C6xC19:C3 - GroupNames

(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	3A	3B	3C	···	3H	6A	6B	6C	···	6H	19A	···	19F	38A	···	38F	57A	···	57L	114A	···	114L
order	1	2	3	3	3	···	3	6	6	6	···	6	19	···	19	38	···	38	57	···	57	114	···	114
size	1	1	1	1	19	···	19	1	1	19	···	19	3	···	3	3	···	3	3	···	3	3	···	3

54 irreducible representations

dim	1	1	1	1	1	1	3	3	3	3
type	+	+
image	C₁	C₂	C₃	C₃	C₆	C₆	C₁₉⋊C₃	C₂×C₁₉⋊C₃	C₃×C₁₉⋊C₃	C₆×C₁₉⋊C₃
kernel	C₆×C₁₉⋊C₃	C₃×C₁₉⋊C₃	C₂×C₁₉⋊C₃	C₁₁₄	C₁₉⋊C₃	C₅₇	C₆	C₃	C₂	C₁
# reps	1	1	6	2	6	2	6	6	12	12

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

3	0	0
0	3	0
0	0	3

5	1	0
1	5	1
4	2	0

1	6	0
0	6	1
0	6	0

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

G = C6×C19⋊C3 order 342 = 2·32·19

Direct product of C6 and C19⋊C3

G = C₆×C₁₉⋊C₃ order 342 = 2·3²·19

Direct product of C₆ and C₁₉⋊C₃