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## G = C10×C5⋊D5order 500 = 22·53

### Direct product of C10 and C5⋊D5

Aliases: C10×C5⋊D5, C527D10, C535C22, C10⋊(C5×D5), C52(D5×C10), (C5×C10)⋊3D5, (C5×C10)⋊4C10, C525(C2×C10), (C52×C10)⋊2C2, SmallGroup(500,54)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C10×C5⋊D5
 Chief series C1 — C5 — C52 — C53 — C5×C5⋊D5 — C10×C5⋊D5
 Lower central C52 — C10×C5⋊D5
 Upper central C1 — C10

Generators and relations for C10×C5⋊D5
G = < a,b,c,d | a10=b5=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 464 in 128 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C22, C5, C5, C5, D5, C10, C10, C10, D10, C2×C10, C52, C52, C52, C5×D5, C5⋊D5, C5×C10, C5×C10, C5×C10, D5×C10, C2×C5⋊D5, C53, C5×C5⋊D5, C52×C10, C10×C5⋊D5
Quotients: C1, C2, C22, C5, D5, C10, D10, C2×C10, C5×D5, C5⋊D5, D5×C10, C2×C5⋊D5, C5×C5⋊D5, C10×C5⋊D5

Smallest permutation representation of C10×C5⋊D5
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 80 68 90 53)(2 71 69 81 54)(3 72 70 82 55)(4 73 61 83 56)(5 74 62 84 57)(6 75 63 85 58)(7 76 64 86 59)(8 77 65 87 60)(9 78 66 88 51)(10 79 67 89 52)(11 25 49 38 98)(12 26 50 39 99)(13 27 41 40 100)(14 28 42 31 91)(15 29 43 32 92)(16 30 44 33 93)(17 21 45 34 94)(18 22 46 35 95)(19 23 47 36 96)(20 24 48 37 97)
(1 84 78 55 64)(2 85 79 56 65)(3 86 80 57 66)(4 87 71 58 67)(5 88 72 59 68)(6 89 73 60 69)(7 90 74 51 70)(8 81 75 52 61)(9 82 76 53 62)(10 83 77 54 63)(11 34 27 96 43)(12 35 28 97 44)(13 36 29 98 45)(14 37 30 99 46)(15 38 21 100 47)(16 39 22 91 48)(17 40 23 92 49)(18 31 24 93 50)(19 32 25 94 41)(20 33 26 95 42)
(1 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 99)(8 100)(9 91)(10 92)(11 58)(12 59)(13 60)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 78)(32 79)(33 80)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 61)(48 62)(49 63)(50 64)

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 5A 5B 5C 5D 5E ··· 5BL 10A 10B 10C 10D 10E ··· 10BL 10BM ··· 10BT order 1 2 2 2 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 10 ··· 10 size 1 1 25 25 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 25 ··· 25

140 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C5 C10 C10 D5 D10 C5×D5 D5×C10 kernel C10×C5⋊D5 C5×C5⋊D5 C52×C10 C2×C5⋊D5 C5⋊D5 C5×C10 C5×C10 C52 C10 C5 # reps 1 2 1 4 8 4 12 12 48 48

Matrix representation of C10×C5⋊D5 in GL4(𝔽11) generated by

 5 0 0 0 0 5 0 0 0 0 10 0 0 0 0 10
,
 3 0 0 0 0 4 0 0 0 0 0 1 0 0 10 7
,
 1 0 0 0 0 1 0 0 0 0 4 4 0 0 7 10
,
 0 7 0 0 8 0 0 0 0 0 7 7 0 0 1 4
G:=sub<GL(4,GF(11))| [5,0,0,0,0,5,0,0,0,0,10,0,0,0,0,10],[3,0,0,0,0,4,0,0,0,0,0,10,0,0,1,7],[1,0,0,0,0,1,0,0,0,0,4,7,0,0,4,10],[0,8,0,0,7,0,0,0,0,0,7,1,0,0,7,4] >;

C10×C5⋊D5 in GAP, Magma, Sage, TeX

C_{10}\times C_5\rtimes D_5
% in TeX

G:=Group("C10xC5:D5");
// GroupNames label

G:=SmallGroup(500,54);
// by ID

G=gap.SmallGroup(500,54);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,1603,10004]);
// Polycyclic

G:=Group<a,b,c,d|a^10=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

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