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G = C25⋊D4order 200 = 23·52

The semidirect product of C25 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C252D4, C22⋊D25, D502C2, Dic25⋊C2, C2.5D50, C10.10D10, C50.5C22, (C2×C50)⋊2C2, C5.(C5⋊D4), (C2×C10).2D5, SmallGroup(200,8)

Series: Derived Chief Lower central Upper central

C1C50 — C25⋊D4
C1C5C25C50D50 — C25⋊D4
C25C50 — C25⋊D4
C1C2C22

Generators and relations for C25⋊D4
 G = < a,b,c | a25=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
50C2
25C4
25C22
2C10
10D5
25D4
5Dic5
5D10
2D25
2C50
5C5⋊D4

Smallest permutation representation of C25⋊D4
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 64 31 81)(2 63 32 80)(3 62 33 79)(4 61 34 78)(5 60 35 77)(6 59 36 76)(7 58 37 100)(8 57 38 99)(9 56 39 98)(10 55 40 97)(11 54 41 96)(12 53 42 95)(13 52 43 94)(14 51 44 93)(15 75 45 92)(16 74 46 91)(17 73 47 90)(18 72 48 89)(19 71 49 88)(20 70 50 87)(21 69 26 86)(22 68 27 85)(23 67 28 84)(24 66 29 83)(25 65 30 82)
(2 25)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)(13 14)(26 36)(27 35)(28 34)(29 33)(30 32)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 84)(62 83)(63 82)(64 81)(65 80)(66 79)(67 78)(68 77)(69 76)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)

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

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

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

C25⋊D4 is a maximal subgroup of   D1005C2  D4×D25  D42D25
C25⋊D4 is a maximal quotient of   C50.D4  D50⋊C4  D4.D25  D4⋊D25  C25⋊Q16  Q8⋊D25  C23.D25

53 conjugacy classes

class 1 2A2B2C 4 5A5B10A···10F25A···25J50A···50AD
order122245510···1025···2550···50
size1125050222···22···22···2

53 irreducible representations

dim11112222222
type+++++++++
imageC1C2C2C2D4D5D10C5⋊D4D25D50C25⋊D4
kernelC25⋊D4Dic25D50C2×C50C25C2×C10C10C5C22C2C1
# reps11111224101020

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

3211
9072
,
3036
9071
,
5013
251
G:=sub<GL(2,GF(101))| [32,90,11,72],[30,90,36,71],[50,2,13,51] >;

C25⋊D4 in GAP, Magma, Sage, TeX

C_{25}\rtimes D_4
% in TeX

G:=Group("C25:D4");
// GroupNames label

G:=SmallGroup(200,8);
// by ID

G=gap.SmallGroup(200,8);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,61,1443,418,4004]);
// Polycyclic

G:=Group<a,b,c|a^25=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C25⋊D4 in TeX

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