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## G = SL2(𝔽3).F5order 480 = 25·3·5

### The non-split extension by SL2(𝔽3) of F5 acting through Inn(SL2(𝔽3))

Aliases: SL2(𝔽3).F5, C5⋊C8.A4, C5⋊(C8.A4), Q8.F5⋊C3, Q8.(C3×F5), (C5×Q8).C12, C2.2(A4×F5), C10.1(C4×A4), Q82D5.2C6, Dic5.4(C2×A4), (C5×SL2(𝔽3)).C4, Dic5.A4.3C2, SmallGroup(480,964)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×Q8 — SL2(𝔽3).F5
 Chief series C1 — C2 — C10 — C5×Q8 — Q8⋊2D5 — Dic5.A4 — SL2(𝔽3).F5
 Lower central C5×Q8 — SL2(𝔽3).F5
 Upper central C1 — C2

Generators and relations for SL2(𝔽3).F5
G = < a,b,c,d,e | a4=c3=d5=1, b2=e4=a2, bab-1=a-1, cac-1=b, ad=da, ae=ea, cbc-1=ab, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Smallest permutation representation of SL2(𝔽3).F5
On 160 points
Generators in S160
(1 35 5 39)(2 36 6 40)(3 37 7 33)(4 38 8 34)(9 147 13 151)(10 148 14 152)(11 149 15 145)(12 150 16 146)(17 65 21 69)(18 66 22 70)(19 67 23 71)(20 68 24 72)(25 127 29 123)(26 128 30 124)(27 121 31 125)(28 122 32 126)(41 143 45 139)(42 144 46 140)(43 137 47 141)(44 138 48 142)(49 107 53 111)(50 108 54 112)(51 109 55 105)(52 110 56 106)(57 87 61 83)(58 88 62 84)(59 81 63 85)(60 82 64 86)(73 131 77 135)(74 132 78 136)(75 133 79 129)(76 134 80 130)(89 117 93 113)(90 118 94 114)(91 119 95 115)(92 120 96 116)(97 155 101 159)(98 156 102 160)(99 157 103 153)(100 158 104 154)
(1 139 5 143)(2 140 6 144)(3 141 7 137)(4 142 8 138)(9 81 13 85)(10 82 14 86)(11 83 15 87)(12 84 16 88)(17 103 21 99)(18 104 22 100)(19 97 23 101)(20 98 24 102)(25 89 29 93)(26 90 30 94)(27 91 31 95)(28 92 32 96)(33 43 37 47)(34 44 38 48)(35 45 39 41)(36 46 40 42)(49 73 53 77)(50 74 54 78)(51 75 55 79)(52 76 56 80)(57 149 61 145)(58 150 62 146)(59 151 63 147)(60 152 64 148)(65 157 69 153)(66 158 70 154)(67 159 71 155)(68 160 72 156)(105 133 109 129)(106 134 110 130)(107 135 111 131)(108 136 112 132)(113 123 117 127)(114 124 118 128)(115 125 119 121)(116 126 120 122)
(9 147 85)(10 148 86)(11 149 87)(12 150 88)(13 151 81)(14 152 82)(15 145 83)(16 146 84)(17 103 153)(18 104 154)(19 97 155)(20 98 156)(21 99 157)(22 100 158)(23 101 159)(24 102 160)(25 113 123)(26 114 124)(27 115 125)(28 116 126)(29 117 127)(30 118 128)(31 119 121)(32 120 122)(33 43 137)(34 44 138)(35 45 139)(36 46 140)(37 47 141)(38 48 142)(39 41 143)(40 42 144)(49 135 111)(50 136 112)(51 129 105)(52 130 106)(53 131 107)(54 132 108)(55 133 109)(56 134 110)
(1 63 71 79 95)(2 80 64 96 72)(3 89 73 65 57)(4 66 90 58 74)(5 59 67 75 91)(6 76 60 92 68)(7 93 77 69 61)(8 70 94 62 78)(9 97 105 125 45)(10 126 98 46 106)(11 47 127 107 99)(12 108 48 100 128)(13 101 109 121 41)(14 122 102 42 110)(15 43 123 111 103)(16 112 44 104 124)(17 83 33 113 135)(18 114 84 136 34)(19 129 115 35 85)(20 36 130 86 116)(21 87 37 117 131)(22 118 88 132 38)(23 133 119 39 81)(24 40 134 82 120)(25 49 153 145 137)(26 146 50 138 154)(27 139 147 155 51)(28 156 140 52 148)(29 53 157 149 141)(30 150 54 142 158)(31 143 151 159 55)(32 160 144 56 152)
(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)(153 154 155 156 157 158 159 160)

G:=sub<Sym(160)| (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,147,13,151)(10,148,14,152)(11,149,15,145)(12,150,16,146)(17,65,21,69)(18,66,22,70)(19,67,23,71)(20,68,24,72)(25,127,29,123)(26,128,30,124)(27,121,31,125)(28,122,32,126)(41,143,45,139)(42,144,46,140)(43,137,47,141)(44,138,48,142)(49,107,53,111)(50,108,54,112)(51,109,55,105)(52,110,56,106)(57,87,61,83)(58,88,62,84)(59,81,63,85)(60,82,64,86)(73,131,77,135)(74,132,78,136)(75,133,79,129)(76,134,80,130)(89,117,93,113)(90,118,94,114)(91,119,95,115)(92,120,96,116)(97,155,101,159)(98,156,102,160)(99,157,103,153)(100,158,104,154), (1,139,5,143)(2,140,6,144)(3,141,7,137)(4,142,8,138)(9,81,13,85)(10,82,14,86)(11,83,15,87)(12,84,16,88)(17,103,21,99)(18,104,22,100)(19,97,23,101)(20,98,24,102)(25,89,29,93)(26,90,30,94)(27,91,31,95)(28,92,32,96)(33,43,37,47)(34,44,38,48)(35,45,39,41)(36,46,40,42)(49,73,53,77)(50,74,54,78)(51,75,55,79)(52,76,56,80)(57,149,61,145)(58,150,62,146)(59,151,63,147)(60,152,64,148)(65,157,69,153)(66,158,70,154)(67,159,71,155)(68,160,72,156)(105,133,109,129)(106,134,110,130)(107,135,111,131)(108,136,112,132)(113,123,117,127)(114,124,118,128)(115,125,119,121)(116,126,120,122), (9,147,85)(10,148,86)(11,149,87)(12,150,88)(13,151,81)(14,152,82)(15,145,83)(16,146,84)(17,103,153)(18,104,154)(19,97,155)(20,98,156)(21,99,157)(22,100,158)(23,101,159)(24,102,160)(25,113,123)(26,114,124)(27,115,125)(28,116,126)(29,117,127)(30,118,128)(31,119,121)(32,120,122)(33,43,137)(34,44,138)(35,45,139)(36,46,140)(37,47,141)(38,48,142)(39,41,143)(40,42,144)(49,135,111)(50,136,112)(51,129,105)(52,130,106)(53,131,107)(54,132,108)(55,133,109)(56,134,110), (1,63,71,79,95)(2,80,64,96,72)(3,89,73,65,57)(4,66,90,58,74)(5,59,67,75,91)(6,76,60,92,68)(7,93,77,69,61)(8,70,94,62,78)(9,97,105,125,45)(10,126,98,46,106)(11,47,127,107,99)(12,108,48,100,128)(13,101,109,121,41)(14,122,102,42,110)(15,43,123,111,103)(16,112,44,104,124)(17,83,33,113,135)(18,114,84,136,34)(19,129,115,35,85)(20,36,130,86,116)(21,87,37,117,131)(22,118,88,132,38)(23,133,119,39,81)(24,40,134,82,120)(25,49,153,145,137)(26,146,50,138,154)(27,139,147,155,51)(28,156,140,52,148)(29,53,157,149,141)(30,150,54,142,158)(31,143,151,159,55)(32,160,144,56,152), (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)(153,154,155,156,157,158,159,160)>;

G:=Group( (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,147,13,151)(10,148,14,152)(11,149,15,145)(12,150,16,146)(17,65,21,69)(18,66,22,70)(19,67,23,71)(20,68,24,72)(25,127,29,123)(26,128,30,124)(27,121,31,125)(28,122,32,126)(41,143,45,139)(42,144,46,140)(43,137,47,141)(44,138,48,142)(49,107,53,111)(50,108,54,112)(51,109,55,105)(52,110,56,106)(57,87,61,83)(58,88,62,84)(59,81,63,85)(60,82,64,86)(73,131,77,135)(74,132,78,136)(75,133,79,129)(76,134,80,130)(89,117,93,113)(90,118,94,114)(91,119,95,115)(92,120,96,116)(97,155,101,159)(98,156,102,160)(99,157,103,153)(100,158,104,154), (1,139,5,143)(2,140,6,144)(3,141,7,137)(4,142,8,138)(9,81,13,85)(10,82,14,86)(11,83,15,87)(12,84,16,88)(17,103,21,99)(18,104,22,100)(19,97,23,101)(20,98,24,102)(25,89,29,93)(26,90,30,94)(27,91,31,95)(28,92,32,96)(33,43,37,47)(34,44,38,48)(35,45,39,41)(36,46,40,42)(49,73,53,77)(50,74,54,78)(51,75,55,79)(52,76,56,80)(57,149,61,145)(58,150,62,146)(59,151,63,147)(60,152,64,148)(65,157,69,153)(66,158,70,154)(67,159,71,155)(68,160,72,156)(105,133,109,129)(106,134,110,130)(107,135,111,131)(108,136,112,132)(113,123,117,127)(114,124,118,128)(115,125,119,121)(116,126,120,122), (9,147,85)(10,148,86)(11,149,87)(12,150,88)(13,151,81)(14,152,82)(15,145,83)(16,146,84)(17,103,153)(18,104,154)(19,97,155)(20,98,156)(21,99,157)(22,100,158)(23,101,159)(24,102,160)(25,113,123)(26,114,124)(27,115,125)(28,116,126)(29,117,127)(30,118,128)(31,119,121)(32,120,122)(33,43,137)(34,44,138)(35,45,139)(36,46,140)(37,47,141)(38,48,142)(39,41,143)(40,42,144)(49,135,111)(50,136,112)(51,129,105)(52,130,106)(53,131,107)(54,132,108)(55,133,109)(56,134,110), (1,63,71,79,95)(2,80,64,96,72)(3,89,73,65,57)(4,66,90,58,74)(5,59,67,75,91)(6,76,60,92,68)(7,93,77,69,61)(8,70,94,62,78)(9,97,105,125,45)(10,126,98,46,106)(11,47,127,107,99)(12,108,48,100,128)(13,101,109,121,41)(14,122,102,42,110)(15,43,123,111,103)(16,112,44,104,124)(17,83,33,113,135)(18,114,84,136,34)(19,129,115,35,85)(20,36,130,86,116)(21,87,37,117,131)(22,118,88,132,38)(23,133,119,39,81)(24,40,134,82,120)(25,49,153,145,137)(26,146,50,138,154)(27,139,147,155,51)(28,156,140,52,148)(29,53,157,149,141)(30,150,54,142,158)(31,143,151,159,55)(32,160,144,56,152), (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)(153,154,155,156,157,158,159,160) );

G=PermutationGroup([[(1,35,5,39),(2,36,6,40),(3,37,7,33),(4,38,8,34),(9,147,13,151),(10,148,14,152),(11,149,15,145),(12,150,16,146),(17,65,21,69),(18,66,22,70),(19,67,23,71),(20,68,24,72),(25,127,29,123),(26,128,30,124),(27,121,31,125),(28,122,32,126),(41,143,45,139),(42,144,46,140),(43,137,47,141),(44,138,48,142),(49,107,53,111),(50,108,54,112),(51,109,55,105),(52,110,56,106),(57,87,61,83),(58,88,62,84),(59,81,63,85),(60,82,64,86),(73,131,77,135),(74,132,78,136),(75,133,79,129),(76,134,80,130),(89,117,93,113),(90,118,94,114),(91,119,95,115),(92,120,96,116),(97,155,101,159),(98,156,102,160),(99,157,103,153),(100,158,104,154)], [(1,139,5,143),(2,140,6,144),(3,141,7,137),(4,142,8,138),(9,81,13,85),(10,82,14,86),(11,83,15,87),(12,84,16,88),(17,103,21,99),(18,104,22,100),(19,97,23,101),(20,98,24,102),(25,89,29,93),(26,90,30,94),(27,91,31,95),(28,92,32,96),(33,43,37,47),(34,44,38,48),(35,45,39,41),(36,46,40,42),(49,73,53,77),(50,74,54,78),(51,75,55,79),(52,76,56,80),(57,149,61,145),(58,150,62,146),(59,151,63,147),(60,152,64,148),(65,157,69,153),(66,158,70,154),(67,159,71,155),(68,160,72,156),(105,133,109,129),(106,134,110,130),(107,135,111,131),(108,136,112,132),(113,123,117,127),(114,124,118,128),(115,125,119,121),(116,126,120,122)], [(9,147,85),(10,148,86),(11,149,87),(12,150,88),(13,151,81),(14,152,82),(15,145,83),(16,146,84),(17,103,153),(18,104,154),(19,97,155),(20,98,156),(21,99,157),(22,100,158),(23,101,159),(24,102,160),(25,113,123),(26,114,124),(27,115,125),(28,116,126),(29,117,127),(30,118,128),(31,119,121),(32,120,122),(33,43,137),(34,44,138),(35,45,139),(36,46,140),(37,47,141),(38,48,142),(39,41,143),(40,42,144),(49,135,111),(50,136,112),(51,129,105),(52,130,106),(53,131,107),(54,132,108),(55,133,109),(56,134,110)], [(1,63,71,79,95),(2,80,64,96,72),(3,89,73,65,57),(4,66,90,58,74),(5,59,67,75,91),(6,76,60,92,68),(7,93,77,69,61),(8,70,94,62,78),(9,97,105,125,45),(10,126,98,46,106),(11,47,127,107,99),(12,108,48,100,128),(13,101,109,121,41),(14,122,102,42,110),(15,43,123,111,103),(16,112,44,104,124),(17,83,33,113,135),(18,114,84,136,34),(19,129,115,35,85),(20,36,130,86,116),(21,87,37,117,131),(22,118,88,132,38),(23,133,119,39,81),(24,40,134,82,120),(25,49,153,145,137),(26,146,50,138,154),(27,139,147,155,51),(28,156,140,52,148),(29,53,157,149,141),(30,150,54,142,158),(31,143,151,159,55),(32,160,144,56,152)], [(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),(153,154,155,156,157,158,159,160)]])

35 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 5 6A 6B 8A 8B 8C 8D 8E 8F 10 12A 12B 12C 12D 15A 15B 20 24A ··· 24H 30A 30B order 1 2 2 3 3 4 4 4 5 6 6 8 8 8 8 8 8 10 12 12 12 12 15 15 20 24 ··· 24 30 30 size 1 1 30 4 4 5 5 6 4 4 4 5 5 5 5 30 30 4 20 20 20 20 16 16 24 20 ··· 20 16 16

35 irreducible representations

 dim 1 1 1 1 1 1 12 2 3 3 3 4 4 8 8 type + + + + + + + image C1 C2 C3 C4 C6 C12 A4×F5 C8.A4 A4 C2×A4 C4×A4 F5 C3×F5 SL2(𝔽3).F5 SL2(𝔽3).F5 kernel SL2(𝔽3).F5 Dic5.A4 Q8.F5 C5×SL2(𝔽3) Q8⋊2D5 C5×Q8 C2 C5 C5⋊C8 Dic5 C10 SL2(𝔽3) Q8 C1 C1 # reps 1 1 2 2 2 4 1 12 1 1 2 1 2 1 2

Matrix representation of SL2(𝔽3).F5 in GL6(𝔽241)

 16 15 0 0 0 0 15 225 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 15 225 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 240 0 0 1 0 0 240 0 0 0 1 0 240 0 0 0 0 1 240
,
 30 0 0 0 0 0 0 30 0 0 0 0 0 0 83 115 232 211 0 0 74 85 73 53 0 0 156 168 188 44 0 0 30 159 158 126

G:=sub<GL(6,GF(241))| [16,15,0,0,0,0,15,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,15,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,240,240,240],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,83,74,156,30,0,0,115,85,168,159,0,0,232,73,188,158,0,0,211,53,44,126] >;

SL2(𝔽3).F5 in GAP, Magma, Sage, TeX

{\rm SL}_2({\mathbb F}_3).F_5
% in TeX

G:=Group("SL(2,3).F5");
// GroupNames label

G:=SmallGroup(480,964);
// by ID

G=gap.SmallGroup(480,964);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-5,-2,42,856,514,584,221,795,382,4037,1363]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^3=d^5=1,b^2=e^4=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,a*d=d*a,a*e=e*a,c*b*c^-1=a*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations

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