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G = C19×C7⋊C3order 399 = 3·7·19

Direct product of C19 and C7⋊C3

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

Aliases: C19×C7⋊C3, C7⋊C57, C1331C3, SmallGroup(399,1)

Series: Derived Chief Lower central Upper central

C1C7 — C19×C7⋊C3
C1C7C133 — C19×C7⋊C3
C7 — C19×C7⋊C3
C1C19

Generators and relations for C19×C7⋊C3
 G = < a,b,c | a19=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

7C3
7C57

Smallest permutation representation of C19×C7⋊C3
On 133 points
Generators in S133
(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)(115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133)
(1 20 65 110 90 46 130)(2 21 66 111 91 47 131)(3 22 67 112 92 48 132)(4 23 68 113 93 49 133)(5 24 69 114 94 50 115)(6 25 70 96 95 51 116)(7 26 71 97 77 52 117)(8 27 72 98 78 53 118)(9 28 73 99 79 54 119)(10 29 74 100 80 55 120)(11 30 75 101 81 56 121)(12 31 76 102 82 57 122)(13 32 58 103 83 39 123)(14 33 59 104 84 40 124)(15 34 60 105 85 41 125)(16 35 61 106 86 42 126)(17 36 62 107 87 43 127)(18 37 63 108 88 44 128)(19 38 64 109 89 45 129)
(20 65 90)(21 66 91)(22 67 92)(23 68 93)(24 69 94)(25 70 95)(26 71 77)(27 72 78)(28 73 79)(29 74 80)(30 75 81)(31 76 82)(32 58 83)(33 59 84)(34 60 85)(35 61 86)(36 62 87)(37 63 88)(38 64 89)(39 103 123)(40 104 124)(41 105 125)(42 106 126)(43 107 127)(44 108 128)(45 109 129)(46 110 130)(47 111 131)(48 112 132)(49 113 133)(50 114 115)(51 96 116)(52 97 117)(53 98 118)(54 99 119)(55 100 120)(56 101 121)(57 102 122)

G:=sub<Sym(133)| (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)(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133), (1,20,65,110,90,46,130)(2,21,66,111,91,47,131)(3,22,67,112,92,48,132)(4,23,68,113,93,49,133)(5,24,69,114,94,50,115)(6,25,70,96,95,51,116)(7,26,71,97,77,52,117)(8,27,72,98,78,53,118)(9,28,73,99,79,54,119)(10,29,74,100,80,55,120)(11,30,75,101,81,56,121)(12,31,76,102,82,57,122)(13,32,58,103,83,39,123)(14,33,59,104,84,40,124)(15,34,60,105,85,41,125)(16,35,61,106,86,42,126)(17,36,62,107,87,43,127)(18,37,63,108,88,44,128)(19,38,64,109,89,45,129), (20,65,90)(21,66,91)(22,67,92)(23,68,93)(24,69,94)(25,70,95)(26,71,77)(27,72,78)(28,73,79)(29,74,80)(30,75,81)(31,76,82)(32,58,83)(33,59,84)(34,60,85)(35,61,86)(36,62,87)(37,63,88)(38,64,89)(39,103,123)(40,104,124)(41,105,125)(42,106,126)(43,107,127)(44,108,128)(45,109,129)(46,110,130)(47,111,131)(48,112,132)(49,113,133)(50,114,115)(51,96,116)(52,97,117)(53,98,118)(54,99,119)(55,100,120)(56,101,121)(57,102,122)>;

G:=Group( (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)(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133), (1,20,65,110,90,46,130)(2,21,66,111,91,47,131)(3,22,67,112,92,48,132)(4,23,68,113,93,49,133)(5,24,69,114,94,50,115)(6,25,70,96,95,51,116)(7,26,71,97,77,52,117)(8,27,72,98,78,53,118)(9,28,73,99,79,54,119)(10,29,74,100,80,55,120)(11,30,75,101,81,56,121)(12,31,76,102,82,57,122)(13,32,58,103,83,39,123)(14,33,59,104,84,40,124)(15,34,60,105,85,41,125)(16,35,61,106,86,42,126)(17,36,62,107,87,43,127)(18,37,63,108,88,44,128)(19,38,64,109,89,45,129), (20,65,90)(21,66,91)(22,67,92)(23,68,93)(24,69,94)(25,70,95)(26,71,77)(27,72,78)(28,73,79)(29,74,80)(30,75,81)(31,76,82)(32,58,83)(33,59,84)(34,60,85)(35,61,86)(36,62,87)(37,63,88)(38,64,89)(39,103,123)(40,104,124)(41,105,125)(42,106,126)(43,107,127)(44,108,128)(45,109,129)(46,110,130)(47,111,131)(48,112,132)(49,113,133)(50,114,115)(51,96,116)(52,97,117)(53,98,118)(54,99,119)(55,100,120)(56,101,121)(57,102,122) );

G=PermutationGroup([(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),(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133)], [(1,20,65,110,90,46,130),(2,21,66,111,91,47,131),(3,22,67,112,92,48,132),(4,23,68,113,93,49,133),(5,24,69,114,94,50,115),(6,25,70,96,95,51,116),(7,26,71,97,77,52,117),(8,27,72,98,78,53,118),(9,28,73,99,79,54,119),(10,29,74,100,80,55,120),(11,30,75,101,81,56,121),(12,31,76,102,82,57,122),(13,32,58,103,83,39,123),(14,33,59,104,84,40,124),(15,34,60,105,85,41,125),(16,35,61,106,86,42,126),(17,36,62,107,87,43,127),(18,37,63,108,88,44,128),(19,38,64,109,89,45,129)], [(20,65,90),(21,66,91),(22,67,92),(23,68,93),(24,69,94),(25,70,95),(26,71,77),(27,72,78),(28,73,79),(29,74,80),(30,75,81),(31,76,82),(32,58,83),(33,59,84),(34,60,85),(35,61,86),(36,62,87),(37,63,88),(38,64,89),(39,103,123),(40,104,124),(41,105,125),(42,106,126),(43,107,127),(44,108,128),(45,109,129),(46,110,130),(47,111,131),(48,112,132),(49,113,133),(50,114,115),(51,96,116),(52,97,117),(53,98,118),(54,99,119),(55,100,120),(56,101,121),(57,102,122)])

95 conjugacy classes

class 1 3A3B7A7B19A···19R57A···57AJ133A···133AJ
order1337719···1957···57133···133
size177331···17···73···3

95 irreducible representations

dim111133
type+
imageC1C3C19C57C7⋊C3C19×C7⋊C3
kernelC19×C7⋊C3C133C7⋊C3C7C19C1
# reps121836236

Matrix representation of C19×C7⋊C3 in GL3(𝔽1597) generated by

8100
0810
0081
,
5711540
101596
011596
,
159601
5611541
159600
G:=sub<GL(3,GF(1597))| [81,0,0,0,81,0,0,0,81],[57,1,0,1,0,1,1540,1596,1596],[1596,56,1596,0,1,0,1,1541,0] >;

C19×C7⋊C3 in GAP, Magma, Sage, TeX

C_{19}\times C_7\rtimes C_3
% in TeX

G:=Group("C19xC7:C3");
// GroupNames label

G:=SmallGroup(399,1);
// by ID

G=gap.SmallGroup(399,1);
# by ID

G:=PCGroup([3,-3,-19,-7,1028]);
// Polycyclic

G:=Group<a,b,c|a^19=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

Export

Subgroup lattice of C19×C7⋊C3 in TeX

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