Copied to
clipboard

G = C22×D25order 200 = 23·52

Direct product of C22 and D25

Aliases: C22×D25, C25⋊C23, C50⋊C22, C10.11D10, (C2×C50)⋊3C2, C5.(C22×D5), (C2×C10).3D5, SmallGroup(200,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — C22×D25
 Chief series C1 — C5 — C25 — D25 — D50 — C22×D25
 Lower central C25 — C22×D25
 Upper central C1 — C22

Generators and relations for C22×D25
G = < a,b,c,d | a2=b2=c25=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of C22×D25
On 100 points
Generators in S100
(1 72)(2 73)(3 74)(4 75)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 71)(26 80)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 76)(48 77)(49 78)(50 79)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(51 97)(52 98)(53 99)(54 100)(55 76)(56 77)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)
(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)(2 91)(3 90)(4 89)(5 88)(6 87)(7 86)(8 85)(9 84)(10 83)(11 82)(12 81)(13 80)(14 79)(15 78)(16 77)(17 76)(18 100)(19 99)(20 98)(21 97)(22 96)(23 95)(24 94)(25 93)(26 59)(27 58)(28 57)(29 56)(30 55)(31 54)(32 53)(33 52)(34 51)(35 75)(36 74)(37 73)(38 72)(39 71)(40 70)(41 69)(42 68)(43 67)(44 66)(45 65)(46 64)(47 63)(48 62)(49 61)(50 60)

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

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

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

C22×D25 is a maximal subgroup of   D50⋊C4  D25.D4
C22×D25 is a maximal quotient of   D1005C2  D42D25  Q82D25

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 5A 5B 10A ··· 10F 25A ··· 25J 50A ··· 50AD order 1 2 2 2 2 2 2 2 5 5 10 ··· 10 25 ··· 25 50 ··· 50 size 1 1 1 1 25 25 25 25 2 2 2 ··· 2 2 ··· 2 2 ··· 2

56 irreducible representations

 dim 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 D5 D10 D25 D50 kernel C22×D25 D50 C2×C50 C2×C10 C10 C22 C2 # reps 1 6 1 2 6 10 30

Matrix representation of C22×D25 in GL4(𝔽101) generated by

 100 0 0 0 0 100 0 0 0 0 1 0 0 0 0 1
,
 100 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 50 55 0 0 44 6
,
 100 0 0 0 0 1 0 0 0 0 33 4 0 0 31 68
G:=sub<GL(4,GF(101))| [100,0,0,0,0,100,0,0,0,0,1,0,0,0,0,1],[100,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,50,44,0,0,55,6],[100,0,0,0,0,1,0,0,0,0,33,31,0,0,4,68] >;

C22×D25 in GAP, Magma, Sage, TeX

C_2^2\times D_{25}
% in TeX

G:=Group("C2^2xD25");
// GroupNames label

G:=SmallGroup(200,13);
// by ID

G=gap.SmallGroup(200,13);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,1443,418,4004]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^25=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

׿
×
𝔽