direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C25, C22⋊C75, (C2×C50)⋊C3, C5.(C5×A4), (C5×A4).C5, (C2×C10).C15, SmallGroup(300,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C25 |
Generators and relations for A4×C25
G = < a,b,c,d | a25=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×C25
►On 100 points
Generators in S100
(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)
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 99)(42 100)(43 76)(44 77)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 85)(23 86)(24 87)(25 88)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)
(26 65 84)(27 66 85)(28 67 86)(29 68 87)(30 69 88)(31 70 89)(32 71 90)(33 72 91)(34 73 92)(35 74 93)(36 75 94)(37 51 95)(38 52 96)(39 53 97)(40 54 98)(41 55 99)(42 56 100)(43 57 76)(44 58 77)(45 59 78)(46 60 79)(47 61 80)(48 62 81)(49 63 82)(50 64 83)
G:=sub<Sym(100)| (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), (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,99)(42,100)(43,76)(44,77)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64), (26,65,84)(27,66,85)(28,67,86)(29,68,87)(30,69,88)(31,70,89)(32,71,90)(33,72,91)(34,73,92)(35,74,93)(36,75,94)(37,51,95)(38,52,96)(39,53,97)(40,54,98)(41,55,99)(42,56,100)(43,57,76)(44,58,77)(45,59,78)(46,60,79)(47,61,80)(48,62,81)(49,63,82)(50,64,83)>;
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), (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,99)(42,100)(43,76)(44,77)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64), (26,65,84)(27,66,85)(28,67,86)(29,68,87)(30,69,88)(31,70,89)(32,71,90)(33,72,91)(34,73,92)(35,74,93)(36,75,94)(37,51,95)(38,52,96)(39,53,97)(40,54,98)(41,55,99)(42,56,100)(43,57,76)(44,58,77)(45,59,78)(46,60,79)(47,61,80)(48,62,81)(49,63,82)(50,64,83) );
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)], [(1,70),(2,71),(3,72),(4,73),(5,74),(6,75),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,65),(22,66),(23,67),(24,68),(25,69),(26,84),(27,85),(28,86),(29,87),(30,88),(31,89),(32,90),(33,91),(34,92),(35,93),(36,94),(37,95),(38,96),(39,97),(40,98),(41,99),(42,100),(43,76),(44,77),(45,78),(46,79),(47,80),(48,81),(49,82),(50,83)], [(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,97),(10,98),(11,99),(12,100),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,85),(23,86),(24,87),(25,88),(26,65),(27,66),(28,67),(29,68),(30,69),(31,70),(32,71),(33,72),(34,73),(35,74),(36,75),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(49,63),(50,64)], [(26,65,84),(27,66,85),(28,67,86),(29,68,87),(30,69,88),(31,70,89),(32,71,90),(33,72,91),(34,73,92),(35,74,93),(36,75,94),(37,51,95),(38,52,96),(39,53,97),(40,54,98),(41,55,99),(42,56,100),(43,57,76),(44,58,77),(45,59,78),(46,60,79),(47,61,80),(48,62,81),(49,63,82),(50,64,83)]])
100 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 15A | ··· | 15H | 25A | ··· | 25T | 50A | ··· | 50T | 75A | ··· | 75AN |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 25 | ··· | 25 | 50 | ··· | 50 | 75 | ··· | 75 |
| size | 1 | 3 | 4 | 4 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | + | |||||||
| image | C1 | C3 | C5 | C15 | C25 | C75 | A4 | C5×A4 | A4×C25 |
| kernel | A4×C25 | C2×C50 | C5×A4 | C2×C10 | A4 | C22 | C25 | C5 | C1 |
| # reps | 1 | 2 | 4 | 8 | 20 | 40 | 1 | 4 | 20 |
Matrix representation of A4×C25 ►in GL4(𝔽151) generated by
| 127 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 150 | 0 | 0 |
| 0 | 150 | 0 | 1 |
| 0 | 150 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 150 |
| 0 | 1 | 0 | 150 |
| 0 | 0 | 0 | 150 |
| 118 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(151))| [127,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,150,150,150,0,0,0,1,0,0,1,0],[1,0,0,0,0,0,1,0,0,1,0,0,0,150,150,150],[118,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;
A4×C25 in GAP, Magma, Sage, TeX
A_4\times C_{25} % in TeX
G:=Group("A4xC25"); // GroupNames label
G:=SmallGroup(300,8);
// by ID
G=gap.SmallGroup(300,8);
# by ID
G:=PCGroup([5,-3,-5,-5,-2,2,56,3003,5629]);
// Polycyclic
G:=Group<a,b,c,d|a^25=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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