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## G = C4×C5⋊D5order 200 = 23·52

### Direct product of C4 and C5⋊D5

Aliases: C4×C5⋊D5, C202D5, C10.13D10, C53(C4×D5), (C5×C20)⋊4C2, C5210(C2×C4), C526C44C2, (C5×C10).12C22, C2.1(C2×C5⋊D5), (C2×C5⋊D5).4C2, SmallGroup(200,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C4×C5⋊D5
 Chief series C1 — C5 — C52 — C5×C10 — C2×C5⋊D5 — C4×C5⋊D5
 Lower central C52 — C4×C5⋊D5
 Upper central C1 — C4

Generators and relations for C4×C5⋊D5
G = < a,b,c,d | a4=b5=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 304 in 64 conjugacy classes, 29 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, D5, C10, Dic5, C20, D10, C52, C4×D5, C5⋊D5, C5×C10, C526C4, C5×C20, C2×C5⋊D5, C4×C5⋊D5
Quotients: C1, C2, C4, C22, C2×C4, D5, D10, C4×D5, C5⋊D5, C2×C5⋊D5, C4×C5⋊D5

Smallest permutation representation of C4×C5⋊D5
On 100 points
Generators in S100
(1 79 29 54)(2 80 30 55)(3 76 26 51)(4 77 27 52)(5 78 28 53)(6 81 31 56)(7 82 32 57)(8 83 33 58)(9 84 34 59)(10 85 35 60)(11 86 36 61)(12 87 37 62)(13 88 38 63)(14 89 39 64)(15 90 40 65)(16 91 41 66)(17 92 42 67)(18 93 43 68)(19 94 44 69)(20 95 45 70)(21 96 46 71)(22 97 47 72)(23 98 48 73)(24 99 49 74)(25 100 50 75)
(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)| (1,79,29,54)(2,80,30,55)(3,76,26,51)(4,77,27,52)(5,78,28,53)(6,81,31,56)(7,82,32,57)(8,83,33,58)(9,84,34,59)(10,85,35,60)(11,86,36,61)(12,87,37,62)(13,88,38,63)(14,89,39,64)(15,90,40,65)(16,91,41,66)(17,92,42,67)(18,93,43,68)(19,94,44,69)(20,95,45,70)(21,96,46,71)(22,97,47,72)(23,98,48,73)(24,99,49,74)(25,100,50,75), (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( (1,79,29,54)(2,80,30,55)(3,76,26,51)(4,77,27,52)(5,78,28,53)(6,81,31,56)(7,82,32,57)(8,83,33,58)(9,84,34,59)(10,85,35,60)(11,86,36,61)(12,87,37,62)(13,88,38,63)(14,89,39,64)(15,90,40,65)(16,91,41,66)(17,92,42,67)(18,93,43,68)(19,94,44,69)(20,95,45,70)(21,96,46,71)(22,97,47,72)(23,98,48,73)(24,99,49,74)(25,100,50,75), (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([[(1,79,29,54),(2,80,30,55),(3,76,26,51),(4,77,27,52),(5,78,28,53),(6,81,31,56),(7,82,32,57),(8,83,33,58),(9,84,34,59),(10,85,35,60),(11,86,36,61),(12,87,37,62),(13,88,38,63),(14,89,39,64),(15,90,40,65),(16,91,41,66),(17,92,42,67),(18,93,43,68),(19,94,44,69),(20,95,45,70),(21,96,46,71),(22,97,47,72),(23,98,48,73),(24,99,49,74),(25,100,50,75)], [(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)]])

C4×C5⋊D5 is a maximal subgroup of
C20.29D10  C20.31D10  C40⋊D5  C20.F5  C527M4(2)  C20⋊F5  C20.11F5  C528M4(2)  C202F5  D20⋊D5  Dic10⋊D5  D10.9D10  C4×D52  C20⋊D10  C20.50D10  C20.D10  C20.26D10
C4×C5⋊D5 is a maximal quotient of
C40⋊D5  C102.22C22  C10.11D20

56 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A ··· 5L 10A ··· 10L 20A ··· 20X order 1 2 2 2 4 4 4 4 5 ··· 5 10 ··· 10 20 ··· 20 size 1 1 25 25 1 1 25 25 2 ··· 2 2 ··· 2 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 D5 D10 C4×D5 kernel C4×C5⋊D5 C52⋊6C4 C5×C20 C2×C5⋊D5 C5⋊D5 C20 C10 C5 # reps 1 1 1 1 4 12 12 24

Matrix representation of C4×C5⋊D5 in GL4(𝔽41) generated by

 32 0 0 0 0 32 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 40 1 0 0 5 35
,
 0 1 0 0 40 6 0 0 0 0 0 7 0 0 35 6
,
 0 1 0 0 1 0 0 0 0 0 35 7 0 0 36 6
G:=sub<GL(4,GF(41))| [32,0,0,0,0,32,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,5,0,0,1,35],[0,40,0,0,1,6,0,0,0,0,0,35,0,0,7,6],[0,1,0,0,1,0,0,0,0,0,35,36,0,0,7,6] >;

C4×C5⋊D5 in GAP, Magma, Sage, TeX

C_4\times C_5\rtimes D_5
% in TeX

G:=Group("C4xC5:D5");
// GroupNames label

G:=SmallGroup(200,33);
// by ID

G=gap.SmallGroup(200,33);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,26,643,4004]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^5=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽