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G = C19⋊C8order 152 = 23·19

The semidirect product of C19 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C19⋊C8, C38.C4, C76.2C2, C4.2D19, C2.Dic19, SmallGroup(152,1)

Series: Derived Chief Lower central Upper central

C1C19 — C19⋊C8
C1C19C38C76 — C19⋊C8
C19 — C19⋊C8
C1C4

Generators and relations for C19⋊C8
 G = < a,b | a19=b8=1, bab-1=a-1 >

19C8

Smallest permutation representation of C19⋊C8
Regular action on 152 points
Generators in S152
(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)(134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152)
(1 134 64 96 36 115 51 77)(2 152 65 114 37 133 52 95)(3 151 66 113 38 132 53 94)(4 150 67 112 20 131 54 93)(5 149 68 111 21 130 55 92)(6 148 69 110 22 129 56 91)(7 147 70 109 23 128 57 90)(8 146 71 108 24 127 39 89)(9 145 72 107 25 126 40 88)(10 144 73 106 26 125 41 87)(11 143 74 105 27 124 42 86)(12 142 75 104 28 123 43 85)(13 141 76 103 29 122 44 84)(14 140 58 102 30 121 45 83)(15 139 59 101 31 120 46 82)(16 138 60 100 32 119 47 81)(17 137 61 99 33 118 48 80)(18 136 62 98 34 117 49 79)(19 135 63 97 35 116 50 78)

G:=sub<Sym(152)| (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)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (1,134,64,96,36,115,51,77)(2,152,65,114,37,133,52,95)(3,151,66,113,38,132,53,94)(4,150,67,112,20,131,54,93)(5,149,68,111,21,130,55,92)(6,148,69,110,22,129,56,91)(7,147,70,109,23,128,57,90)(8,146,71,108,24,127,39,89)(9,145,72,107,25,126,40,88)(10,144,73,106,26,125,41,87)(11,143,74,105,27,124,42,86)(12,142,75,104,28,123,43,85)(13,141,76,103,29,122,44,84)(14,140,58,102,30,121,45,83)(15,139,59,101,31,120,46,82)(16,138,60,100,32,119,47,81)(17,137,61,99,33,118,48,80)(18,136,62,98,34,117,49,79)(19,135,63,97,35,116,50,78)>;

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)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (1,134,64,96,36,115,51,77)(2,152,65,114,37,133,52,95)(3,151,66,113,38,132,53,94)(4,150,67,112,20,131,54,93)(5,149,68,111,21,130,55,92)(6,148,69,110,22,129,56,91)(7,147,70,109,23,128,57,90)(8,146,71,108,24,127,39,89)(9,145,72,107,25,126,40,88)(10,144,73,106,26,125,41,87)(11,143,74,105,27,124,42,86)(12,142,75,104,28,123,43,85)(13,141,76,103,29,122,44,84)(14,140,58,102,30,121,45,83)(15,139,59,101,31,120,46,82)(16,138,60,100,32,119,47,81)(17,137,61,99,33,118,48,80)(18,136,62,98,34,117,49,79)(19,135,63,97,35,116,50,78) );

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),(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)], [(1,134,64,96,36,115,51,77),(2,152,65,114,37,133,52,95),(3,151,66,113,38,132,53,94),(4,150,67,112,20,131,54,93),(5,149,68,111,21,130,55,92),(6,148,69,110,22,129,56,91),(7,147,70,109,23,128,57,90),(8,146,71,108,24,127,39,89),(9,145,72,107,25,126,40,88),(10,144,73,106,26,125,41,87),(11,143,74,105,27,124,42,86),(12,142,75,104,28,123,43,85),(13,141,76,103,29,122,44,84),(14,140,58,102,30,121,45,83),(15,139,59,101,31,120,46,82),(16,138,60,100,32,119,47,81),(17,137,61,99,33,118,48,80),(18,136,62,98,34,117,49,79),(19,135,63,97,35,116,50,78)])

C19⋊C8 is a maximal subgroup of
C8×D19  C8⋊D19  C76.C4  D4⋊D19  D4.D19  Q8⋊D19  C19⋊Q16  C19⋊C24  C57⋊C8
C19⋊C8 is a maximal quotient of
C19⋊C16  C57⋊C8

44 conjugacy classes

class 1  2 4A4B8A8B8C8D19A···19I38A···38I76A···76R
order1244888819···1938···3876···76
size1111191919192···22···22···2

44 irreducible representations

dim1111222
type+++-
imageC1C2C4C8D19Dic19C19⋊C8
kernelC19⋊C8C76C38C19C4C2C1
# reps11249918

Matrix representation of C19⋊C8 in GL2(𝔽37) generated by

628
201
,
06
10
G:=sub<GL(2,GF(37))| [6,20,28,1],[0,1,6,0] >;

C19⋊C8 in GAP, Magma, Sage, TeX

C_{19}\rtimes C_8
% in TeX

G:=Group("C19:C8");
// GroupNames label

G:=SmallGroup(152,1);
// by ID

G=gap.SmallGroup(152,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-19,8,21,2307]);
// Polycyclic

G:=Group<a,b|a^19=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C19⋊C8 in TeX

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