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## G = D4×C52order 200 = 23·52

### Direct product of C52 and D4

direct product, metacyclic, nilpotent (class 2), monomial

Aliases: D4×C52, C203C10, C1021C2, C2.1C102, C4⋊(C5×C10), (C5×C20)⋊5C2, C22⋊(C5×C10), (C2×C10)⋊1C10, C10.8(C2×C10), (C5×C10).16C22, SmallGroup(200,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C52
 Chief series C1 — C2 — C10 — C5×C10 — C102 — D4×C52
 Lower central C1 — C2 — D4×C52
 Upper central C1 — C5×C10 — D4×C52

Generators and relations for D4×C52
G = < a,b,c,d | a5=b5=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 80 in 64 conjugacy classes, 48 normal (8 characteristic)
C1, C2, C2, C4, C22, C5, D4, C10, C10, C20, C2×C10, C52, C5×D4, C5×C10, C5×C10, C5×C20, C102, D4×C52
Quotients: C1, C2, C22, C5, D4, C10, C2×C10, C52, C5×D4, C5×C10, C102, D4×C52

Smallest permutation representation of D4×C52
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 60 68 10 20)(2 56 69 6 16)(3 57 70 7 17)(4 58 66 8 18)(5 59 67 9 19)(11 24 95 77 90)(12 25 91 78 86)(13 21 92 79 87)(14 22 93 80 88)(15 23 94 76 89)(26 72 85 42 32)(27 73 81 43 33)(28 74 82 44 34)(29 75 83 45 35)(30 71 84 41 31)(36 49 53 61 98)(37 50 54 62 99)(38 46 55 63 100)(39 47 51 64 96)(40 48 52 65 97)
(1 90 82 48)(2 86 83 49)(3 87 84 50)(4 88 85 46)(5 89 81 47)(6 91 29 98)(7 92 30 99)(8 93 26 100)(9 94 27 96)(10 95 28 97)(11 44 52 60)(12 45 53 56)(13 41 54 57)(14 42 55 58)(15 43 51 59)(16 78 75 36)(17 79 71 37)(18 80 72 38)(19 76 73 39)(20 77 74 40)(21 31 62 70)(22 32 63 66)(23 33 64 67)(24 34 65 68)(25 35 61 69)
(11 52)(12 53)(13 54)(14 55)(15 51)(21 62)(22 63)(23 64)(24 65)(25 61)(36 78)(37 79)(38 80)(39 76)(40 77)(46 88)(47 89)(48 90)(49 86)(50 87)(91 98)(92 99)(93 100)(94 96)(95 97)

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

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

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

D4×C52 is a maximal subgroup of   C527D8  C528SD16  C20.D10

125 conjugacy classes

 class 1 2A 2B 2C 4 5A ··· 5X 10A ··· 10X 10Y ··· 10BT 20A ··· 20X order 1 2 2 2 4 5 ··· 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

125 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C5 C10 C10 D4 C5×D4 kernel D4×C52 C5×C20 C102 C5×D4 C20 C2×C10 C52 C5 # reps 1 1 2 24 24 48 1 24

Matrix representation of D4×C52 in GL3(𝔽41) generated by

 16 0 0 0 10 0 0 0 10
,
 18 0 0 0 10 0 0 0 10
,
 40 0 0 0 10 30 0 39 31
,
 40 0 0 0 1 10 0 0 40
G:=sub<GL(3,GF(41))| [16,0,0,0,10,0,0,0,10],[18,0,0,0,10,0,0,0,10],[40,0,0,0,10,39,0,30,31],[40,0,0,0,1,0,0,10,40] >;

D4×C52 in GAP, Magma, Sage, TeX

D_4\times C_5^2
% in TeX

G:=Group("D4xC5^2");
// GroupNames label

G:=SmallGroup(200,38);
// by ID

G=gap.SmallGroup(200,38);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-2,1021]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^4=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

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