non-abelian, soluble, monomial
Aliases: C20.1S4, C22⋊Dic30, A4⋊2Dic10, C23.1D30, C4.1(C5⋊S4), C5⋊2(A4⋊Q8), (C5×A4)⋊2Q8, (C4×A4).1D5, C10.16(C2×S4), (A4×C20).1C2, (C2×A4).8D10, (C2×C10)⋊3Dic6, A4⋊Dic5.1C2, (C22×C20).2S3, (C22×C4).2D15, (C10×A4).8C22, (C22×C10).13D6, C2.3(C2×C5⋊S4), SmallGroup(480,1024)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.1S4
G = < a,b,c,d,e | a20=b2=c2=d3=1, e2=a10, ab=ba, ac=ca, ad=da, eae-1=a-1, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >
Subgroups: 588 in 84 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C2×C4, Q8, C23, C10, C10, Dic3, C12, A4, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, Dic6, C2×A4, C30, C22⋊Q8, Dic10, C2×Dic5, C2×C20, C22×C10, A4⋊C4, C4×A4, Dic15, C60, C5×A4, C10.D4, C4⋊Dic5, C23.D5, C2×Dic10, C22×C20, A4⋊Q8, Dic30, C10×A4, C20.48D4, A4⋊Dic5, A4×C20, C20.1S4
Quotients: C1, C2, C22, S3, Q8, D5, D6, D10, Dic6, S4, D15, Dic10, C2×S4, D30, A4⋊Q8, Dic30, C5⋊S4, C2×C5⋊S4, C20.1S4
►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)
(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)(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 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(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 60 110)(2 41 111)(3 42 112)(4 43 113)(5 44 114)(6 45 115)(7 46 116)(8 47 117)(9 48 118)(10 49 119)(11 50 120)(12 51 101)(13 52 102)(14 53 103)(15 54 104)(16 55 105)(17 56 106)(18 57 107)(19 58 108)(20 59 109)(21 64 86)(22 65 87)(23 66 88)(24 67 89)(25 68 90)(26 69 91)(27 70 92)(28 71 93)(29 72 94)(30 73 95)(31 74 96)(32 75 97)(33 76 98)(34 77 99)(35 78 100)(36 79 81)(37 80 82)(38 61 83)(39 62 84)(40 63 85)
(1 99 11 89)(2 98 12 88)(3 97 13 87)(4 96 14 86)(5 95 15 85)(6 94 16 84)(7 93 17 83)(8 92 18 82)(9 91 19 81)(10 90 20 100)(21 113 31 103)(22 112 32 102)(23 111 33 101)(24 110 34 120)(25 109 35 119)(26 108 36 118)(27 107 37 117)(28 106 38 116)(29 105 39 115)(30 104 40 114)(41 76 51 66)(42 75 52 65)(43 74 53 64)(44 73 54 63)(45 72 55 62)(46 71 56 61)(47 70 57 80)(48 69 58 79)(49 68 59 78)(50 67 60 77)
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), (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)(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), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(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), (1,60,110)(2,41,111)(3,42,112)(4,43,113)(5,44,114)(6,45,115)(7,46,116)(8,47,117)(9,48,118)(10,49,119)(11,50,120)(12,51,101)(13,52,102)(14,53,103)(15,54,104)(16,55,105)(17,56,106)(18,57,107)(19,58,108)(20,59,109)(21,64,86)(22,65,87)(23,66,88)(24,67,89)(25,68,90)(26,69,91)(27,70,92)(28,71,93)(29,72,94)(30,73,95)(31,74,96)(32,75,97)(33,76,98)(34,77,99)(35,78,100)(36,79,81)(37,80,82)(38,61,83)(39,62,84)(40,63,85), (1,99,11,89)(2,98,12,88)(3,97,13,87)(4,96,14,86)(5,95,15,85)(6,94,16,84)(7,93,17,83)(8,92,18,82)(9,91,19,81)(10,90,20,100)(21,113,31,103)(22,112,32,102)(23,111,33,101)(24,110,34,120)(25,109,35,119)(26,108,36,118)(27,107,37,117)(28,106,38,116)(29,105,39,115)(30,104,40,114)(41,76,51,66)(42,75,52,65)(43,74,53,64)(44,73,54,63)(45,72,55,62)(46,71,56,61)(47,70,57,80)(48,69,58,79)(49,68,59,78)(50,67,60,77)>;
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), (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)(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), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(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), (1,60,110)(2,41,111)(3,42,112)(4,43,113)(5,44,114)(6,45,115)(7,46,116)(8,47,117)(9,48,118)(10,49,119)(11,50,120)(12,51,101)(13,52,102)(14,53,103)(15,54,104)(16,55,105)(17,56,106)(18,57,107)(19,58,108)(20,59,109)(21,64,86)(22,65,87)(23,66,88)(24,67,89)(25,68,90)(26,69,91)(27,70,92)(28,71,93)(29,72,94)(30,73,95)(31,74,96)(32,75,97)(33,76,98)(34,77,99)(35,78,100)(36,79,81)(37,80,82)(38,61,83)(39,62,84)(40,63,85), (1,99,11,89)(2,98,12,88)(3,97,13,87)(4,96,14,86)(5,95,15,85)(6,94,16,84)(7,93,17,83)(8,92,18,82)(9,91,19,81)(10,90,20,100)(21,113,31,103)(22,112,32,102)(23,111,33,101)(24,110,34,120)(25,109,35,119)(26,108,36,118)(27,107,37,117)(28,106,38,116)(29,105,39,115)(30,104,40,114)(41,76,51,66)(42,75,52,65)(43,74,53,64)(44,73,54,63)(45,72,55,62)(46,71,56,61)(47,70,57,80)(48,69,58,79)(49,68,59,78)(50,67,60,77) );
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)], [(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),(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)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(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)], [(1,60,110),(2,41,111),(3,42,112),(4,43,113),(5,44,114),(6,45,115),(7,46,116),(8,47,117),(9,48,118),(10,49,119),(11,50,120),(12,51,101),(13,52,102),(14,53,103),(15,54,104),(16,55,105),(17,56,106),(18,57,107),(19,58,108),(20,59,109),(21,64,86),(22,65,87),(23,66,88),(24,67,89),(25,68,90),(26,69,91),(27,70,92),(28,71,93),(29,72,94),(30,73,95),(31,74,96),(32,75,97),(33,76,98),(34,77,99),(35,78,100),(36,79,81),(37,80,82),(38,61,83),(39,62,84),(40,63,85)], [(1,99,11,89),(2,98,12,88),(3,97,13,87),(4,96,14,86),(5,95,15,85),(6,94,16,84),(7,93,17,83),(8,92,18,82),(9,91,19,81),(10,90,20,100),(21,113,31,103),(22,112,32,102),(23,111,33,101),(24,110,34,120),(25,109,35,119),(26,108,36,118),(27,107,37,117),(28,106,38,116),(29,105,39,115),(30,104,40,114),(41,76,51,66),(42,75,52,65),(43,74,53,64),(44,73,54,63),(45,72,55,62),(46,71,56,61),(47,70,57,80),(48,69,58,79),(49,68,59,78),(50,67,60,77)]])
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6 | 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 | 4 | 4 | 5 | 5 | 6 | 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 | 2 | 6 | 60 | 60 | 60 | 60 | 2 | 2 | 8 | 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 |
46 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | + | + | - | + | - | + | - | + | + | - | + | + | - |
| image | C1 | C2 | C2 | S3 | Q8 | D5 | D6 | D10 | Dic6 | D15 | Dic10 | D30 | Dic30 | S4 | C2×S4 | A4⋊Q8 | C5⋊S4 | C2×C5⋊S4 | C20.1S4 |
| kernel | C20.1S4 | A4⋊Dic5 | A4×C20 | C22×C20 | C5×A4 | C4×A4 | C22×C10 | C2×A4 | C2×C10 | C22×C4 | A4 | C23 | C22 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 2 | 2 | 1 | 2 | 2 | 4 |
Matrix representation of C20.1S4 ►in GL5(𝔽61)
| 32 | 13 | 0 | 0 | 0 |
| 9 | 38 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 | 1 |
| 0 | 0 | 60 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 1 | 0 | 60 |
| 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 47 | 51 | 0 | 0 | 0 |
| 38 | 14 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(5,GF(61))| [32,9,0,0,0,13,38,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[47,38,0,0,0,51,14,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,60] >;
C20.1S4 in GAP, Magma, Sage, TeX
C_{20}._1S_4 % in TeX
G:=Group("C20.1S4"); // GroupNames label
G:=SmallGroup(480,1024);
// by ID
G=gap.SmallGroup(480,1024);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,28,85,36,451,3364,10085,1286,5886,2232]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=b^2=c^2=d^3=1,e^2=a^10,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,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