metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C25⋊2D4, C22⋊D25, D50⋊2C2, Dic25⋊C2, C2.5D50, C10.10D10, C50.5C22, (C2×C50)⋊2C2, C5.(C5⋊D4), (C2×C10).2D5, SmallGroup(200,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C25⋊D4
G = < a,b,c | a25=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C25⋊D4
►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 68 47 93)(2 67 48 92)(3 66 49 91)(4 65 50 90)(5 64 26 89)(6 63 27 88)(7 62 28 87)(8 61 29 86)(9 60 30 85)(10 59 31 84)(11 58 32 83)(12 57 33 82)(13 56 34 81)(14 55 35 80)(15 54 36 79)(16 53 37 78)(17 52 38 77)(18 51 39 76)(19 75 40 100)(20 74 41 99)(21 73 42 98)(22 72 43 97)(23 71 44 96)(24 70 45 95)(25 69 46 94)
(2 25)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)(13 14)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(44 50)(45 49)(46 48)(51 85)(52 84)(53 83)(54 82)(55 81)(56 80)(57 79)(58 78)(59 77)(60 76)(61 100)(62 99)(63 98)(64 97)(65 96)(66 95)(67 94)(68 93)(69 92)(70 91)(71 90)(72 89)(73 88)(74 87)(75 86)
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), (1,68,47,93)(2,67,48,92)(3,66,49,91)(4,65,50,90)(5,64,26,89)(6,63,27,88)(7,62,28,87)(8,61,29,86)(9,60,30,85)(10,59,31,84)(11,58,32,83)(12,57,33,82)(13,56,34,81)(14,55,35,80)(15,54,36,79)(16,53,37,78)(17,52,38,77)(18,51,39,76)(19,75,40,100)(20,74,41,99)(21,73,42,98)(22,72,43,97)(23,71,44,96)(24,70,45,95)(25,69,46,94), (2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(44,50)(45,49)(46,48)(51,85)(52,84)(53,83)(54,82)(55,81)(56,80)(57,79)(58,78)(59,77)(60,76)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)>;
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), (1,68,47,93)(2,67,48,92)(3,66,49,91)(4,65,50,90)(5,64,26,89)(6,63,27,88)(7,62,28,87)(8,61,29,86)(9,60,30,85)(10,59,31,84)(11,58,32,83)(12,57,33,82)(13,56,34,81)(14,55,35,80)(15,54,36,79)(16,53,37,78)(17,52,38,77)(18,51,39,76)(19,75,40,100)(20,74,41,99)(21,73,42,98)(22,72,43,97)(23,71,44,96)(24,70,45,95)(25,69,46,94), (2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(44,50)(45,49)(46,48)(51,85)(52,84)(53,83)(54,82)(55,81)(56,80)(57,79)(58,78)(59,77)(60,76)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86) );
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)], [(1,68,47,93),(2,67,48,92),(3,66,49,91),(4,65,50,90),(5,64,26,89),(6,63,27,88),(7,62,28,87),(8,61,29,86),(9,60,30,85),(10,59,31,84),(11,58,32,83),(12,57,33,82),(13,56,34,81),(14,55,35,80),(15,54,36,79),(16,53,37,78),(17,52,38,77),(18,51,39,76),(19,75,40,100),(20,74,41,99),(21,73,42,98),(22,72,43,97),(23,71,44,96),(24,70,45,95),(25,69,46,94)], [(2,25),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,18),(10,17),(11,16),(12,15),(13,14),(26,43),(27,42),(28,41),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(44,50),(45,49),(46,48),(51,85),(52,84),(53,83),(54,82),(55,81),(56,80),(57,79),(58,78),(59,77),(60,76),(61,100),(62,99),(63,98),(64,97),(65,96),(66,95),(67,94),(68,93),(69,92),(70,91),(71,90),(72,89),(73,88),(74,87),(75,86)]])
C25⋊D4 is a maximal subgroup of
D100⋊5C2 D4×D25 D4⋊2D25
C25⋊D4 is a maximal quotient of C50.D4 D50⋊C4 D4.D25 D4⋊D25 C25⋊Q16 Q8⋊D25 C23.D25
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 10A | ··· | 10F | 25A | ··· | 25J | 50A | ··· | 50AD |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 10 | ··· | 10 | 25 | ··· | 25 | 50 | ··· | 50 |
| size | 1 | 1 | 2 | 50 | 50 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | D5 | D10 | C5⋊D4 | D25 | D50 | C25⋊D4 |
| kernel | C25⋊D4 | Dic25 | D50 | C2×C50 | C25 | C2×C10 | C10 | C5 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 10 | 10 | 20 |
Matrix representation of C25⋊D4 ►in GL2(𝔽101) generated by
| 32 | 11 |
| 90 | 72 |
| 30 | 36 |
| 90 | 71 |
| 50 | 13 |
| 2 | 51 |
G:=sub<GL(2,GF(101))| [32,90,11,72],[30,90,36,71],[50,2,13,51] >;
C25⋊D4 in GAP, Magma, Sage, TeX
C_{25}\rtimes D_4 % in TeX
G:=Group("C25:D4"); // GroupNames label
G:=SmallGroup(200,8);
// by ID
G=gap.SmallGroup(200,8);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,61,1443,418,4004]);
// Polycyclic
G:=Group<a,b,c|a^25=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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