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## G = Dic5×SL2(𝔽3)  order 480 = 25·3·5

### Direct product of Dic5 and SL2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×Q8 — Dic5×SL2(𝔽3)
 Chief series C1 — C2 — C10 — C5×Q8 — Q8×C10 — C10×SL2(𝔽3) — Dic5×SL2(𝔽3)
 Lower central C5×Q8 — Dic5×SL2(𝔽3)
 Upper central C1 — C22

Generators and relations for Dic5×SL2(𝔽3)
G = < a,b,c,d,e | a10=c4=e3=1, b2=a5, d2=c2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

Subgroups: 274 in 60 conjugacy classes, 23 normal (21 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, Q8, Q8, C10, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C2×C10, SL2(𝔽3), C2×C12, C30, C4×Q8, C2×Dic5, C2×Dic5, C2×C20, C5×Q8, C5×Q8, C2×SL2(𝔽3), C3×Dic5, C2×C30, C4×Dic5, C4⋊Dic5, Q8×C10, C4×SL2(𝔽3), C5×SL2(𝔽3), C6×Dic5, Q8×Dic5, C10×SL2(𝔽3), Dic5×SL2(𝔽3)
Quotients: C1, C2, C3, C4, C6, D5, C12, A4, Dic5, SL2(𝔽3), C2×A4, C3×D5, C4×A4, C2×SL2(𝔽3), C4.A4, C3×Dic5, C4×SL2(𝔽3), D5×A4, Dic5.A4, D5×SL2(𝔽3), A4×Dic5, Dic5×SL2(𝔽3)

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 6 6 type + + + - - + + + - + - image C1 C2 C3 C4 C6 C12 D5 Dic5 SL2(𝔽3) SL2(𝔽3) C3×D5 C4.A4 C3×Dic5 A4 C2×A4 C4×A4 Dic5.A4 Dic5.A4 D5×SL2(𝔽3) D5×SL2(𝔽3) D5×A4 A4×Dic5 kernel Dic5×SL2(𝔽3) C10×SL2(𝔽3) Q8×Dic5 C5×SL2(𝔽3) Q8×C10 C5×Q8 C2×SL2(𝔽3) SL2(𝔽3) Dic5 Dic5 C2×Q8 C10 Q8 C2×Dic5 C2×C10 C10 C2 C2 C2 C2 C22 C2 # reps 1 1 2 2 2 4 2 2 2 4 4 6 4 1 1 2 2 4 2 4 2 2

Matrix representation of Dic5×SL2(𝔽3) in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 44 60 0 0 1 0
,
 11 0 0 0 0 11 0 0 0 0 58 30 0 0 20 3
,
 0 1 0 0 60 0 0 0 0 0 1 0 0 0 0 1
,
 13 47 0 0 47 48 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 13 47 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,44,1,0,0,60,0],[11,0,0,0,0,11,0,0,0,0,58,20,0,0,30,3],[0,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[13,47,0,0,47,48,0,0,0,0,1,0,0,0,0,1],[1,13,0,0,0,47,0,0,0,0,1,0,0,0,0,1] >;

Dic5×SL2(𝔽3) in GAP, Magma, Sage, TeX

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

G:=Group("Dic5xSL(2,3)");
// GroupNames label

G:=SmallGroup(480,266);
// by ID

G=gap.SmallGroup(480,266);
# by ID

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

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

׿
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