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
Subgroups: 588 in 84 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×5], C22, C22 [×2], C5, C6, C2×C4 [×6], Q8 [×2], C23, C10, C10 [×2], Dic3 [×2], C12, A4, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×4], C20, C20, C2×C10, C2×C10 [×2], Dic6, C2×A4, C30, C22⋊Q8, Dic10 [×2], C2×Dic5 [×4], C2×C20 [×2], C22×C10, A4⋊C4 [×2], C4×A4, Dic15 [×2], C60, C5×A4, C10.D4 [×2], C4⋊Dic5, C23.D5 [×2], C2×Dic10, C22×C20, A4⋊Q8, Dic30, C10×A4, C20.48D4, A4⋊Dic5 [×2], A4×C20, C20.1S4
Quotients:
C1, C2 [×3], C22, S3, Q8, D5, D6, D10, Dic6, S4, D15, Dic10, C2×S4, D30, A4⋊Q8, Dic30, C5⋊S4, C2×C5⋊S4, C20.1S4
Generators and relations
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 >
►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)(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)(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)
(1 103 48)(2 104 49)(3 105 50)(4 106 51)(5 107 52)(6 108 53)(7 109 54)(8 110 55)(9 111 56)(10 112 57)(11 113 58)(12 114 59)(13 115 60)(14 116 41)(15 117 42)(16 118 43)(17 119 44)(18 120 45)(19 101 46)(20 102 47)(21 71 86)(22 72 87)(23 73 88)(24 74 89)(25 75 90)(26 76 91)(27 77 92)(28 78 93)(29 79 94)(30 80 95)(31 61 96)(32 62 97)(33 63 98)(34 64 99)(35 65 100)(36 66 81)(37 67 82)(38 68 83)(39 69 84)(40 70 85)
(1 33 11 23)(2 32 12 22)(3 31 13 21)(4 30 14 40)(5 29 15 39)(6 28 16 38)(7 27 17 37)(8 26 18 36)(9 25 19 35)(10 24 20 34)(41 70 51 80)(42 69 52 79)(43 68 53 78)(44 67 54 77)(45 66 55 76)(46 65 56 75)(47 64 57 74)(48 63 58 73)(49 62 59 72)(50 61 60 71)(81 110 91 120)(82 109 92 119)(83 108 93 118)(84 107 94 117)(85 106 95 116)(86 105 96 115)(87 104 97 114)(88 103 98 113)(89 102 99 112)(90 101 100 111)
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)(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)(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), (1,103,48)(2,104,49)(3,105,50)(4,106,51)(5,107,52)(6,108,53)(7,109,54)(8,110,55)(9,111,56)(10,112,57)(11,113,58)(12,114,59)(13,115,60)(14,116,41)(15,117,42)(16,118,43)(17,119,44)(18,120,45)(19,101,46)(20,102,47)(21,71,86)(22,72,87)(23,73,88)(24,74,89)(25,75,90)(26,76,91)(27,77,92)(28,78,93)(29,79,94)(30,80,95)(31,61,96)(32,62,97)(33,63,98)(34,64,99)(35,65,100)(36,66,81)(37,67,82)(38,68,83)(39,69,84)(40,70,85), (1,33,11,23)(2,32,12,22)(3,31,13,21)(4,30,14,40)(5,29,15,39)(6,28,16,38)(7,27,17,37)(8,26,18,36)(9,25,19,35)(10,24,20,34)(41,70,51,80)(42,69,52,79)(43,68,53,78)(44,67,54,77)(45,66,55,76)(46,65,56,75)(47,64,57,74)(48,63,58,73)(49,62,59,72)(50,61,60,71)(81,110,91,120)(82,109,92,119)(83,108,93,118)(84,107,94,117)(85,106,95,116)(86,105,96,115)(87,104,97,114)(88,103,98,113)(89,102,99,112)(90,101,100,111)>;
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)(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)(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), (1,103,48)(2,104,49)(3,105,50)(4,106,51)(5,107,52)(6,108,53)(7,109,54)(8,110,55)(9,111,56)(10,112,57)(11,113,58)(12,114,59)(13,115,60)(14,116,41)(15,117,42)(16,118,43)(17,119,44)(18,120,45)(19,101,46)(20,102,47)(21,71,86)(22,72,87)(23,73,88)(24,74,89)(25,75,90)(26,76,91)(27,77,92)(28,78,93)(29,79,94)(30,80,95)(31,61,96)(32,62,97)(33,63,98)(34,64,99)(35,65,100)(36,66,81)(37,67,82)(38,68,83)(39,69,84)(40,70,85), (1,33,11,23)(2,32,12,22)(3,31,13,21)(4,30,14,40)(5,29,15,39)(6,28,16,38)(7,27,17,37)(8,26,18,36)(9,25,19,35)(10,24,20,34)(41,70,51,80)(42,69,52,79)(43,68,53,78)(44,67,54,77)(45,66,55,76)(46,65,56,75)(47,64,57,74)(48,63,58,73)(49,62,59,72)(50,61,60,71)(81,110,91,120)(82,109,92,119)(83,108,93,118)(84,107,94,117)(85,106,95,116)(86,105,96,115)(87,104,97,114)(88,103,98,113)(89,102,99,112)(90,101,100,111) );
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),(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),(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)], [(1,103,48),(2,104,49),(3,105,50),(4,106,51),(5,107,52),(6,108,53),(7,109,54),(8,110,55),(9,111,56),(10,112,57),(11,113,58),(12,114,59),(13,115,60),(14,116,41),(15,117,42),(16,118,43),(17,119,44),(18,120,45),(19,101,46),(20,102,47),(21,71,86),(22,72,87),(23,73,88),(24,74,89),(25,75,90),(26,76,91),(27,77,92),(28,78,93),(29,79,94),(30,80,95),(31,61,96),(32,62,97),(33,63,98),(34,64,99),(35,65,100),(36,66,81),(37,67,82),(38,68,83),(39,69,84),(40,70,85)], [(1,33,11,23),(2,32,12,22),(3,31,13,21),(4,30,14,40),(5,29,15,39),(6,28,16,38),(7,27,17,37),(8,26,18,36),(9,25,19,35),(10,24,20,34),(41,70,51,80),(42,69,52,79),(43,68,53,78),(44,67,54,77),(45,66,55,76),(46,65,56,75),(47,64,57,74),(48,63,58,73),(49,62,59,72),(50,61,60,71),(81,110,91,120),(82,109,92,119),(83,108,93,118),(84,107,94,117),(85,106,95,116),(86,105,96,115),(87,104,97,114),(88,103,98,113),(89,102,99,112),(90,101,100,111)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀