non-abelian, soluble, monomial
Aliases: C20.4S4, C23.Dic15, A4⋊(C5⋊2C8), C5⋊2(A4⋊C8), (C5×A4)⋊3C8, C4.4(C5⋊S4), (C4×A4).2D5, (C2×A4).Dic5, (C10×A4).3C4, (A4×C20).2C2, C22⋊(C15⋊3C8), C10.4(A4⋊C4), (C22×C20).3S3, (C22×C4).1D15, C2.1(A4⋊Dic5), (C22×C10).3Dic3, (C2×C10)⋊3(C3⋊C8), SmallGroup(480,259)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×A4 — C20.S4 |
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 >
3C2
3C2
4C3
3C4
3C22
3C22
4C6
3C10
3C10
4C15
30C8
30C8
4C12
3C20
4C30
15C2×C8
15C2×C8
20C3⋊C8
4C60
15C22⋊C8
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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