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G = C19⋊C24order 456 = 23·3·19

The semidirect product of C19 and C24 acting via C24/C4=C6

metacyclic, supersoluble, monomial, Z-group

Aliases: C19⋊C24, C38.C12, C76.2C6, C19⋊C8⋊C3, C19⋊C3⋊C8, C2.(C19⋊C12), C4.2(C19⋊C6), (C2×C19⋊C3).C4, (C4×C19⋊C3).2C2, SmallGroup(456,1)

Series: Derived Chief Lower central Upper central

C1C19 — C19⋊C24
C1C19C38C76C4×C19⋊C3 — C19⋊C24
C19 — C19⋊C24
C1C4

Generators and relations for C19⋊C24
 G = < a,b | a19=b24=1, bab-1=a12 >

19C3
19C6
19C8
19C12
19C24

Smallest permutation representation of C19⋊C24
On 152 points
Generators in S152
(1 126 52 44 19 92 11 110 140 27 72 118 132 80 36 64 100 84 148)(2 133 93 85 73 45 65 141 127 57 12 149 119 20 101 28 53 37 111)(3 120 46 38 13 86 29 128 134 21 66 112 150 74 54 58 94 102 142)(4 151 87 103 67 39 59 135 121 75 30 143 113 14 95 22 47 55 105)(5 114 40 56 31 104 23 122 152 15 60 106 144 68 48 76 88 96 136)(6 145 81 97 61 33 77 129 115 69 24 137 107 32 89 16 41 49 123)(7 108 34 50 25 98 17 116 146 9 78 124 138 62 42 70 82 90 130)(8 139 99 91 79 51 71 147 109 63 18 131 125 26 83 10 35 43 117)
(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)

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

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

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

36 conjugacy classes

class 1  2 3A3B4A4B6A6B8A8B8C8D12A12B12C12D19A19B19C24A···24H38A38B38C76A···76F
order1233446688881212121219191924···2438383876···76
size111919111919191919191919191966619···196666···6

36 irreducible representations

dim11111111666
type+++-
imageC1C2C3C4C6C8C12C24C19⋊C6C19⋊C12C19⋊C24
kernelC19⋊C24C4×C19⋊C3C19⋊C8C2×C19⋊C3C76C19⋊C3C38C19C4C2C1
# reps11222448336

Matrix representation of C19⋊C24 in GL6(𝔽457)

010000
001000
000100
000010
000001
4564617126717146
,
1594155241371436
279333532193185
213864440542109
26638012845328640
117190353379236416
19213881111919

G:=sub<GL(6,GF(457))| [0,0,0,0,0,456,1,0,0,0,0,46,0,1,0,0,0,171,0,0,1,0,0,267,0,0,0,1,0,171,0,0,0,0,1,46],[159,279,21,266,117,19,415,3,386,380,190,21,52,335,44,128,353,388,413,32,405,453,379,111,71,193,42,286,236,19,436,185,109,40,416,19] >;

C19⋊C24 in GAP, Magma, Sage, TeX

C_{19}\rtimes C_{24}
% in TeX

G:=Group("C19:C24");
// GroupNames label

G:=SmallGroup(456,1);
// by ID

G=gap.SmallGroup(456,1);
# by ID

G:=PCGroup([5,-2,-3,-2,-2,-19,30,42,10804,2109]);
// Polycyclic

G:=Group<a,b|a^19=b^24=1,b*a*b^-1=a^12>;
// generators/relations

Export

Subgroup lattice of C19⋊C24 in TeX

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