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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))

non-abelian, soluble

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

C1C2C5×Q8 — SL2(𝔽3).F5
C1C2C10C5×Q8Q82D5Dic5.A4 — SL2(𝔽3).F5
C5×Q8 — SL2(𝔽3).F5
C1C2

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 >

30C2
4C3
3C4
5C4
15C22
4C6
6D5
4C15
5C8
15C8
15C2×C4
15D4
20C12
3D10
3C20
4C30
5C4○D4
15C2×C8
15M4(2)
20C24
3D20
3C5⋊C8
3C4×D5
4C3×Dic5
5C8○D4
5C4.A4
3D5⋊C8
3C4.F5
4C3×C5⋊C8
5C8.A4

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 2A2B3A3B4A4B4C 5 6A6B8A8B8C8D8E8F 10 12A12B12C12D15A15B 20 24A···24H30A30B
order12233444566888888101212121215152024···243030
size1130445564445555303042020202016162420···201616

35 irreducible representations

dim1111111223334488
type+++++++
imageC1C2C3C4C6C12A4×F5C8.A4A4C2×A4C4×A4F5C3×F5SL2(𝔽3).F5SL2(𝔽3).F5
kernelSL2(𝔽3).F5Dic5.A4Q8.F5C5×SL2(𝔽3)Q82D5C5×Q8C2C5C5⋊C8Dic5C10SL2(𝔽3)Q8C1C1
# reps1122241121121212

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

16150000
152250000
001000
000100
000010
000001
,
010000
24000000
001000
000100
000010
000001
,
100000
152250000
001000
000100
000010
000001
,
100000
010000
00000240
00100240
00010240
00001240
,
3000000
0300000
0083115232211
0074857353
0015616818844
0030159158126

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

Export

Subgroup lattice of SL2(𝔽3).F5 in TeX

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