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G = C3×C36order 108 = 22·33

Abelian group of type [3,36]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C36, SmallGroup(108,12)

Series: Derived Chief Lower central Upper central

C1 — C3×C36
C1C3C6C3×C6C3×C18 — C3×C36
C1 — C3×C36
C1 — C3×C36

Generators and relations for C3×C36
 G = < a,b | a3=b36=1, ab=ba >


Smallest permutation representation of C3×C36
Regular action on 108 points
Generators in S108
(1 82 60)(2 83 61)(3 84 62)(4 85 63)(5 86 64)(6 87 65)(7 88 66)(8 89 67)(9 90 68)(10 91 69)(11 92 70)(12 93 71)(13 94 72)(14 95 37)(15 96 38)(16 97 39)(17 98 40)(18 99 41)(19 100 42)(20 101 43)(21 102 44)(22 103 45)(23 104 46)(24 105 47)(25 106 48)(26 107 49)(27 108 50)(28 73 51)(29 74 52)(30 75 53)(31 76 54)(32 77 55)(33 78 56)(34 79 57)(35 80 58)(36 81 59)
(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)

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

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

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

C3×C36 is a maximal subgroup of   C36.S3  C12.D9  C36⋊S3

108 conjugacy classes

class 1  2 3A···3H4A4B6A···6H9A···9R12A···12P18A···18R36A···36AJ
order123···3446···69···912···1218···1836···36
size111···1111···11···11···11···11···1

108 irreducible representations

dim111111111111
type++
imageC1C2C3C3C4C6C6C9C12C12C18C36
kernelC3×C36C3×C18C36C3×C12C3×C9C18C3×C6C12C9C32C6C3
# reps1162262181241836

Matrix representation of C3×C36 in GL2(𝔽37) generated by

100
01
,
30
020
G:=sub<GL(2,GF(37))| [10,0,0,1],[3,0,0,20] >;

C3×C36 in GAP, Magma, Sage, TeX

C_3\times C_{36}
% in TeX

G:=Group("C3xC36");
// GroupNames label

G:=SmallGroup(108,12);
// by ID

G=gap.SmallGroup(108,12);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,-3,90,186]);
// Polycyclic

G:=Group<a,b|a^3=b^36=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C36 in TeX

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