direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D25, C100⋊2C2, C20.5D5, C2.1D50, D50.2C2, C10.7D10, Dic25⋊2C2, C50.2C22, C5.(C4×D5), C25⋊2(C2×C4), SmallGroup(200,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C4×D25 |
Generators and relations for C4×D25
G = < a,b,c | a4=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D25
►On 100 points
Generators in S100
(1 76 50 71)(2 77 26 72)(3 78 27 73)(4 79 28 74)(5 80 29 75)(6 81 30 51)(7 82 31 52)(8 83 32 53)(9 84 33 54)(10 85 34 55)(11 86 35 56)(12 87 36 57)(13 88 37 58)(14 89 38 59)(15 90 39 60)(16 91 40 61)(17 92 41 62)(18 93 42 63)(19 94 43 64)(20 95 44 65)(21 96 45 66)(22 97 46 67)(23 98 47 68)(24 99 48 69)(25 100 49 70)
(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 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(25 50)(51 95)(52 94)(53 93)(54 92)(55 91)(56 90)(57 89)(58 88)(59 87)(60 86)(61 85)(62 84)(63 83)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 100)(72 99)(73 98)(74 97)(75 96)
G:=sub<Sym(100)| (1,76,50,71)(2,77,26,72)(3,78,27,73)(4,79,28,74)(5,80,29,75)(6,81,30,51)(7,82,31,52)(8,83,32,53)(9,84,33,54)(10,85,34,55)(11,86,35,56)(12,87,36,57)(13,88,37,58)(14,89,38,59)(15,90,39,60)(16,91,40,61)(17,92,41,62)(18,93,42,63)(19,94,43,64)(20,95,44,65)(21,96,45,66)(22,97,46,67)(23,98,47,68)(24,99,48,69)(25,100,49,70), (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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(25,50)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,100)(72,99)(73,98)(74,97)(75,96)>;
G:=Group( (1,76,50,71)(2,77,26,72)(3,78,27,73)(4,79,28,74)(5,80,29,75)(6,81,30,51)(7,82,31,52)(8,83,32,53)(9,84,33,54)(10,85,34,55)(11,86,35,56)(12,87,36,57)(13,88,37,58)(14,89,38,59)(15,90,39,60)(16,91,40,61)(17,92,41,62)(18,93,42,63)(19,94,43,64)(20,95,44,65)(21,96,45,66)(22,97,46,67)(23,98,47,68)(24,99,48,69)(25,100,49,70), (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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(25,50)(51,95)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,100)(72,99)(73,98)(74,97)(75,96) );
G=PermutationGroup([[(1,76,50,71),(2,77,26,72),(3,78,27,73),(4,79,28,74),(5,80,29,75),(6,81,30,51),(7,82,31,52),(8,83,32,53),(9,84,33,54),(10,85,34,55),(11,86,35,56),(12,87,36,57),(13,88,37,58),(14,89,38,59),(15,90,39,60),(16,91,40,61),(17,92,41,62),(18,93,42,63),(19,94,43,64),(20,95,44,65),(21,96,45,66),(22,97,46,67),(23,98,47,68),(24,99,48,69),(25,100,49,70)], [(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)], [(1,49),(2,48),(3,47),(4,46),(5,45),(6,44),(7,43),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,35),(16,34),(17,33),(18,32),(19,31),(20,30),(21,29),(22,28),(23,27),(24,26),(25,50),(51,95),(52,94),(53,93),(54,92),(55,91),(56,90),(57,89),(58,88),(59,87),(60,86),(61,85),(62,84),(63,83),(64,82),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,100),(72,99),(73,98),(74,97),(75,96)]])
C4×D25 is a maximal subgroup of
C8⋊D25 D25⋊C8 C100.C4 C100⋊C4 D100⋊5C2 D4⋊2D25 Q8⋊2D25
C4×D25 is a maximal quotient of C8⋊D25 C50.D4 D50⋊C4
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 10A | 10B | 20A | 20B | 20C | 20D | 25A | ··· | 25J | 50A | ··· | 50J | 100A | ··· | 100T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | 10 | 20 | 20 | 20 | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 25 | 25 | 1 | 1 | 25 | 25 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D5 | D10 | C4×D5 | D25 | D50 | C4×D25 |
| kernel | C4×D25 | Dic25 | C100 | D50 | D25 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 10 | 10 | 20 |
Matrix representation of C4×D25 ►in GL3(𝔽101) generated by
| 91 | 0 | 0 |
| 0 | 100 | 0 |
| 0 | 0 | 100 |
| 1 | 0 | 0 |
| 0 | 51 | 93 |
| 0 | 91 | 61 |
| 1 | 0 | 0 |
| 0 | 29 | 43 |
| 0 | 11 | 72 |
G:=sub<GL(3,GF(101))| [91,0,0,0,100,0,0,0,100],[1,0,0,0,51,91,0,93,61],[1,0,0,0,29,11,0,43,72] >;
C4×D25 in GAP, Magma, Sage, TeX
C_4\times D_{25} % in TeX
G:=Group("C4xD25"); // GroupNames label
G:=SmallGroup(200,5);
// by ID
G=gap.SmallGroup(200,5);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,26,1443,418,4004]);
// Polycyclic
G:=Group<a,b,c|a^4=b^25=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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