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G = C103⋊C3order 309 = 3·103

The semidirect product of C103 and C3 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C103⋊C3, SmallGroup(309,1)

Series: Derived Chief Lower central Upper central

C1C103 — C103⋊C3
C1C103 — C103⋊C3
C103 — C103⋊C3
C1

Generators and relations for C103⋊C3
 G = < a,b | a103=b3=1, bab-1=a46 >

103C3

Smallest permutation representation of C103⋊C3
On 103 points: primitive
Generators in S103
(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 99 100 101 102 103)
(2 57 47)(3 10 93)(4 66 36)(5 19 82)(6 75 25)(7 28 71)(8 84 14)(9 37 60)(11 46 49)(12 102 95)(13 55 38)(15 64 27)(16 17 73)(18 26 62)(20 35 51)(21 91 97)(22 44 40)(23 100 86)(24 53 29)(30 80 99)(31 33 42)(32 89 88)(34 98 77)(39 69 101)(41 78 90)(43 87 79)(45 96 68)(48 58 103)(50 67 92)(52 76 81)(54 85 70)(56 94 59)(61 65 83)(63 74 72)

G:=sub<Sym(103)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103), (2,57,47)(3,10,93)(4,66,36)(5,19,82)(6,75,25)(7,28,71)(8,84,14)(9,37,60)(11,46,49)(12,102,95)(13,55,38)(15,64,27)(16,17,73)(18,26,62)(20,35,51)(21,91,97)(22,44,40)(23,100,86)(24,53,29)(30,80,99)(31,33,42)(32,89,88)(34,98,77)(39,69,101)(41,78,90)(43,87,79)(45,96,68)(48,58,103)(50,67,92)(52,76,81)(54,85,70)(56,94,59)(61,65,83)(63,74,72)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103), (2,57,47)(3,10,93)(4,66,36)(5,19,82)(6,75,25)(7,28,71)(8,84,14)(9,37,60)(11,46,49)(12,102,95)(13,55,38)(15,64,27)(16,17,73)(18,26,62)(20,35,51)(21,91,97)(22,44,40)(23,100,86)(24,53,29)(30,80,99)(31,33,42)(32,89,88)(34,98,77)(39,69,101)(41,78,90)(43,87,79)(45,96,68)(48,58,103)(50,67,92)(52,76,81)(54,85,70)(56,94,59)(61,65,83)(63,74,72) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103)], [(2,57,47),(3,10,93),(4,66,36),(5,19,82),(6,75,25),(7,28,71),(8,84,14),(9,37,60),(11,46,49),(12,102,95),(13,55,38),(15,64,27),(16,17,73),(18,26,62),(20,35,51),(21,91,97),(22,44,40),(23,100,86),(24,53,29),(30,80,99),(31,33,42),(32,89,88),(34,98,77),(39,69,101),(41,78,90),(43,87,79),(45,96,68),(48,58,103),(50,67,92),(52,76,81),(54,85,70),(56,94,59),(61,65,83),(63,74,72)])

37 conjugacy classes

class 1 3A3B103A···103AH
order133103···103
size11031033···3

37 irreducible representations

dim113
type+
imageC1C3C103⋊C3
kernelC103⋊C3C103C1
# reps1234

Matrix representation of C103⋊C3 in GL3(𝔽619) generated by

010
001
173184
,
100
332371426
73497247
G:=sub<GL(3,GF(619))| [0,0,1,1,0,73,0,1,184],[1,332,73,0,371,497,0,426,247] >;

C103⋊C3 in GAP, Magma, Sage, TeX

C_{103}\rtimes C_3
% in TeX

G:=Group("C103:C3");
// GroupNames label

G:=SmallGroup(309,1);
// by ID

G=gap.SmallGroup(309,1);
# by ID

G:=PCGroup([2,-3,-103,673]);
// Polycyclic

G:=Group<a,b|a^103=b^3=1,b*a*b^-1=a^46>;
// generators/relations

Export

Subgroup lattice of C103⋊C3 in TeX

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