Copied to
clipboard

G = C6×D19order 228 = 22·3·19

Direct product of C6 and D19

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

Aliases: C6×D19, C383C6, C1142C2, C573C22, C193(C2×C6), SmallGroup(228,12)

Series: Derived Chief Lower central Upper central

C1C19 — C6×D19
C1C19C57C3×D19 — C6×D19
C19 — C6×D19
C1C6

Generators and relations for C6×D19
 G = < a,b,c | a6=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >

19C2
19C2
19C22
19C6
19C6
19C2×C6

Smallest permutation representation of C6×D19
On 114 points
Generators in S114
(1 77 43 62 22 108)(2 78 44 63 23 109)(3 79 45 64 24 110)(4 80 46 65 25 111)(5 81 47 66 26 112)(6 82 48 67 27 113)(7 83 49 68 28 114)(8 84 50 69 29 96)(9 85 51 70 30 97)(10 86 52 71 31 98)(11 87 53 72 32 99)(12 88 54 73 33 100)(13 89 55 74 34 101)(14 90 56 75 35 102)(15 91 57 76 36 103)(16 92 39 58 37 104)(17 93 40 59 38 105)(18 94 41 60 20 106)(19 95 42 61 21 107)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114)
(1 61)(2 60)(3 59)(4 58)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 78)(21 77)(22 95)(23 94)(24 93)(25 92)(26 91)(27 90)(28 89)(29 88)(30 87)(31 86)(32 85)(33 84)(34 83)(35 82)(36 81)(37 80)(38 79)(39 111)(40 110)(41 109)(42 108)(43 107)(44 106)(45 105)(46 104)(47 103)(48 102)(49 101)(50 100)(51 99)(52 98)(53 97)(54 96)(55 114)(56 113)(57 112)

G:=sub<Sym(114)| (1,77,43,62,22,108)(2,78,44,63,23,109)(3,79,45,64,24,110)(4,80,46,65,25,111)(5,81,47,66,26,112)(6,82,48,67,27,113)(7,83,49,68,28,114)(8,84,50,69,29,96)(9,85,51,70,30,97)(10,86,52,71,31,98)(11,87,53,72,32,99)(12,88,54,73,33,100)(13,89,55,74,34,101)(14,90,56,75,35,102)(15,91,57,76,36,103)(16,92,39,58,37,104)(17,93,40,59,38,105)(18,94,41,60,20,106)(19,95,42,61,21,107), (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)(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114), (1,61)(2,60)(3,59)(4,58)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,78)(21,77)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,111)(40,110)(41,109)(42,108)(43,107)(44,106)(45,105)(46,104)(47,103)(48,102)(49,101)(50,100)(51,99)(52,98)(53,97)(54,96)(55,114)(56,113)(57,112)>;

G:=Group( (1,77,43,62,22,108)(2,78,44,63,23,109)(3,79,45,64,24,110)(4,80,46,65,25,111)(5,81,47,66,26,112)(6,82,48,67,27,113)(7,83,49,68,28,114)(8,84,50,69,29,96)(9,85,51,70,30,97)(10,86,52,71,31,98)(11,87,53,72,32,99)(12,88,54,73,33,100)(13,89,55,74,34,101)(14,90,56,75,35,102)(15,91,57,76,36,103)(16,92,39,58,37,104)(17,93,40,59,38,105)(18,94,41,60,20,106)(19,95,42,61,21,107), (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)(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114), (1,61)(2,60)(3,59)(4,58)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,78)(21,77)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,111)(40,110)(41,109)(42,108)(43,107)(44,106)(45,105)(46,104)(47,103)(48,102)(49,101)(50,100)(51,99)(52,98)(53,97)(54,96)(55,114)(56,113)(57,112) );

G=PermutationGroup([(1,77,43,62,22,108),(2,78,44,63,23,109),(3,79,45,64,24,110),(4,80,46,65,25,111),(5,81,47,66,26,112),(6,82,48,67,27,113),(7,83,49,68,28,114),(8,84,50,69,29,96),(9,85,51,70,30,97),(10,86,52,71,31,98),(11,87,53,72,32,99),(12,88,54,73,33,100),(13,89,55,74,34,101),(14,90,56,75,35,102),(15,91,57,76,36,103),(16,92,39,58,37,104),(17,93,40,59,38,105),(18,94,41,60,20,106),(19,95,42,61,21,107)], [(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),(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114)], [(1,61),(2,60),(3,59),(4,58),(5,76),(6,75),(7,74),(8,73),(9,72),(10,71),(11,70),(12,69),(13,68),(14,67),(15,66),(16,65),(17,64),(18,63),(19,62),(20,78),(21,77),(22,95),(23,94),(24,93),(25,92),(26,91),(27,90),(28,89),(29,88),(30,87),(31,86),(32,85),(33,84),(34,83),(35,82),(36,81),(37,80),(38,79),(39,111),(40,110),(41,109),(42,108),(43,107),(44,106),(45,105),(46,104),(47,103),(48,102),(49,101),(50,100),(51,99),(52,98),(53,97),(54,96),(55,114),(56,113),(57,112)])

C6×D19 is a maximal subgroup of   C57⋊D4  C3⋊D76

66 conjugacy classes

class 1 2A2B2C3A3B6A6B6C6D6E6F19A···19I38A···38I57A···57R114A···114R
order12223366666619···1938···3857···57114···114
size1119191111191919192···22···22···22···2

66 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6D19D38C3×D19C6×D19
kernelC6×D19C3×D19C114D38D19C38C6C3C2C1
# reps121242991818

Matrix representation of C6×D19 in GL2(𝔽37) generated by

110
011
,
114
3436
,
10
3436
G:=sub<GL(2,GF(37))| [11,0,0,11],[11,34,4,36],[1,34,0,36] >;

C6×D19 in GAP, Magma, Sage, TeX

C_6\times D_{19}
% in TeX

G:=Group("C6xD19");
// GroupNames label

G:=SmallGroup(228,12);
// by ID

G=gap.SmallGroup(228,12);
# by ID

G:=PCGroup([4,-2,-2,-3,-19,3459]);
// Polycyclic

G:=Group<a,b,c|a^6=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C6×D19 in TeX

׿
×
𝔽