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## G = C4×D25order 200 = 23·52

### Direct product of C4 and D25

Aliases: C4×D25, C1002C2, C20.5D5, C2.1D50, D50.2C2, C10.7D10, Dic252C2, C50.2C22, C5.(C4×D5), C252(C2×C4), SmallGroup(200,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — C4×D25
 Chief series C1 — C5 — C25 — C50 — D50 — C4×D25
 Lower central C25 — C4×D25
 Upper central C1 — C4

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

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

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

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

C4×D25 is a maximal subgroup of   C8⋊D25  D25⋊C8  C100.C4  C100⋊C4  D1005C2  D42D25  Q82D25
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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