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## G = C20.2S4order 480 = 25·3·5

### 2nd non-split extension by C20 of S4 acting via S4/A4=C2

Aliases: C20.2S4, Q8.3D30, SL2(𝔽3).9D10, C4.2(C5⋊S4), C4.A4.1D5, C10.22(C2×S4), C52(C4.S4), C4○D4.2D15, Q8.D152C2, (C5×Q8).10D6, (C5×SL2(𝔽3)).9C22, C2.8(C2×C5⋊S4), (C5×C4○D4).2S3, (C5×C4.A4).1C2, SmallGroup(480,1030)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — C20.2S4
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — Q8.D15 — C20.2S4
 Lower central C5×SL2(𝔽3) — C20.2S4
 Upper central C1 — C2 — C4

Generators and relations for C20.2S4
G = < a,b,c,d,e | a20=d3=1, b2=c2=e2=a10, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a10b, dbd-1=a10bc, ebe-1=bc, dcd-1=b, ece-1=a10c, ede-1=d-1 >

Subgroups: 506 in 72 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, D4, Q8, Q8, C10, C10, Dic3, C12, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, SL2(𝔽3), Dic6, C30, C8.C22, C52C8, Dic10, C2×Dic5, C2×C20, C5×D4, C5×Q8, CSU2(𝔽3), C4.A4, Dic15, C60, C4.Dic5, D4.D5, C5⋊Q16, C2×Dic10, C5×C4○D4, C4.S4, C5×SL2(𝔽3), Dic30, D4.9D10, Q8.D15, C5×C4.A4, C20.2S4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, D15, C2×S4, D30, C4.S4, C5⋊S4, C2×C5⋊S4, C20.2S4

Smallest permutation representation of C20.2S4
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)
(1 102 11 112)(2 103 12 113)(3 104 13 114)(4 105 14 115)(5 106 15 116)(6 107 16 117)(7 108 17 118)(8 109 18 119)(9 110 19 120)(10 111 20 101)(21 130 31 140)(22 131 32 121)(23 132 33 122)(24 133 34 123)(25 134 35 124)(26 135 36 125)(27 136 37 126)(28 137 38 127)(29 138 39 128)(30 139 40 129)(41 74 51 64)(42 75 52 65)(43 76 53 66)(44 77 54 67)(45 78 55 68)(46 79 56 69)(47 80 57 70)(48 61 58 71)(49 62 59 72)(50 63 60 73)(81 150 91 160)(82 151 92 141)(83 152 93 142)(84 153 94 143)(85 154 95 144)(86 155 96 145)(87 156 97 146)(88 157 98 147)(89 158 99 148)(90 159 100 149)
(1 90 11 100)(2 91 12 81)(3 92 13 82)(4 93 14 83)(5 94 15 84)(6 95 16 85)(7 96 17 86)(8 97 18 87)(9 98 19 88)(10 99 20 89)(21 79 31 69)(22 80 32 70)(23 61 33 71)(24 62 34 72)(25 63 35 73)(26 64 36 74)(27 65 37 75)(28 66 38 76)(29 67 39 77)(30 68 40 78)(41 135 51 125)(42 136 52 126)(43 137 53 127)(44 138 54 128)(45 139 55 129)(46 140 56 130)(47 121 57 131)(48 122 58 132)(49 123 59 133)(50 124 60 134)(101 148 111 158)(102 149 112 159)(103 150 113 160)(104 151 114 141)(105 152 115 142)(106 153 116 143)(107 154 117 144)(108 155 118 145)(109 156 119 146)(110 157 120 147)
(21 130 46)(22 131 47)(23 132 48)(24 133 49)(25 134 50)(26 135 51)(27 136 52)(28 137 53)(29 138 54)(30 139 55)(31 140 56)(32 121 57)(33 122 58)(34 123 59)(35 124 60)(36 125 41)(37 126 42)(38 127 43)(39 128 44)(40 129 45)(81 150 113)(82 151 114)(83 152 115)(84 153 116)(85 154 117)(86 155 118)(87 156 119)(88 157 120)(89 158 101)(90 159 102)(91 160 103)(92 141 104)(93 142 105)(94 143 106)(95 144 107)(96 145 108)(97 146 109)(98 147 110)(99 148 111)(100 149 112)
(1 78 11 68)(2 77 12 67)(3 76 13 66)(4 75 14 65)(5 74 15 64)(6 73 16 63)(7 72 17 62)(8 71 18 61)(9 70 19 80)(10 69 20 79)(21 99 31 89)(22 98 32 88)(23 97 33 87)(24 96 34 86)(25 95 35 85)(26 94 36 84)(27 93 37 83)(28 92 38 82)(29 91 39 81)(30 90 40 100)(41 153 51 143)(42 152 52 142)(43 151 53 141)(44 150 54 160)(45 149 55 159)(46 148 56 158)(47 147 57 157)(48 146 58 156)(49 145 59 155)(50 144 60 154)(101 130 111 140)(102 129 112 139)(103 128 113 138)(104 127 114 137)(105 126 115 136)(106 125 116 135)(107 124 117 134)(108 123 118 133)(109 122 119 132)(110 121 120 131)

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

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

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

41 conjugacy classes

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

41 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 3 3 4 4 6 6 type + + + + + + + + + + + - - + + image C1 C2 C2 S3 D5 D6 D10 D15 D30 S4 C2×S4 C4.S4 C20.2S4 C5⋊S4 C2×C5⋊S4 kernel C20.2S4 Q8.D15 C5×C4.A4 C5×C4○D4 C4.A4 C5×Q8 SL2(𝔽3) C4○D4 Q8 C20 C10 C5 C1 C4 C2 # reps 1 2 1 1 2 1 2 4 4 2 2 3 12 2 2

Matrix representation of C20.2S4 in GL6(𝔽241)

 22 8 0 0 0 0 233 30 0 0 0 0 0 0 0 142 99 142 0 0 99 0 142 142 0 0 142 99 0 142 0 0 99 99 99 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 240 0 0 0 0 0 0 240 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 240 0 0 0 0 1 0 0 0 0 240 0 0 0 0 1 0 0 0
,
 0 240 0 0 0 0 1 240 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 240 0 0 0 0 0 0 240 0
,
 152 170 0 0 0 0 81 89 0 0 0 0 0 0 239 221 20 221 0 0 221 20 239 221 0 0 20 239 221 221 0 0 221 221 221 2

G:=sub<GL(6,GF(241))| [22,233,0,0,0,0,8,30,0,0,0,0,0,0,0,99,142,99,0,0,142,0,99,99,0,0,99,142,0,99,0,0,142,142,142,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,240,0,0,0,0,1,0,0,0,0,240,0,0,0],[0,1,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,1,0,0],[152,81,0,0,0,0,170,89,0,0,0,0,0,0,239,221,20,221,0,0,221,20,239,221,0,0,20,239,221,221,0,0,221,221,221,2] >;

C20.2S4 in GAP, Magma, Sage, TeX

C_{20}._2S_4
% in TeX

G:=Group("C20.2S4");
// GroupNames label

G:=SmallGroup(480,1030);
// by ID

G=gap.SmallGroup(480,1030);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1680,3389,1688,170,1347,4204,3168,172,2525,1909,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^20=d^3=1,b^2=c^2=e^2=a^10,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c^-1=a^10*b,d*b*d^-1=a^10*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^10*c,e*d*e^-1=d^-1>;
// generators/relations

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