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G = C6×D13order 156 = 22·3·13

Direct product of C6 and D13

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

Aliases: C6×D13, C782C2, C263C6, C393C22, C133(C2×C6), SmallGroup(156,15)

Series: Derived Chief Lower central Upper central

C1C13 — C6×D13
C1C13C39C3×D13 — C6×D13
C13 — C6×D13
C1C6

Generators and relations for C6×D13
 G = < a,b,c | a6=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

13C2
13C2
13C22
13C6
13C6
13C2×C6

Smallest permutation representation of C6×D13
On 78 points
Generators in S78
(1 64 28 48 18 70)(2 65 29 49 19 71)(3 53 30 50 20 72)(4 54 31 51 21 73)(5 55 32 52 22 74)(6 56 33 40 23 75)(7 57 34 41 24 76)(8 58 35 42 25 77)(9 59 36 43 26 78)(10 60 37 44 14 66)(11 61 38 45 15 67)(12 62 39 46 16 68)(13 63 27 47 17 69)
(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)
(1 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 52)(10 51)(11 50)(12 49)(13 48)(14 54)(15 53)(16 65)(17 64)(18 63)(19 62)(20 61)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 70)(28 69)(29 68)(30 67)(31 66)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)

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

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

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

C6×D13 is a maximal subgroup of   C39⋊D4  C3⋊D52

48 conjugacy classes

class 1 2A2B2C3A3B6A6B6C6D6E6F13A···13F26A···26F39A···39L78A···78L
order12223366666613···1326···2639···3978···78
size1113131111131313132···22···22···22···2

48 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6D13D26C3×D13C6×D13
kernelC6×D13C3×D13C78D26D13C26C6C3C2C1
# reps121242661212

Matrix representation of C6×D13 in GL3(𝔽79) generated by

7800
0550
0055
,
100
001
0786
,
100
001
010
G:=sub<GL(3,GF(79))| [78,0,0,0,55,0,0,0,55],[1,0,0,0,0,78,0,1,6],[1,0,0,0,0,1,0,1,0] >;

C6×D13 in GAP, Magma, Sage, TeX

C_6\times D_{13}
% in TeX

G:=Group("C6xD13");
// GroupNames label

G:=SmallGroup(156,15);
// by ID

G=gap.SmallGroup(156,15);
# by ID

G:=PCGroup([4,-2,-2,-3,-13,2307]);
// Polycyclic

G:=Group<a,b,c|a^6=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C6×D13 in TeX

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