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G = C8⋊D19order 304 = 24·19

3rd semidirect product of C8 and D19 acting via D19/C19=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C83D19, D38.C4, C1524C2, C4.13D38, Dic19.C4, C191M4(2), C76.13C22, C19⋊C84C2, C38.2(C2×C4), C2.3(C4×D19), (C4×D19).2C2, SmallGroup(304,4)

Series: Derived Chief Lower central Upper central

C1C38 — C8⋊D19
C1C19C38C76C4×D19 — C8⋊D19
C19C38 — C8⋊D19
C1C4C8

Generators and relations for C8⋊D19
 G = < a,b,c | a8=b19=c2=1, ab=ba, cac=a5, cbc=b-1 >

38C2
19C22
19C4
2D19
19C2×C4
19C8
19M4(2)

Smallest permutation representation of C8⋊D19
On 152 points
Generators in S152
(1 135 64 106 23 117 43 85)(2 136 65 107 24 118 44 86)(3 137 66 108 25 119 45 87)(4 138 67 109 26 120 46 88)(5 139 68 110 27 121 47 89)(6 140 69 111 28 122 48 90)(7 141 70 112 29 123 49 91)(8 142 71 113 30 124 50 92)(9 143 72 114 31 125 51 93)(10 144 73 96 32 126 52 94)(11 145 74 97 33 127 53 95)(12 146 75 98 34 128 54 77)(13 147 76 99 35 129 55 78)(14 148 58 100 36 130 56 79)(15 149 59 101 37 131 57 80)(16 150 60 102 38 132 39 81)(17 151 61 103 20 133 40 82)(18 152 62 104 21 115 41 83)(19 134 63 105 22 116 42 84)
(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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 25)(21 24)(22 23)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(39 46)(40 45)(41 44)(42 43)(47 57)(48 56)(49 55)(50 54)(51 53)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(70 76)(71 75)(72 74)(77 113)(78 112)(79 111)(80 110)(81 109)(82 108)(83 107)(84 106)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(95 114)(115 136)(116 135)(117 134)(118 152)(119 151)(120 150)(121 149)(122 148)(123 147)(124 146)(125 145)(126 144)(127 143)(128 142)(129 141)(130 140)(131 139)(132 138)(133 137)

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

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

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

82 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D19A···19I38A···38I76A···76R152A···152AJ
order122444888819···1938···3876···76152···152
size113811382238382···22···22···22···2

82 irreducible representations

dim11111122222
type++++++
imageC1C2C2C2C4C4M4(2)D19D38C4×D19C8⋊D19
kernelC8⋊D19C19⋊C8C152C4×D19Dic19D38C19C8C4C2C1
# reps1111222991836

Matrix representation of C8⋊D19 in GL2(𝔽37) generated by

1028
2027
,
729
2618
,
1926
2618
G:=sub<GL(2,GF(37))| [10,20,28,27],[7,26,29,18],[19,26,26,18] >;

C8⋊D19 in GAP, Magma, Sage, TeX

C_8\rtimes D_{19}
% in TeX

G:=Group("C8:D19");
// GroupNames label

G:=SmallGroup(304,4);
// by ID

G=gap.SmallGroup(304,4);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-19,101,26,42,7204]);
// Polycyclic

G:=Group<a,b,c|a^8=b^19=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C8⋊D19 in TeX

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