C2^2xC5:D5 - GroupNames

G = C₂²×C₅⋊D₅ order 200 = 2³·5²

Direct product of C₂² and C₅⋊D₅

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₂²×C₅⋊D₅, C₁₀⋊₂D₁₀, C₁₀²⋊₅C₂, C₅²⋊₃C₂³, (C₂×C₁₀)⋊₃D₅, C₅⋊₂(C₂²×D₅), (C₅×C₁₀)⋊₃C₂², SmallGroup(200,51)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅² — C₂²×C₅⋊D₅

Chief series

C₁ — C₅ — C₅² — C₅⋊D₅ — C₂×C₅⋊D₅ — C₂²×C₅⋊D₅

Lower central

C₅² — C₂²×C₅⋊D₅

Upper central

C₁ — C₂²

Generators and relations for C₂²×C₅⋊D₅
G = < a,b,c,d,e | a²=b²=c⁵=d⁵=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c^-1, ede=d^-1 >

Subgroups: 656 in 128 conjugacy classes, 51 normal (5 characteristic)
C₁, C₂, C₂, C₂², C₂², C₅, C₂³, D₅, C₁₀, D₁₀, C₂×C₁₀, C₅², C₂²×D₅, C₅⋊D₅, C₅×C₁₀, C₂×C₅⋊D₅, C₁₀², C₂²×C₅⋊D₅
Quotients: C₁, C₂, C₂², C₂³, D₅, D₁₀, C₂²×D₅, C₅⋊D₅, C₂×C₅⋊D₅, C₂²×C₅⋊D₅

Smallest permutation representation of C₂²×C₅⋊D₅
►On 100 points

Generators in S₁₀₀

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

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

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

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

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

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

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

C₂²×C₅⋊D₅ is a maximal subgroup of C₁₀.D₂₀ C₁₀.₁₁D₂₀ C₁₀²⋊C₄ C₁₀²⋊₄C₄ D₁₀⋊D₁₀ C₂²×D₅²
C₂²×C₅⋊D₅ is a maximal quotient of C₂₀.₅₀D₁₀ C₂₀.D₁₀ C₂₀.₂₆D₁₀

56 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	5A	···	5L	10A	···	10AJ
order	1	2	2	2	2	2	2	2	5	···	5	10	···	10
size	1	1	1	1	25	25	25	25	2	···	2	2	···	2

56 irreducible representations

dim	1	1	1	2	2
type	+	+	+	+	+
image	C₁	C₂	C₂	D₅	D₁₀
kernel	C₂²×C₅⋊D₅	C₂×C₅⋊D₅	C₁₀²	C₂×C₁₀	C₁₀
# reps	1	6	1	12	36

Matrix representation of C₂²×C₅⋊D₅ ►in GL₅(𝔽₁₁)

10	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

10	0	0	0	0
0	10	0	0	0
0	0	10	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	10	1
0	0	0	2	8

1	0	0	0	0
0	0	1	0	0
0	10	3	0	0
0	0	0	10	1
0	0	0	2	8

10	0	0	0	0
0	0	1	0	0
0	1	0	0	0
0	0	0	10	0
0	0	0	2	1

G:=sub<GL(5,GF(11))| [10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[10,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,10,2,0,0,0,1,8],[1,0,0,0,0,0,0,10,0,0,0,1,3,0,0,0,0,0,10,2,0,0,0,1,8],[10,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,10,2,0,0,0,0,1] >;

C₂²×C₅⋊D₅ in GAP, Magma, Sage, TeX

C_2^2\times C_5\rtimes D_5

% in TeX

G:=Group("C2^2xC5:D5");

// GroupNames label

G:=SmallGroup(200,51);

// by ID

G=gap.SmallGroup(200,51);

# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,643,4004]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^5=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;

// generators/relations

G = C22×C5⋊D5 order 200 = 23·52

Direct product of C22 and C5⋊D5

G = C₂²×C₅⋊D₅ order 200 = 2³·5²

Direct product of C₂² and C₅⋊D₅