direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C25, C4⋊C50, C22⋊C50, C100⋊3C2, C20.3C10, C50.6C22, C5.(C5×D4), (C5×D4)○C50, (C5×D4).C5, (C2×C50)⋊1C2, C2.1(C2×C50), C10.6(C2×C10), (C2×C10).1C10, SmallGroup(200,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C25
G = < a,b,c | a25=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×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 92 41 62)(2 93 42 63)(3 94 43 64)(4 95 44 65)(5 96 45 66)(6 97 46 67)(7 98 47 68)(8 99 48 69)(9 100 49 70)(10 76 50 71)(11 77 26 72)(12 78 27 73)(13 79 28 74)(14 80 29 75)(15 81 30 51)(16 82 31 52)(17 83 32 53)(18 84 33 54)(19 85 34 55)(20 86 35 56)(21 87 36 57)(22 88 37 58)(23 89 38 59)(24 90 39 60)(25 91 40 61)
(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 76)(72 77)(73 78)(74 79)(75 80)
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), (1,92,41,62)(2,93,42,63)(3,94,43,64)(4,95,44,65)(5,96,45,66)(6,97,46,67)(7,98,47,68)(8,99,48,69)(9,100,49,70)(10,76,50,71)(11,77,26,72)(12,78,27,73)(13,79,28,74)(14,80,29,75)(15,81,30,51)(16,82,31,52)(17,83,32,53)(18,84,33,54)(19,85,34,55)(20,86,35,56)(21,87,36,57)(22,88,37,58)(23,89,38,59)(24,90,39,60)(25,91,40,61), (51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,76)(72,77)(73,78)(74,79)(75,80)>;
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), (1,92,41,62)(2,93,42,63)(3,94,43,64)(4,95,44,65)(5,96,45,66)(6,97,46,67)(7,98,47,68)(8,99,48,69)(9,100,49,70)(10,76,50,71)(11,77,26,72)(12,78,27,73)(13,79,28,74)(14,80,29,75)(15,81,30,51)(16,82,31,52)(17,83,32,53)(18,84,33,54)(19,85,34,55)(20,86,35,56)(21,87,36,57)(22,88,37,58)(23,89,38,59)(24,90,39,60)(25,91,40,61), (51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,76)(72,77)(73,78)(74,79)(75,80) );
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)], [(1,92,41,62),(2,93,42,63),(3,94,43,64),(4,95,44,65),(5,96,45,66),(6,97,46,67),(7,98,47,68),(8,99,48,69),(9,100,49,70),(10,76,50,71),(11,77,26,72),(12,78,27,73),(13,79,28,74),(14,80,29,75),(15,81,30,51),(16,82,31,52),(17,83,32,53),(18,84,33,54),(19,85,34,55),(20,86,35,56),(21,87,36,57),(22,88,37,58),(23,89,38,59),(24,90,39,60),(25,91,40,61)], [(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,76),(72,77),(73,78),(74,79),(75,80)]])
D4×C25 is a maximal subgroup of
D4.D25 D4⋊D25 D4⋊2D25
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 20A | 20B | 20C | 20D | 25A | ··· | 25T | 50A | ··· | 50T | 50U | ··· | 50BH | 100A | ··· | 100T |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | C25 | C50 | C50 | D4 | C5×D4 | D4×C25 |
| kernel | D4×C25 | C100 | C2×C50 | C5×D4 | C20 | C2×C10 | D4 | C4 | C22 | C25 | C5 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 20 | 20 | 40 | 1 | 4 | 20 |
Matrix representation of D4×C25 ►in GL2(𝔽101) generated by
| 88 | 0 |
| 0 | 88 |
| 1 | 8 |
| 25 | 100 |
| 1 | 8 |
| 0 | 100 |
G:=sub<GL(2,GF(101))| [88,0,0,88],[1,25,8,100],[1,0,8,100] >;
D4×C25 in GAP, Magma, Sage, TeX
D_4\times C_{25} % in TeX
G:=Group("D4xC25"); // GroupNames label
G:=SmallGroup(200,10);
// by ID
G=gap.SmallGroup(200,10);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-5,221,162]);
// Polycyclic
G:=Group<a,b,c|a^25=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export