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G = C4×D25order 200 = 23·52

Direct product of C4 and D25

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C25 — C4×D25
C1C5C25C50D50 — C4×D25
C25 — C4×D25
C1C4

Generators and relations for C4×D25
 G = < a,b,c | a4=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >

25C2
25C2
25C22
25C4
5D5
5D5
25C2×C4
5D10
5Dic5
5C4×D5

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 2A2B2C4A4B4C4D5A5B10A10B20A20B20C20D25A···25J50A···50J100A···100T
order122244445510102020202025···2550···50100···100
size112525112525222222222···22···22···2

56 irreducible representations

dim11111222222
type++++++++
imageC1C2C2C2C4D5D10C4×D5D25D50C4×D25
kernelC4×D25Dic25C100D50D25C20C10C5C4C2C1
# reps11114224101020

Matrix representation of C4×D25 in GL3(𝔽101) generated by

9100
01000
00100
,
100
05193
09161
,
100
02943
01172
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

Export

Subgroup lattice of C4×D25 in TeX

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