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## G = D4×C25order 200 = 23·52

### Direct product of C25 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C25, C4⋊C50, C22⋊C50, C1003C2, 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

 Derived series C1 — C2 — D4×C25
 Chief series C1 — C5 — C10 — C50 — C2×C50 — D4×C25
 Lower central C1 — C2 — D4×C25
 Upper central C1 — C50 — D4×C25

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  D42D25

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

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