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G = C3×D31order 186 = 2·3·31

Direct product of C3 and D31

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3×D31, C313C6, C932C2, SmallGroup(186,4)

Series: Derived Chief Lower central Upper central

C1C31 — C3×D31
C1C31C93 — C3×D31
C31 — C3×D31
C1C3

Generators and relations for C3×D31
 G = < a,b,c | a3=b31=c2=1, ab=ba, ac=ca, cbc=b-1 >

31C2
31C6

Smallest permutation representation of C3×D31
On 93 points
Generators in S93
(1 77 41)(2 78 42)(3 79 43)(4 80 44)(5 81 45)(6 82 46)(7 83 47)(8 84 48)(9 85 49)(10 86 50)(11 87 51)(12 88 52)(13 89 53)(14 90 54)(15 91 55)(16 92 56)(17 93 57)(18 63 58)(19 64 59)(20 65 60)(21 66 61)(22 67 62)(23 68 32)(24 69 33)(25 70 34)(26 71 35)(27 72 36)(28 73 37)(29 74 38)(30 75 39)(31 76 40)
(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)
(1 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(91 93)

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

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

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

51 conjugacy classes

class 1  2 3A3B6A6B31A···31O93A···93AD
order12336631···3193···93
size1311131312···22···2

51 irreducible representations

dim111122
type+++
imageC1C2C3C6D31C3×D31
kernelC3×D31C93D31C31C3C1
# reps11221530

Matrix representation of C3×D31 in GL2(𝔽373) generated by

880
088
,
1781
318138
,
34316
354339
G:=sub<GL(2,GF(373))| [88,0,0,88],[178,318,1,138],[34,354,316,339] >;

C3×D31 in GAP, Magma, Sage, TeX

C_3\times D_{31}
% in TeX

G:=Group("C3xD31");
// GroupNames label

G:=SmallGroup(186,4);
// by ID

G=gap.SmallGroup(186,4);
# by ID

G:=PCGroup([3,-2,-3,-31,1622]);
// Polycyclic

G:=Group<a,b,c|a^3=b^31=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D31 in TeX

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