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G = C6×C19⋊C3order 342 = 2·32·19

Direct product of C6 and C19⋊C3

direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary

Aliases: C6×C19⋊C3, C114⋊C3, C38⋊C32, C574C6, C192(C3×C6), SmallGroup(342,12)

Series: Derived Chief Lower central Upper central

C1C19 — C6×C19⋊C3
C1C19C57C3×C19⋊C3 — C6×C19⋊C3
C19 — C6×C19⋊C3
C1C6

Generators and relations for C6×C19⋊C3
 G = < a,b,c | a6=b19=c3=1, ab=ba, ac=ca, cbc-1=b11 >

19C3
19C3
19C3
19C6
19C6
19C6
19C32
19C3×C6

Smallest permutation representation of C6×C19⋊C3
On 114 points
Generators in S114
(1 77 39 58 20 96)(2 78 40 59 21 97)(3 79 41 60 22 98)(4 80 42 61 23 99)(5 81 43 62 24 100)(6 82 44 63 25 101)(7 83 45 64 26 102)(8 84 46 65 27 103)(9 85 47 66 28 104)(10 86 48 67 29 105)(11 87 49 68 30 106)(12 88 50 69 31 107)(13 89 51 70 32 108)(14 90 52 71 33 109)(15 91 53 72 34 110)(16 92 54 73 35 111)(17 93 55 74 36 112)(18 94 56 75 37 113)(19 95 57 76 38 114)
(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 39 20)(2 46 31)(3 53 23)(4 41 34)(5 48 26)(6 55 37)(7 43 29)(8 50 21)(9 57 32)(10 45 24)(11 52 35)(12 40 27)(13 47 38)(14 54 30)(15 42 22)(16 49 33)(17 56 25)(18 44 36)(19 51 28)(58 96 77)(59 103 88)(60 110 80)(61 98 91)(62 105 83)(63 112 94)(64 100 86)(65 107 78)(66 114 89)(67 102 81)(68 109 92)(69 97 84)(70 104 95)(71 111 87)(72 99 79)(73 106 90)(74 113 82)(75 101 93)(76 108 85)

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

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

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

54 conjugacy classes

class 1  2 3A3B3C···3H6A6B6C···6H19A···19F38A···38F57A···57L114A···114L
order12333···3666···619···1938···3857···57114···114
size111119···191119···193···33···33···33···3

54 irreducible representations

dim1111113333
type++
imageC1C2C3C3C6C6C19⋊C3C2×C19⋊C3C3×C19⋊C3C6×C19⋊C3
kernelC6×C19⋊C3C3×C19⋊C3C2×C19⋊C3C114C19⋊C3C57C6C3C2C1
# reps116262661212

Matrix representation of C6×C19⋊C3 in GL3(𝔽7) generated by

300
030
003
,
510
151
420
,
160
061
060
G:=sub<GL(3,GF(7))| [3,0,0,0,3,0,0,0,3],[5,1,4,1,5,2,0,1,0],[1,0,0,6,6,6,0,1,0] >;

C6×C19⋊C3 in GAP, Magma, Sage, TeX

C_6\times C_{19}\rtimes C_3
% in TeX

G:=Group("C6xC19:C3");
// GroupNames label

G:=SmallGroup(342,12);
// by ID

G=gap.SmallGroup(342,12);
# by ID

G:=PCGroup([4,-2,-3,-3,-19,1015]);
// Polycyclic

G:=Group<a,b,c|a^6=b^19=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^11>;
// generators/relations

Export

Subgroup lattice of C6×C19⋊C3 in TeX

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