\times D_{25}
% in TeX

G:=Group("S3xD25");
// GroupNames label

G:=SmallGroup(300,7);
// by ID

G=gap.SmallGroup(300,7);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-5,67,2163,418,6004]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^25=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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