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## G = C2×C10×SL2(𝔽3)  order 480 = 25·3·5

### Direct product of C2×C10 and SL2(𝔽3)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C10×SL2(𝔽3)
G = < a,b,c,d,e | a2=b10=c4=e3=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

Subgroups: 294 in 122 conjugacy classes, 52 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C2×C4, Q8, Q8, C23, C10, C10, C2×C6, C15, C22×C4, C2×Q8, C2×Q8, C20, C2×C10, SL2(𝔽3), C22×C6, C30, C22×Q8, C2×C20, C5×Q8, C5×Q8, C22×C10, C2×SL2(𝔽3), C2×C30, C22×C20, Q8×C10, Q8×C10, C22×SL2(𝔽3), C5×SL2(𝔽3), C22×C30, Q8×C2×C10, C10×SL2(𝔽3), C2×C10×SL2(𝔽3)
Quotients: C1, C2, C3, C22, C5, C6, C10, A4, C2×C6, C15, C2×C10, SL2(𝔽3), C2×A4, C30, C2×SL2(𝔽3), C22×A4, C5×A4, C2×C30, C22×SL2(𝔽3), C5×SL2(𝔽3), C10×A4, C10×SL2(𝔽3), A4×C2×C10, C2×C10×SL2(𝔽3)

Smallest permutation representation of C2×C10×SL2(𝔽3)
On 160 points
Generators in S160
(1 152)(2 153)(3 154)(4 155)(5 156)(6 157)(7 158)(8 159)(9 160)(10 151)(11 143)(12 144)(13 145)(14 146)(15 147)(16 148)(17 149)(18 150)(19 141)(20 142)(21 137)(22 138)(23 139)(24 140)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 106)(32 107)(33 108)(34 109)(35 110)(36 101)(37 102)(38 103)(39 104)(40 105)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 88)(52 89)(53 90)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 125)(72 126)(73 127)(74 128)(75 129)(76 130)(77 121)(78 122)(79 123)(80 124)
(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 98 125 88)(2 99 126 89)(3 100 127 90)(4 91 128 81)(5 92 129 82)(6 93 130 83)(7 94 121 84)(8 95 122 85)(9 96 123 86)(10 97 124 87)(11 35 24 63)(12 36 25 64)(13 37 26 65)(14 38 27 66)(15 39 28 67)(16 40 29 68)(17 31 30 69)(18 32 21 70)(19 33 22 61)(20 34 23 62)(41 74 54 155)(42 75 55 156)(43 76 56 157)(44 77 57 158)(45 78 58 159)(46 79 59 160)(47 80 60 151)(48 71 51 152)(49 72 52 153)(50 73 53 154)(101 131 114 144)(102 132 115 145)(103 133 116 146)(104 134 117 147)(105 135 118 148)(106 136 119 149)(107 137 120 150)(108 138 111 141)(109 139 112 142)(110 140 113 143)
(1 120 125 107)(2 111 126 108)(3 112 127 109)(4 113 128 110)(5 114 129 101)(6 115 130 102)(7 116 121 103)(8 117 122 104)(9 118 123 105)(10 119 124 106)(11 41 24 54)(12 42 25 55)(13 43 26 56)(14 44 27 57)(15 45 28 58)(16 46 29 59)(17 47 30 60)(18 48 21 51)(19 49 22 52)(20 50 23 53)(31 151 69 80)(32 152 70 71)(33 153 61 72)(34 154 62 73)(35 155 63 74)(36 156 64 75)(37 157 65 76)(38 158 66 77)(39 159 67 78)(40 160 68 79)(81 143 91 140)(82 144 92 131)(83 145 93 132)(84 146 94 133)(85 147 95 134)(86 148 96 135)(87 149 97 136)(88 150 98 137)(89 141 99 138)(90 142 100 139)
(11 35 54)(12 36 55)(13 37 56)(14 38 57)(15 39 58)(16 40 59)(17 31 60)(18 32 51)(19 33 52)(20 34 53)(21 70 48)(22 61 49)(23 62 50)(24 63 41)(25 64 42)(26 65 43)(27 66 44)(28 67 45)(29 68 46)(30 69 47)(81 143 110)(82 144 101)(83 145 102)(84 146 103)(85 147 104)(86 148 105)(87 149 106)(88 150 107)(89 141 108)(90 142 109)(91 140 113)(92 131 114)(93 132 115)(94 133 116)(95 134 117)(96 135 118)(97 136 119)(98 137 120)(99 138 111)(100 139 112)

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

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

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

140 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A 4B 4C 4D 5A 5B 5C 5D 6A ··· 6N 10A ··· 10AB 15A ··· 15H 20A ··· 20P 30A ··· 30BD order 1 2 ··· 2 3 3 4 4 4 4 5 5 5 5 6 ··· 6 10 ··· 10 15 ··· 15 20 ··· 20 30 ··· 30 size 1 1 ··· 1 4 4 6 6 6 6 1 1 1 1 4 ··· 4 1 ··· 1 4 ··· 4 6 ··· 6 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 type + + - + + image C1 C2 C3 C5 C6 C10 C15 C30 SL2(𝔽3) SL2(𝔽3) C5×SL2(𝔽3) A4 C2×A4 C5×A4 C10×A4 kernel C2×C10×SL2(𝔽3) C10×SL2(𝔽3) Q8×C2×C10 C22×SL2(𝔽3) Q8×C10 C2×SL2(𝔽3) C22×Q8 C2×Q8 C2×C10 C2×C10 C22 C22×C10 C2×C10 C23 C22 # reps 1 3 2 4 6 12 8 24 4 8 48 1 3 4 12

Matrix representation of C2×C10×SL2(𝔽3) in GL4(𝔽61) generated by

 60 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 14 48 0 0 48 47
,
 1 0 0 0 0 1 0 0 0 0 0 60 0 0 1 0
,
 47 0 0 0 0 13 0 0 0 0 1 0 0 0 48 47
G:=sub<GL(4,GF(61))| [60,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,14,48,0,0,48,47],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,0],[47,0,0,0,0,13,0,0,0,0,1,48,0,0,0,47] >;

C2×C10×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_2\times C_{10}\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C2xC10xSL(2,3)");
// GroupNames label

G:=SmallGroup(480,1128);
// by ID

G=gap.SmallGroup(480,1128);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1068,172,1909,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=c^4=e^3=1,d^2=c^2,a*b=b*a,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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