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G = C20.S4order 480 = 25·3·5

4th non-split extension by C20 of S4 acting via S4/A4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×A4 — C20.S4
 Chief series C1 — C22 — C2×C10 — C5×A4 — C10×A4 — A4×C20 — C20.S4
 Lower central C5×A4 — C20.S4
 Upper central C1 — C4

Generators and relations for C20.S4
G = < a,b,c,d,e | a20=b2=c2=d3=1, e2=a5, ab=ba, ac=ca, ad=da, eae-1=a9, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >

Smallest permutation representation of C20.S4
On 120 points
Generators in S120
```(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)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)
(1 42 78)(2 43 79)(3 44 80)(4 45 61)(5 46 62)(6 47 63)(7 48 64)(8 49 65)(9 50 66)(10 51 67)(11 52 68)(12 53 69)(13 54 70)(14 55 71)(15 56 72)(16 57 73)(17 58 74)(18 59 75)(19 60 76)(20 41 77)(21 119 82)(22 120 83)(23 101 84)(24 102 85)(25 103 86)(26 104 87)(27 105 88)(28 106 89)(29 107 90)(30 108 91)(31 109 92)(32 110 93)(33 111 94)(34 112 95)(35 113 96)(36 114 97)(37 115 98)(38 116 99)(39 117 100)(40 118 81)
(1 113 6 118 11 103 16 108)(2 102 7 107 12 112 17 117)(3 111 8 116 13 101 18 106)(4 120 9 105 14 110 19 115)(5 109 10 114 15 119 20 104)(21 41 26 46 31 51 36 56)(22 50 27 55 32 60 37 45)(23 59 28 44 33 49 38 54)(24 48 29 53 34 58 39 43)(25 57 30 42 35 47 40 52)(61 83 66 88 71 93 76 98)(62 92 67 97 72 82 77 87)(63 81 68 86 73 91 78 96)(64 90 69 95 74 100 79 85)(65 99 70 84 75 89 80 94)```

`G:=sub<Sym(120)| (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), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (1,42,78)(2,43,79)(3,44,80)(4,45,61)(5,46,62)(6,47,63)(7,48,64)(8,49,65)(9,50,66)(10,51,67)(11,52,68)(12,53,69)(13,54,70)(14,55,71)(15,56,72)(16,57,73)(17,58,74)(18,59,75)(19,60,76)(20,41,77)(21,119,82)(22,120,83)(23,101,84)(24,102,85)(25,103,86)(26,104,87)(27,105,88)(28,106,89)(29,107,90)(30,108,91)(31,109,92)(32,110,93)(33,111,94)(34,112,95)(35,113,96)(36,114,97)(37,115,98)(38,116,99)(39,117,100)(40,118,81), (1,113,6,118,11,103,16,108)(2,102,7,107,12,112,17,117)(3,111,8,116,13,101,18,106)(4,120,9,105,14,110,19,115)(5,109,10,114,15,119,20,104)(21,41,26,46,31,51,36,56)(22,50,27,55,32,60,37,45)(23,59,28,44,33,49,38,54)(24,48,29,53,34,58,39,43)(25,57,30,42,35,47,40,52)(61,83,66,88,71,93,76,98)(62,92,67,97,72,82,77,87)(63,81,68,86,73,91,78,96)(64,90,69,95,74,100,79,85)(65,99,70,84,75,89,80,94)>;`

`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), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (1,42,78)(2,43,79)(3,44,80)(4,45,61)(5,46,62)(6,47,63)(7,48,64)(8,49,65)(9,50,66)(10,51,67)(11,52,68)(12,53,69)(13,54,70)(14,55,71)(15,56,72)(16,57,73)(17,58,74)(18,59,75)(19,60,76)(20,41,77)(21,119,82)(22,120,83)(23,101,84)(24,102,85)(25,103,86)(26,104,87)(27,105,88)(28,106,89)(29,107,90)(30,108,91)(31,109,92)(32,110,93)(33,111,94)(34,112,95)(35,113,96)(36,114,97)(37,115,98)(38,116,99)(39,117,100)(40,118,81), (1,113,6,118,11,103,16,108)(2,102,7,107,12,112,17,117)(3,111,8,116,13,101,18,106)(4,120,9,105,14,110,19,115)(5,109,10,114,15,119,20,104)(21,41,26,46,31,51,36,56)(22,50,27,55,32,60,37,45)(23,59,28,44,33,49,38,54)(24,48,29,53,34,58,39,43)(25,57,30,42,35,47,40,52)(61,83,66,88,71,93,76,98)(62,92,67,97,72,82,77,87)(63,81,68,86,73,91,78,96)(64,90,69,95,74,100,79,85)(65,99,70,84,75,89,80,94) );`

`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)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120)], [(1,42,78),(2,43,79),(3,44,80),(4,45,61),(5,46,62),(6,47,63),(7,48,64),(8,49,65),(9,50,66),(10,51,67),(11,52,68),(12,53,69),(13,54,70),(14,55,71),(15,56,72),(16,57,73),(17,58,74),(18,59,75),(19,60,76),(20,41,77),(21,119,82),(22,120,83),(23,101,84),(24,102,85),(25,103,86),(26,104,87),(27,105,88),(28,106,89),(29,107,90),(30,108,91),(31,109,92),(32,110,93),(33,111,94),(34,112,95),(35,113,96),(36,114,97),(37,115,98),(38,116,99),(39,117,100),(40,118,81)], [(1,113,6,118,11,103,16,108),(2,102,7,107,12,112,17,117),(3,111,8,116,13,101,18,106),(4,120,9,105,14,110,19,115),(5,109,10,114,15,119,20,104),(21,41,26,46,31,51,36,56),(22,50,27,55,32,60,37,45),(23,59,28,44,33,49,38,54),(24,48,29,53,34,58,39,43),(25,57,30,42,35,47,40,52),(61,83,66,88,71,93,76,98),(62,92,67,97,72,82,77,87),(63,81,68,86,73,91,78,96),(64,90,69,95,74,100,79,85),(65,99,70,84,75,89,80,94)]])`

52 conjugacy classes

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

52 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 6 6 6 type + + + + - - + - + + - image C1 C2 C4 C8 S3 D5 Dic3 Dic5 C3⋊C8 D15 C5⋊2C8 Dic15 C15⋊3C8 S4 A4⋊C4 A4⋊C8 C5⋊S4 A4⋊Dic5 C20.S4 kernel C20.S4 A4×C20 C10×A4 C5×A4 C22×C20 C4×A4 C22×C10 C2×A4 C2×C10 C22×C4 A4 C23 C22 C20 C10 C5 C4 C2 C1 # reps 1 1 2 4 1 2 1 2 2 4 4 4 8 2 2 4 2 2 4

Matrix representation of C20.S4 in GL5(𝔽241)

 131 64 0 0 0 67 64 0 0 0 0 0 64 0 0 0 0 0 64 0 0 0 0 0 64
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 231 240 0 0 0 9 0 240
,
 1 0 0 0 0 0 1 0 0 0 0 0 240 0 0 0 0 10 1 0 0 0 0 0 240
,
 93 211 0 0 0 187 147 0 0 0 0 0 9 0 239 0 0 75 0 10 0 0 166 1 232
,
 112 76 0 0 0 240 129 0 0 0 0 0 72 0 225 0 0 162 8 80 0 0 79 0 169

`G:=sub<GL(5,GF(241))| [131,67,0,0,0,64,64,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,1,231,9,0,0,0,240,0,0,0,0,0,240],[1,0,0,0,0,0,1,0,0,0,0,0,240,10,0,0,0,0,1,0,0,0,0,0,240],[93,187,0,0,0,211,147,0,0,0,0,0,9,75,166,0,0,0,0,1,0,0,239,10,232],[112,240,0,0,0,76,129,0,0,0,0,0,72,162,79,0,0,0,8,0,0,0,225,80,169] >;`

C20.S4 in GAP, Magma, Sage, TeX

`C_{20}.S_4`
`% in TeX`

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

`G:=SmallGroup(480,259);`
`// by ID`

`G=gap.SmallGroup(480,259);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,14,36,451,3364,10085,1286,5886,2232]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^20=b^2=c^2=d^3=1,e^2=a^5,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^9,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=d^-1>;`
`// generators/relations`

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