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G = C19×SL2(𝔽3)  order 456 = 23·3·19

Direct product of C19 and SL2(𝔽3)

direct product, non-abelian, soluble

Aliases: C19×SL2(𝔽3), Q8⋊C57, C38.2A4, C2.(A4×C19), (Q8×C19)⋊1C3, SmallGroup(456,22)

Series: Derived Chief Lower central Upper central

C1C2Q8 — C19×SL2(𝔽3)
C1C2Q8Q8×C19 — C19×SL2(𝔽3)
Q8 — C19×SL2(𝔽3)
C1C38

Generators and relations for C19×SL2(𝔽3)
 G = < a,b,c,d | a19=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

4C3
3C4
4C6
4C57
3C76
4C114

Smallest permutation representation of C19×SL2(𝔽3)
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)
(1 106 60 137)(2 107 61 138)(3 108 62 139)(4 109 63 140)(5 110 64 141)(6 111 65 142)(7 112 66 143)(8 113 67 144)(9 114 68 145)(10 96 69 146)(11 97 70 147)(12 98 71 148)(13 99 72 149)(14 100 73 150)(15 101 74 151)(16 102 75 152)(17 103 76 134)(18 104 58 135)(19 105 59 136)(20 130 86 43)(21 131 87 44)(22 132 88 45)(23 133 89 46)(24 115 90 47)(25 116 91 48)(26 117 92 49)(27 118 93 50)(28 119 94 51)(29 120 95 52)(30 121 77 53)(31 122 78 54)(32 123 79 55)(33 124 80 56)(34 125 81 57)(35 126 82 39)(36 127 83 40)(37 128 84 41)(38 129 85 42)
(1 20 60 86)(2 21 61 87)(3 22 62 88)(4 23 63 89)(5 24 64 90)(6 25 65 91)(7 26 66 92)(8 27 67 93)(9 28 68 94)(10 29 69 95)(11 30 70 77)(12 31 71 78)(13 32 72 79)(14 33 73 80)(15 34 74 81)(16 35 75 82)(17 36 76 83)(18 37 58 84)(19 38 59 85)(39 152 126 102)(40 134 127 103)(41 135 128 104)(42 136 129 105)(43 137 130 106)(44 138 131 107)(45 139 132 108)(46 140 133 109)(47 141 115 110)(48 142 116 111)(49 143 117 112)(50 144 118 113)(51 145 119 114)(52 146 120 96)(53 147 121 97)(54 148 122 98)(55 149 123 99)(56 150 124 100)(57 151 125 101)
(20 106 43)(21 107 44)(22 108 45)(23 109 46)(24 110 47)(25 111 48)(26 112 49)(27 113 50)(28 114 51)(29 96 52)(30 97 53)(31 98 54)(32 99 55)(33 100 56)(34 101 57)(35 102 39)(36 103 40)(37 104 41)(38 105 42)(77 147 121)(78 148 122)(79 149 123)(80 150 124)(81 151 125)(82 152 126)(83 134 127)(84 135 128)(85 136 129)(86 137 130)(87 138 131)(88 139 132)(89 140 133)(90 141 115)(91 142 116)(92 143 117)(93 144 118)(94 145 119)(95 146 120)

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

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

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

133 conjugacy classes

class 1  2 3A3B 4 6A6B19A···19R38A···38R57A···57AJ76A···76R114A···114AJ
order123346619···1938···3857···5776···76114···114
size11446441···11···14···46···64···4

133 irreducible representations

dim111122233
type+-+
imageC1C3C19C57SL2(𝔽3)SL2(𝔽3)C19×SL2(𝔽3)A4A4×C19
kernelC19×SL2(𝔽3)Q8×C19SL2(𝔽3)Q8C19C19C1C38C2
# reps1218361254118

Matrix representation of C19×SL2(𝔽3) in GL2(𝔽229) generated by

1650
0165
,
13494
9495
,
01
2280
,
01
95135
G:=sub<GL(2,GF(229))| [165,0,0,165],[134,94,94,95],[0,228,1,0],[0,95,1,135] >;

C19×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_{19}\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C19xSL(2,3)");
// GroupNames label

G:=SmallGroup(456,22);
// by ID

G=gap.SmallGroup(456,22);
# by ID

G:=PCGroup([5,-3,-19,-2,2,-2,1712,72,3423,133,58]);
// Polycyclic

G:=Group<a,b,c,d|a^19=b^4=d^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations

Export

Subgroup lattice of C19×SL2(𝔽3) in TeX

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