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G = C3×D51order 306 = 2·32·17

Direct product of C3 and D51

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C3×D51, C511C6, C512S3, C321D17, C17⋊(C3×S3), C3⋊(C3×D17), (C3×C51)⋊2C2, SmallGroup(306,7)

Series: Derived Chief Lower central Upper central

C1C51 — C3×D51
C1C17C51C3×C51 — C3×D51
C51 — C3×D51
C1C3

Generators and relations for C3×D51
 G = < a,b,c | a3=b51=c2=1, ab=ba, ac=ca, cbc=b-1 >

51C2
2C3
17S3
51C6
3D17
2C51
17C3×S3
3C3×D17

Smallest permutation representation of C3×D51
On 102 points
Generators in S102
(1 18 35)(2 19 36)(3 20 37)(4 21 38)(5 22 39)(6 23 40)(7 24 41)(8 25 42)(9 26 43)(10 27 44)(11 28 45)(12 29 46)(13 30 47)(14 31 48)(15 32 49)(16 33 50)(17 34 51)(52 86 69)(53 87 70)(54 88 71)(55 89 72)(56 90 73)(57 91 74)(58 92 75)(59 93 76)(60 94 77)(61 95 78)(62 96 79)(63 97 80)(64 98 81)(65 99 82)(66 100 83)(67 101 84)(68 102 85)
(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 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102)
(1 64)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 102)(15 101)(16 100)(17 99)(18 98)(19 97)(20 96)(21 95)(22 94)(23 93)(24 92)(25 91)(26 90)(27 89)(28 88)(29 87)(30 86)(31 85)(32 84)(33 83)(34 82)(35 81)(36 80)(37 79)(38 78)(39 77)(40 76)(41 75)(42 74)(43 73)(44 72)(45 71)(46 70)(47 69)(48 68)(49 67)(50 66)(51 65)

G:=sub<Sym(102)| (1,18,35)(2,19,36)(3,20,37)(4,21,38)(5,22,39)(6,23,40)(7,24,41)(8,25,42)(9,26,43)(10,27,44)(11,28,45)(12,29,46)(13,30,47)(14,31,48)(15,32,49)(16,33,50)(17,34,51)(52,86,69)(53,87,70)(54,88,71)(55,89,72)(56,90,73)(57,91,74)(58,92,75)(59,93,76)(60,94,77)(61,95,78)(62,96,79)(63,97,80)(64,98,81)(65,99,82)(66,100,83)(67,101,84)(68,102,85), (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)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,102)(15,101)(16,100)(17,99)(18,98)(19,97)(20,96)(21,95)(22,94)(23,93)(24,92)(25,91)(26,90)(27,89)(28,88)(29,87)(30,86)(31,85)(32,84)(33,83)(34,82)(35,81)(36,80)(37,79)(38,78)(39,77)(40,76)(41,75)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)>;

G:=Group( (1,18,35)(2,19,36)(3,20,37)(4,21,38)(5,22,39)(6,23,40)(7,24,41)(8,25,42)(9,26,43)(10,27,44)(11,28,45)(12,29,46)(13,30,47)(14,31,48)(15,32,49)(16,33,50)(17,34,51)(52,86,69)(53,87,70)(54,88,71)(55,89,72)(56,90,73)(57,91,74)(58,92,75)(59,93,76)(60,94,77)(61,95,78)(62,96,79)(63,97,80)(64,98,81)(65,99,82)(66,100,83)(67,101,84)(68,102,85), (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)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,102)(15,101)(16,100)(17,99)(18,98)(19,97)(20,96)(21,95)(22,94)(23,93)(24,92)(25,91)(26,90)(27,89)(28,88)(29,87)(30,86)(31,85)(32,84)(33,83)(34,82)(35,81)(36,80)(37,79)(38,78)(39,77)(40,76)(41,75)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65) );

G=PermutationGroup([(1,18,35),(2,19,36),(3,20,37),(4,21,38),(5,22,39),(6,23,40),(7,24,41),(8,25,42),(9,26,43),(10,27,44),(11,28,45),(12,29,46),(13,30,47),(14,31,48),(15,32,49),(16,33,50),(17,34,51),(52,86,69),(53,87,70),(54,88,71),(55,89,72),(56,90,73),(57,91,74),(58,92,75),(59,93,76),(60,94,77),(61,95,78),(62,96,79),(63,97,80),(64,98,81),(65,99,82),(66,100,83),(67,101,84),(68,102,85)], [(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),(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102)], [(1,64),(2,63),(3,62),(4,61),(5,60),(6,59),(7,58),(8,57),(9,56),(10,55),(11,54),(12,53),(13,52),(14,102),(15,101),(16,100),(17,99),(18,98),(19,97),(20,96),(21,95),(22,94),(23,93),(24,92),(25,91),(26,90),(27,89),(28,88),(29,87),(30,86),(31,85),(32,84),(33,83),(34,82),(35,81),(36,80),(37,79),(38,78),(39,77),(40,76),(41,75),(42,74),(43,73),(44,72),(45,71),(46,70),(47,69),(48,68),(49,67),(50,66),(51,65)])

81 conjugacy classes

class 1  2 3A3B3C3D3E6A6B17A···17H51A···51BL
order12333336617···1751···51
size1511122251512···22···2

81 irreducible representations

dim1111222222
type+++++
imageC1C2C3C6S3C3×S3D17C3×D17D51C3×D51
kernelC3×D51C3×C51D51C51C51C17C32C3C3C1
# reps1122128161632

Matrix representation of C3×D51 in GL4(𝔽103) generated by

46000
04600
0010
0001
,
56000
04600
002925
00272
,
04600
56000
005681
009147
G:=sub<GL(4,GF(103))| [46,0,0,0,0,46,0,0,0,0,1,0,0,0,0,1],[56,0,0,0,0,46,0,0,0,0,29,27,0,0,25,2],[0,56,0,0,46,0,0,0,0,0,56,91,0,0,81,47] >;

C3×D51 in GAP, Magma, Sage, TeX

C_3\times D_{51}
% in TeX

G:=Group("C3xD51");
// GroupNames label

G:=SmallGroup(306,7);
// by ID

G=gap.SmallGroup(306,7);
# by ID

G:=PCGroup([4,-2,-3,-3,-17,146,4611]);
// Polycyclic

G:=Group<a,b,c|a^3=b^51=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D51 in TeX

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