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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

C1C2 — D4×C25
C1C5C10C50C2×C50 — D4×C25
C1C2 — D4×C25
C1C50 — D4×C25

Generators and relations for D4×C25
 G = < a,b,c | a25=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C10
2C10
2C50
2C50

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 90 75 48)(2 91 51 49)(3 92 52 50)(4 93 53 26)(5 94 54 27)(6 95 55 28)(7 96 56 29)(8 97 57 30)(9 98 58 31)(10 99 59 32)(11 100 60 33)(12 76 61 34)(13 77 62 35)(14 78 63 36)(15 79 64 37)(16 80 65 38)(17 81 66 39)(18 82 67 40)(19 83 68 41)(20 84 69 42)(21 85 70 43)(22 86 71 44)(23 87 72 45)(24 88 73 46)(25 89 74 47)
(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 91)(50 92)

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), (1,90,75,48)(2,91,51,49)(3,92,52,50)(4,93,53,26)(5,94,54,27)(6,95,55,28)(7,96,56,29)(8,97,57,30)(9,98,58,31)(10,99,59,32)(11,100,60,33)(12,76,61,34)(13,77,62,35)(14,78,63,36)(15,79,64,37)(16,80,65,38)(17,81,66,39)(18,82,67,40)(19,83,68,41)(20,84,69,42)(21,85,70,43)(22,86,71,44)(23,87,72,45)(24,88,73,46)(25,89,74,47), (26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)>;

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), (1,90,75,48)(2,91,51,49)(3,92,52,50)(4,93,53,26)(5,94,54,27)(6,95,55,28)(7,96,56,29)(8,97,57,30)(9,98,58,31)(10,99,59,32)(11,100,60,33)(12,76,61,34)(13,77,62,35)(14,78,63,36)(15,79,64,37)(16,80,65,38)(17,81,66,39)(18,82,67,40)(19,83,68,41)(20,84,69,42)(21,85,70,43)(22,86,71,44)(23,87,72,45)(24,88,73,46)(25,89,74,47), (26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92) );

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)], [(1,90,75,48),(2,91,51,49),(3,92,52,50),(4,93,53,26),(5,94,54,27),(6,95,55,28),(7,96,56,29),(8,97,57,30),(9,98,58,31),(10,99,59,32),(11,100,60,33),(12,76,61,34),(13,77,62,35),(14,78,63,36),(15,79,64,37),(16,80,65,38),(17,81,66,39),(18,82,67,40),(19,83,68,41),(20,84,69,42),(21,85,70,43),(22,86,71,44),(23,87,72,45),(24,88,73,46),(25,89,74,47)], [(26,93),(27,94),(28,95),(29,96),(30,97),(31,98),(32,99),(33,100),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,89),(48,90),(49,91),(50,92)])

D4×C25 is a maximal subgroup of   D4.D25  D4⋊D25  D42D25

125 conjugacy classes

class 1 2A2B2C 4 5A5B5C5D10A10B10C10D10E···10L20A20B20C20D25A···25T50A···50T50U···50BH100A···100T
order1222455551010101010···102020202025···2550···5050···50100···100
size11222111111112···222221···11···12···22···2

125 irreducible representations

dim111111111222
type++++
imageC1C2C2C5C10C10C25C50C50D4C5×D4D4×C25
kernelD4×C25C100C2×C50C5×D4C20C2×C10D4C4C22C25C5C1
# reps1124482020401420

Matrix representation of D4×C25 in GL2(𝔽101) generated by

880
088
,
18
25100
,
18
0100
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

Subgroup lattice of D4×C25 in TeX

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