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## G = C10×CSU2(𝔽3)  order 480 = 25·3·5

### Direct product of C10 and CSU2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C10×CSU2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C5×SL2(𝔽3) — C5×CSU2(𝔽3) — C10×CSU2(𝔽3)
 Lower central SL2(𝔽3) — C10×CSU2(𝔽3)
 Upper central C1 — C2×C10

Generators and relations for C10×CSU2(𝔽3)
G = < a,b,c,d,e | a10=b4=d3=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=bc, ebe-1=b2c, dcd-1=b, ede-1=d-1 >

Subgroups: 226 in 78 conjugacy classes, 24 normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C8, C2×C4, Q8, Q8, C10, C10, Dic3, C2×C6, C15, C2×C8, Q16, C2×Q8, C2×Q8, C20, C2×C10, SL2(𝔽3), C2×Dic3, C30, C2×Q16, C40, C2×C20, C5×Q8, C5×Q8, CSU2(𝔽3), C2×SL2(𝔽3), C5×Dic3, C2×C30, C2×C40, C5×Q16, Q8×C10, Q8×C10, C2×CSU2(𝔽3), C5×SL2(𝔽3), C10×Dic3, C10×Q16, C5×CSU2(𝔽3), C10×SL2(𝔽3), C10×CSU2(𝔽3)
Quotients: C1, C2, C22, C5, S3, C10, D6, C2×C10, S4, C5×S3, CSU2(𝔽3), C2×S4, S3×C10, C2×CSU2(𝔽3), C5×S4, C5×CSU2(𝔽3), C10×S4, C10×CSU2(𝔽3)

Smallest permutation representation of C10×CSU2(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)
(1 111 133 87)(2 112 134 88)(3 113 135 89)(4 114 136 90)(5 115 137 81)(6 116 138 82)(7 117 139 83)(8 118 140 84)(9 119 131 85)(10 120 132 86)(11 62 23 39)(12 63 24 40)(13 64 25 31)(14 65 26 32)(15 66 27 33)(16 67 28 34)(17 68 29 35)(18 69 30 36)(19 70 21 37)(20 61 22 38)(41 59 160 71)(42 60 151 72)(43 51 152 73)(44 52 153 74)(45 53 154 75)(46 54 155 76)(47 55 156 77)(48 56 157 78)(49 57 158 79)(50 58 159 80)(91 145 128 101)(92 146 129 102)(93 147 130 103)(94 148 121 104)(95 149 122 105)(96 150 123 106)(97 141 124 107)(98 142 125 108)(99 143 126 109)(100 144 127 110)
(1 109 133 143)(2 110 134 144)(3 101 135 145)(4 102 136 146)(5 103 137 147)(6 104 138 148)(7 105 139 149)(8 106 140 150)(9 107 131 141)(10 108 132 142)(11 71 23 59)(12 72 24 60)(13 73 25 51)(14 74 26 52)(15 75 27 53)(16 76 28 54)(17 77 29 55)(18 78 30 56)(19 79 21 57)(20 80 22 58)(31 43 64 152)(32 44 65 153)(33 45 66 154)(34 46 67 155)(35 47 68 156)(36 48 69 157)(37 49 70 158)(38 50 61 159)(39 41 62 160)(40 42 63 151)(81 93 115 130)(82 94 116 121)(83 95 117 122)(84 96 118 123)(85 97 119 124)(86 98 120 125)(87 99 111 126)(88 100 112 127)(89 91 113 128)(90 92 114 129)
(11 41 71)(12 42 72)(13 43 73)(14 44 74)(15 45 75)(16 46 76)(17 47 77)(18 48 78)(19 49 79)(20 50 80)(21 158 57)(22 159 58)(23 160 59)(24 151 60)(25 152 51)(26 153 52)(27 154 53)(28 155 54)(29 156 55)(30 157 56)(81 147 93)(82 148 94)(83 149 95)(84 150 96)(85 141 97)(86 142 98)(87 143 99)(88 144 100)(89 145 91)(90 146 92)(101 128 113)(102 129 114)(103 130 115)(104 121 116)(105 122 117)(106 123 118)(107 124 119)(108 125 120)(109 126 111)(110 127 112)
(1 40 133 63)(2 31 134 64)(3 32 135 65)(4 33 136 66)(5 34 137 67)(6 35 138 68)(7 36 139 69)(8 37 140 70)(9 38 131 61)(10 39 132 62)(11 108 23 142)(12 109 24 143)(13 110 25 144)(14 101 26 145)(15 102 27 146)(16 103 28 147)(17 104 29 148)(18 105 30 149)(19 106 21 150)(20 107 22 141)(41 120 160 86)(42 111 151 87)(43 112 152 88)(44 113 153 89)(45 114 154 90)(46 115 155 81)(47 116 156 82)(48 117 157 83)(49 118 158 84)(50 119 159 85)(51 100 73 127)(52 91 74 128)(53 92 75 129)(54 93 76 130)(55 94 77 121)(56 95 78 122)(57 96 79 123)(58 97 80 124)(59 98 71 125)(60 99 72 126)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 5C 5D 6A 6B 6C 8A 8B 8C 8D 10A ··· 10L 15A 15B 15C 15D 20A ··· 20H 20I ··· 20P 30A ··· 30L 40A ··· 40P order 1 2 2 2 3 4 4 4 4 5 5 5 5 6 6 6 8 8 8 8 10 ··· 10 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 40 ··· 40 size 1 1 1 1 8 6 6 12 12 1 1 1 1 8 8 8 6 6 6 6 1 ··· 1 8 8 8 8 6 ··· 6 12 ··· 12 8 ··· 8 6 ··· 6

80 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4 type + + + + + - + + - image C1 C2 C2 C5 C10 C10 S3 D6 C5×S3 CSU2(𝔽3) S3×C10 C5×CSU2(𝔽3) S4 C2×S4 C5×S4 C10×S4 CSU2(𝔽3) C5×CSU2(𝔽3) kernel C10×CSU2(𝔽3) C5×CSU2(𝔽3) C10×SL2(𝔽3) C2×CSU2(𝔽3) CSU2(𝔽3) C2×SL2(𝔽3) Q8×C10 C5×Q8 C2×Q8 C10 Q8 C2 C2×C10 C10 C22 C2 C10 C2 # reps 1 2 1 4 8 4 1 1 4 4 4 16 2 2 8 8 2 8

Matrix representation of C10×CSU2(𝔽3) in GL4(𝔽241) generated by

 143 0 0 0 0 143 0 0 0 0 150 0 0 0 0 150
,
 1 0 0 0 0 1 0 0 0 0 13 12 0 0 26 228
,
 1 0 0 0 0 1 0 0 0 0 13 215 0 0 229 228
,
 0 240 0 0 1 240 0 0 0 0 0 240 0 0 1 240
,
 0 1 0 0 1 0 0 0 0 0 0 177 0 0 177 0
G:=sub<GL(4,GF(241))| [143,0,0,0,0,143,0,0,0,0,150,0,0,0,0,150],[1,0,0,0,0,1,0,0,0,0,13,26,0,0,12,228],[1,0,0,0,0,1,0,0,0,0,13,229,0,0,215,228],[0,1,0,0,240,240,0,0,0,0,0,1,0,0,240,240],[0,1,0,0,1,0,0,0,0,0,0,177,0,0,177,0] >;

C10×CSU2(𝔽3) in GAP, Magma, Sage, TeX

C_{10}\times {\rm CSU}_2({\mathbb F}_3)
% in TeX

G:=Group("C10xCSU(2,3)");
// GroupNames label

G:=SmallGroup(480,1016);
// by ID

G=gap.SmallGroup(480,1016);
# by ID

G:=PCGroup([7,-2,-2,-5,-3,-2,2,-2,1680,1123,4204,655,172,2525,404,285,124]);
// Polycyclic

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

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