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## G = C20×SL2(𝔽3)  order 480 = 25·3·5

### Direct product of C20 and SL2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C20×SL2(𝔽3)
 Chief series C1 — C2 — Q8 — C2×Q8 — Q8×C10 — C10×SL2(𝔽3) — C20×SL2(𝔽3)
 Lower central Q8 — C20×SL2(𝔽3)
 Upper central C1 — C2×C20

Generators and relations for C20×SL2(𝔽3)
G = < a,b,c,d | a20=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

Subgroups: 146 in 60 conjugacy classes, 28 normal (24 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, Q8, C10, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, C20, C20, C2×C10, SL2(𝔽3), C2×C12, C30, C4×Q8, C2×C20, C2×C20, C5×Q8, C5×Q8, C2×SL2(𝔽3), C60, C2×C30, C4×C20, C5×C4⋊C4, Q8×C10, C4×SL2(𝔽3), C5×SL2(𝔽3), C2×C60, Q8×C20, C10×SL2(𝔽3), C20×SL2(𝔽3)
Quotients: C1, C2, C3, C4, C5, C6, C10, C12, A4, C15, C20, SL2(𝔽3), C2×A4, C30, C4×A4, C2×SL2(𝔽3), C4.A4, C60, C5×A4, C4×SL2(𝔽3), C5×SL2(𝔽3), C10×A4, A4×C20, C10×SL2(𝔽3), C5×C4.A4, C20×SL2(𝔽3)

Smallest permutation representation of C20×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 42 102 90)(2 43 103 91)(3 44 104 92)(4 45 105 93)(5 46 106 94)(6 47 107 95)(7 48 108 96)(8 49 109 97)(9 50 110 98)(10 51 111 99)(11 52 112 100)(12 53 113 81)(13 54 114 82)(14 55 115 83)(15 56 116 84)(16 57 117 85)(17 58 118 86)(18 59 119 87)(19 60 120 88)(20 41 101 89)(21 146 71 136)(22 147 72 137)(23 148 73 138)(24 149 74 139)(25 150 75 140)(26 151 76 121)(27 152 77 122)(28 153 78 123)(29 154 79 124)(30 155 80 125)(31 156 61 126)(32 157 62 127)(33 158 63 128)(34 159 64 129)(35 160 65 130)(36 141 66 131)(37 142 67 132)(38 143 68 133)(39 144 69 134)(40 145 70 135)
(1 64 102 34)(2 65 103 35)(3 66 104 36)(4 67 105 37)(5 68 106 38)(6 69 107 39)(7 70 108 40)(8 71 109 21)(9 72 110 22)(10 73 111 23)(11 74 112 24)(12 75 113 25)(13 76 114 26)(14 77 115 27)(15 78 116 28)(16 79 117 29)(17 80 118 30)(18 61 119 31)(19 62 120 32)(20 63 101 33)(41 158 89 128)(42 159 90 129)(43 160 91 130)(44 141 92 131)(45 142 93 132)(46 143 94 133)(47 144 95 134)(48 145 96 135)(49 146 97 136)(50 147 98 137)(51 148 99 138)(52 149 100 139)(53 150 81 140)(54 151 82 121)(55 152 83 122)(56 153 84 123)(57 154 85 124)(58 155 86 125)(59 156 87 126)(60 157 88 127)
(21 97 136)(22 98 137)(23 99 138)(24 100 139)(25 81 140)(26 82 121)(27 83 122)(28 84 123)(29 85 124)(30 86 125)(31 87 126)(32 88 127)(33 89 128)(34 90 129)(35 91 130)(36 92 131)(37 93 132)(38 94 133)(39 95 134)(40 96 135)(41 158 63)(42 159 64)(43 160 65)(44 141 66)(45 142 67)(46 143 68)(47 144 69)(48 145 70)(49 146 71)(50 147 72)(51 148 73)(52 149 74)(53 150 75)(54 151 76)(55 152 77)(56 153 78)(57 154 79)(58 155 80)(59 156 61)(60 157 62)

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 6A ··· 6F 10A ··· 10L 12A ··· 12H 15A ··· 15H 20A ··· 20P 20Q ··· 20AF 30A ··· 30X 60A ··· 60AF order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 5 5 5 5 6 ··· 6 10 ··· 10 12 ··· 12 15 ··· 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 4 4 1 1 1 1 6 6 6 6 1 1 1 1 4 ··· 4 1 ··· 1 4 ··· 4 4 ··· 4 1 ··· 1 6 ··· 6 4 ··· 4 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 type + + - + + image C1 C2 C3 C4 C5 C6 C10 C12 C15 C20 C30 C60 SL2(𝔽3) SL2(𝔽3) C4.A4 C5×SL2(𝔽3) C5×C4.A4 A4 C2×A4 C4×A4 C5×A4 C10×A4 A4×C20 kernel C20×SL2(𝔽3) C10×SL2(𝔽3) Q8×C20 C5×SL2(𝔽3) C4×SL2(𝔽3) Q8×C10 C2×SL2(𝔽3) C5×Q8 C4×Q8 SL2(𝔽3) C2×Q8 Q8 C20 C20 C10 C4 C2 C2×C20 C2×C10 C10 C2×C4 C22 C2 # reps 1 1 2 2 4 2 4 4 8 8 8 16 2 4 6 24 24 1 1 2 4 4 8

Matrix representation of C20×SL2(𝔽3) in GL4(𝔽61) generated by

 50 0 0 0 0 58 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 60 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 47 48 0 0 48 14
,
 13 0 0 0 0 13 0 0 0 0 1 0 0 0 14 13
G:=sub<GL(4,GF(61))| [50,0,0,0,0,58,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,0],[1,0,0,0,0,1,0,0,0,0,47,48,0,0,48,14],[13,0,0,0,0,13,0,0,0,0,1,14,0,0,0,13] >;

C20×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_{20}\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C20xSL(2,3)");
// GroupNames label

G:=SmallGroup(480,655);
// by ID

G=gap.SmallGroup(480,655);
# by ID

G:=PCGroup([7,-2,-3,-5,-2,-2,2,-2,210,2111,172,3792,285,124]);
// Polycyclic

G:=Group<a,b,c,d|a^20=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

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