direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C2×C20, C24.C30, C23⋊2C60, (C23×C20)⋊C3, (C23×C4)⋊C15, C22⋊(C2×C60), (C22×C20)⋊4C6, (C22×C4)⋊2C30, (C22×C10)⋊8C12, C22.7(C10×A4), (C23×C10).2C6, C23.6(C2×C30), (C22×A4).2C10, C10.12(C22×A4), (C10×A4).23C22, C2.1(A4×C2×C10), (A4×C2×C10).4C2, (C2×C10)⋊10(C2×C12), (C2×A4).6(C2×C10), (C2×C10).16(C2×A4), (C22×C10).29(C2×C6), SmallGroup(480,1126)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C2×C20 |
Subgroups: 316 in 132 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C22 [×2], C22 [×11], C5, C6 [×3], C2×C4, C2×C4 [×9], C23, C23 [×2], C23 [×4], C10, C10 [×2], C10 [×4], C12 [×2], A4, C2×C6, C15, C22×C4 [×2], C22×C4 [×4], C24, C20 [×2], C20 [×2], C2×C10 [×2], C2×C10 [×11], C2×C12, C2×A4, C2×A4 [×2], C30 [×3], C23×C4, C2×C20, C2×C20 [×9], C22×C10, C22×C10 [×2], C22×C10 [×4], C4×A4 [×2], C22×A4, C60 [×2], C5×A4, C2×C30, C22×C20 [×2], C22×C20 [×4], C23×C10, C2×C4×A4, C2×C60, C10×A4, C10×A4 [×2], C23×C20, A4×C20 [×2], A4×C2×C10, A4×C2×C20
Quotients:
C1, C2 [×3], C3, C4 [×2], C22, C5, C6 [×3], C2×C4, C10 [×3], C12 [×2], A4, C2×C6, C15, C20 [×2], C2×C10, C2×C12, C2×A4 [×3], C30 [×3], C2×C20, C4×A4 [×2], C22×A4, C60 [×2], C5×A4, C2×C30, C2×C4×A4, C2×C60, C10×A4 [×3], A4×C20 [×2], A4×C2×C10, A4×C2×C20
Generators and relations
G = < a,b,c,d,e | a2=b20=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >
►On 120 points
Generators in S120
(1 97)(2 98)(3 99)(4 100)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)
(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)
(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(37 119)(38 120)(39 101)(40 102)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 61)(60 62)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 81)(16 82)(17 83)(18 84)(19 85)(20 86)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(37 119)(38 120)(39 101)(40 102)
(1 61 106)(2 62 107)(3 63 108)(4 64 109)(5 65 110)(6 66 111)(7 67 112)(8 68 113)(9 69 114)(10 70 115)(11 71 116)(12 72 117)(13 73 118)(14 74 119)(15 75 120)(16 76 101)(17 77 102)(18 78 103)(19 79 104)(20 80 105)(21 84 56)(22 85 57)(23 86 58)(24 87 59)(25 88 60)(26 89 41)(27 90 42)(28 91 43)(29 92 44)(30 93 45)(31 94 46)(32 95 47)(33 96 48)(34 97 49)(35 98 50)(36 99 51)(37 100 52)(38 81 53)(39 82 54)(40 83 55)
G:=sub<Sym(120)| (1,97)(2,98)(3,99)(4,100)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72), (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), (21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,101)(40,102)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,61)(60,62), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,81)(16,82)(17,83)(18,84)(19,85)(20,86)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,101)(40,102), (1,61,106)(2,62,107)(3,63,108)(4,64,109)(5,65,110)(6,66,111)(7,67,112)(8,68,113)(9,69,114)(10,70,115)(11,71,116)(12,72,117)(13,73,118)(14,74,119)(15,75,120)(16,76,101)(17,77,102)(18,78,103)(19,79,104)(20,80,105)(21,84,56)(22,85,57)(23,86,58)(24,87,59)(25,88,60)(26,89,41)(27,90,42)(28,91,43)(29,92,44)(30,93,45)(31,94,46)(32,95,47)(33,96,48)(34,97,49)(35,98,50)(36,99,51)(37,100,52)(38,81,53)(39,82,54)(40,83,55)>;
G:=Group( (1,97)(2,98)(3,99)(4,100)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72), (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), (21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,101)(40,102)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,61)(60,62), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,81)(16,82)(17,83)(18,84)(19,85)(20,86)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,101)(40,102), (1,61,106)(2,62,107)(3,63,108)(4,64,109)(5,65,110)(6,66,111)(7,67,112)(8,68,113)(9,69,114)(10,70,115)(11,71,116)(12,72,117)(13,73,118)(14,74,119)(15,75,120)(16,76,101)(17,77,102)(18,78,103)(19,79,104)(20,80,105)(21,84,56)(22,85,57)(23,86,58)(24,87,59)(25,88,60)(26,89,41)(27,90,42)(28,91,43)(29,92,44)(30,93,45)(31,94,46)(32,95,47)(33,96,48)(34,97,49)(35,98,50)(36,99,51)(37,100,52)(38,81,53)(39,82,54)(40,83,55) );
G=PermutationGroup([(1,97),(2,98),(3,99),(4,100),(5,81),(6,82),(7,83),(8,84),(9,85),(10,86),(11,87),(12,88),(13,89),(14,90),(15,91),(16,92),(17,93),(18,94),(19,95),(20,96),(21,113),(22,114),(23,115),(24,116),(25,117),(26,118),(27,119),(28,120),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72)], [(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)], [(21,103),(22,104),(23,105),(24,106),(25,107),(26,108),(27,109),(28,110),(29,111),(30,112),(31,113),(32,114),(33,115),(34,116),(35,117),(36,118),(37,119),(38,120),(39,101),(40,102),(41,63),(42,64),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70),(49,71),(50,72),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,79),(58,80),(59,61),(60,62)], [(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,99),(14,100),(15,81),(16,82),(17,83),(18,84),(19,85),(20,86),(21,103),(22,104),(23,105),(24,106),(25,107),(26,108),(27,109),(28,110),(29,111),(30,112),(31,113),(32,114),(33,115),(34,116),(35,117),(36,118),(37,119),(38,120),(39,101),(40,102)], [(1,61,106),(2,62,107),(3,63,108),(4,64,109),(5,65,110),(6,66,111),(7,67,112),(8,68,113),(9,69,114),(10,70,115),(11,71,116),(12,72,117),(13,73,118),(14,74,119),(15,75,120),(16,76,101),(17,77,102),(18,78,103),(19,79,104),(20,80,105),(21,84,56),(22,85,57),(23,86,58),(24,87,59),(25,88,60),(26,89,41),(27,90,42),(28,91,43),(29,92,44),(30,93,45),(31,94,46),(32,95,47),(33,96,48),(34,97,49),(35,98,50),(36,99,51),(37,100,52),(38,81,53),(39,82,54),(40,83,55)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 53 | 0 | 0 |
| 0 | 0 | 53 | 0 |
| 0 | 0 | 0 | 53 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,53,0,0,0,0,53,0,0,0,0,53],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,60,0,0,0,0,60,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;
160 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 10A | ··· | 10L | 10M | ··· | 10AB | 12A | ··· | 12H | 15A | ··· | 15H | 20A | ··· | 20P | 20Q | ··· | 20AF | 30A | ··· | 30X | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C60 | A4 | C2×A4 | C2×A4 | C4×A4 | C5×A4 | C10×A4 | C10×A4 | A4×C20 |
| kernel | A4×C2×C20 | A4×C20 | A4×C2×C10 | C23×C20 | C10×A4 | C2×C4×A4 | C22×C20 | C23×C10 | C4×A4 | C22×A4 | C22×C10 | C23×C4 | C2×A4 | C22×C4 | C24 | C23 | C2×C20 | C20 | C2×C10 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 4 | 8 | 8 | 16 | 16 | 8 | 32 | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 |
In GAP, Magma, Sage, TeX
A_4\times C_2\times C_{20} % in TeX
G:=Group("A4xC2xC20"); // GroupNames label
G:=SmallGroup(480,1126);
// by ID
G=gap.SmallGroup(480,1126);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,-2,2,428,2539,4430]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^20=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀