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G = C109⋊C3order 327 = 3·109

The semidirect product of C109 and C3 acting faithfully

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

Aliases: C109⋊C3, SmallGroup(327,1)

Series: Derived Chief Lower central Upper central

C1C109 — C109⋊C3
C1C109 — C109⋊C3
C109 — C109⋊C3
C1

Generators and relations for C109⋊C3
 G = < a,b | a109=b3=1, bab-1=a45 >

109C3

Smallest permutation representation of C109⋊C3
On 109 points: primitive
Generators in S109
(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 99 100 101 102 103 104 105 106 107 108 109)
(2 64 46)(3 18 91)(4 81 27)(5 35 72)(6 98 8)(7 52 53)(9 69 34)(10 23 79)(11 86 15)(12 40 60)(13 103 105)(14 57 41)(16 74 22)(17 28 67)(19 45 48)(20 108 93)(21 62 29)(24 33 55)(25 96 100)(26 50 36)(30 84 107)(31 38 43)(32 101 88)(37 89 95)(39 106 76)(42 77 102)(44 94 83)(47 65 109)(49 82 90)(51 99 71)(54 70 97)(56 87 78)(58 104 59)(61 75 85)(63 92 66)(68 80 73)

G:=sub<Sym(109)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109), (2,64,46)(3,18,91)(4,81,27)(5,35,72)(6,98,8)(7,52,53)(9,69,34)(10,23,79)(11,86,15)(12,40,60)(13,103,105)(14,57,41)(16,74,22)(17,28,67)(19,45,48)(20,108,93)(21,62,29)(24,33,55)(25,96,100)(26,50,36)(30,84,107)(31,38,43)(32,101,88)(37,89,95)(39,106,76)(42,77,102)(44,94,83)(47,65,109)(49,82,90)(51,99,71)(54,70,97)(56,87,78)(58,104,59)(61,75,85)(63,92,66)(68,80,73)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109), (2,64,46)(3,18,91)(4,81,27)(5,35,72)(6,98,8)(7,52,53)(9,69,34)(10,23,79)(11,86,15)(12,40,60)(13,103,105)(14,57,41)(16,74,22)(17,28,67)(19,45,48)(20,108,93)(21,62,29)(24,33,55)(25,96,100)(26,50,36)(30,84,107)(31,38,43)(32,101,88)(37,89,95)(39,106,76)(42,77,102)(44,94,83)(47,65,109)(49,82,90)(51,99,71)(54,70,97)(56,87,78)(58,104,59)(61,75,85)(63,92,66)(68,80,73) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109)], [(2,64,46),(3,18,91),(4,81,27),(5,35,72),(6,98,8),(7,52,53),(9,69,34),(10,23,79),(11,86,15),(12,40,60),(13,103,105),(14,57,41),(16,74,22),(17,28,67),(19,45,48),(20,108,93),(21,62,29),(24,33,55),(25,96,100),(26,50,36),(30,84,107),(31,38,43),(32,101,88),(37,89,95),(39,106,76),(42,77,102),(44,94,83),(47,65,109),(49,82,90),(51,99,71),(54,70,97),(56,87,78),(58,104,59),(61,75,85),(63,92,66),(68,80,73)])

39 conjugacy classes

class 1 3A3B109A···109AJ
order133109···109
size11091093···3

39 irreducible representations

dim113
type+
imageC1C3C109⋊C3
kernelC109⋊C3C109C1
# reps1236

Matrix representation of C109⋊C3 in GL3(𝔽2617) generated by

112910
95601
149814971650
,
2006704986
20425972292
176828631
G:=sub<GL(3,GF(2617))| [1129,956,1498,1,0,1497,0,1,1650],[2006,204,1768,704,2597,28,986,2292,631] >;

C109⋊C3 in GAP, Magma, Sage, TeX

C_{109}\rtimes C_3
% in TeX

G:=Group("C109:C3");
// GroupNames label

G:=SmallGroup(327,1);
// by ID

G=gap.SmallGroup(327,1);
# by ID

G:=PCGroup([2,-3,-109,757]);
// Polycyclic

G:=Group<a,b|a^109=b^3=1,b*a*b^-1=a^45>;
// generators/relations

Export

Subgroup lattice of C109⋊C3 in TeX

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