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## G = SL2(𝔽3).Dic5order 480 = 25·3·5

### The non-split extension by SL2(𝔽3) of Dic5 acting through Inn(SL2(𝔽3))

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×Q8 — SL2(𝔽3).Dic5
 Chief series C1 — C2 — C10 — C5×Q8 — C5×C4○D4 — C5×C4.A4 — SL2(𝔽3).Dic5
 Lower central C5×Q8 — SL2(𝔽3).Dic5
 Upper central C1 — C4

Generators and relations for SL2(𝔽3).Dic5
G = < a,b,c,d,e | a4=c3=1, b2=d10=a2, e2=d5, bab-1=a-1, cac-1=b, ad=da, ae=ea, cbc-1=ab, bd=db, be=eb, cd=dc, ce=ec, ede-1=d9 >

Smallest permutation representation of SL2(𝔽3).Dic5
On 160 points
Generators in S160
(1 118 11 108)(2 119 12 109)(3 120 13 110)(4 101 14 111)(5 102 15 112)(6 103 16 113)(7 104 17 114)(8 105 18 115)(9 106 19 116)(10 107 20 117)(21 158 31 148)(22 159 32 149)(23 160 33 150)(24 141 34 151)(25 142 35 152)(26 143 36 153)(27 144 37 154)(28 145 38 155)(29 146 39 156)(30 147 40 157)(41 126 51 136)(42 127 52 137)(43 128 53 138)(44 129 54 139)(45 130 55 140)(46 131 56 121)(47 132 57 122)(48 133 58 123)(49 134 59 124)(50 135 60 125)(61 87 71 97)(62 88 72 98)(63 89 73 99)(64 90 74 100)(65 91 75 81)(66 92 76 82)(67 93 77 83)(68 94 78 84)(69 95 79 85)(70 96 80 86)
(1 100 11 90)(2 81 12 91)(3 82 13 92)(4 83 14 93)(5 84 15 94)(6 85 16 95)(7 86 17 96)(8 87 18 97)(9 88 19 98)(10 89 20 99)(21 137 31 127)(22 138 32 128)(23 139 33 129)(24 140 34 130)(25 121 35 131)(26 122 36 132)(27 123 37 133)(28 124 38 134)(29 125 39 135)(30 126 40 136)(41 157 51 147)(42 158 52 148)(43 159 53 149)(44 160 54 150)(45 141 55 151)(46 142 56 152)(47 143 57 153)(48 144 58 154)(49 145 59 155)(50 146 60 156)(61 115 71 105)(62 116 72 106)(63 117 73 107)(64 118 74 108)(65 119 75 109)(66 120 76 110)(67 101 77 111)(68 102 78 112)(69 103 79 113)(70 104 80 114)
(21 158 127)(22 159 128)(23 160 129)(24 141 130)(25 142 131)(26 143 132)(27 144 133)(28 145 134)(29 146 135)(30 147 136)(31 148 137)(32 149 138)(33 150 139)(34 151 140)(35 152 121)(36 153 122)(37 154 123)(38 155 124)(39 156 125)(40 157 126)(61 87 105)(62 88 106)(63 89 107)(64 90 108)(65 91 109)(66 92 110)(67 93 111)(68 94 112)(69 95 113)(70 96 114)(71 97 115)(72 98 116)(73 99 117)(74 100 118)(75 81 119)(76 82 120)(77 83 101)(78 84 102)(79 85 103)(80 86 104)
(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)
(1 57 6 42 11 47 16 52)(2 46 7 51 12 56 17 41)(3 55 8 60 13 45 18 50)(4 44 9 49 14 54 19 59)(5 53 10 58 15 43 20 48)(21 64 26 69 31 74 36 79)(22 73 27 78 32 63 37 68)(23 62 28 67 33 72 38 77)(24 71 29 76 34 61 39 66)(25 80 30 65 35 70 40 75)(81 142 86 147 91 152 96 157)(82 151 87 156 92 141 97 146)(83 160 88 145 93 150 98 155)(84 149 89 154 94 159 99 144)(85 158 90 143 95 148 100 153)(101 129 106 134 111 139 116 124)(102 138 107 123 112 128 117 133)(103 127 108 132 113 137 118 122)(104 136 109 121 114 126 119 131)(105 125 110 130 115 135 120 140)

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

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

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

56 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 5A 5B 6A 6B 8A 8B 8C 8D 8E 8F 10A 10B 10C 10D 12A 12B 12C 12D 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 24A ··· 24H 30A 30B 30C 30D 60A ··· 60H order 1 2 2 3 3 4 4 4 5 5 6 6 8 8 8 8 8 8 10 10 10 10 12 12 12 12 15 15 15 15 20 20 20 20 20 20 24 ··· 24 30 30 30 30 60 ··· 60 size 1 1 6 4 4 1 1 6 2 2 4 4 5 5 5 5 30 30 2 2 12 12 4 4 4 4 8 8 8 8 2 2 2 2 12 12 20 ··· 20 8 8 8 8 8 ··· 8

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 3 3 3 4 6 6 type + + + - + + + - image C1 C2 C3 C4 C6 C12 D5 Dic5 C3×D5 C3×Dic5 C8.A4 A4 C2×A4 C4×A4 SL2(𝔽3).Dic5 D5×A4 A4×Dic5 kernel SL2(𝔽3).Dic5 C5×C4.A4 D4.Dic5 C5×SL2(𝔽3) C5×C4○D4 C5×Q8 C4.A4 SL2(𝔽3) C4○D4 Q8 C5 C5⋊2C8 C20 C10 C1 C4 C2 # reps 1 1 2 2 2 4 2 2 4 4 12 1 1 2 12 2 2

Matrix representation of SL2(𝔽3).Dic5 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 16 15 0 0 15 225
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 240 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 15 225
,
 0 240 0 0 1 190 0 0 0 0 177 0 0 0 0 177
,
 31 147 0 0 41 210 0 0 0 0 211 0 0 0 0 211
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,16,15,0,0,15,225],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,1,15,0,0,0,225],[0,1,0,0,240,190,0,0,0,0,177,0,0,0,0,177],[31,41,0,0,147,210,0,0,0,0,211,0,0,0,0,211] >;

SL2(𝔽3).Dic5 in GAP, Magma, Sage, TeX

{\rm SL}_2({\mathbb F}_3).{\rm Dic}_5
% in TeX

G:=Group("SL(2,3).Dic5");
// GroupNames label

G:=SmallGroup(480,267);
// by ID

G=gap.SmallGroup(480,267);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-5,-2,42,856,514,584,221,795,382,8069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^3=1,b^2=d^10=a^2,e^2=d^5,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^9>;
// generators/relations

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