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## G = D4×D25order 400 = 24·52

### Direct product of D4 and D25

Series: Derived Chief Lower central Upper central

 Derived series C1 — C50 — D4×D25
 Chief series C1 — C5 — C25 — C50 — D50 — C22×D25 — D4×D25
 Lower central C25 — C50 — D4×D25
 Upper central C1 — C2 — D4

Generators and relations for D4×D25
G = < a,b,c,d | a4=b2=c25=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 805 in 81 conjugacy classes, 31 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, D4, D4, C23, D5, C10, C10, C2×D4, Dic5, C20, D10, C2×C10, C25, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, D25, D25, C50, C50, D4×D5, Dic25, C100, D50, D50, D50, C2×C50, C4×D25, D100, C25⋊D4, D4×C25, C22×D25, D4×D25
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22×D5, D25, D4×D5, D50, C22×D25, D4×D25

Smallest permutation representation of D4×D25
On 100 points
Generators in S100
(1 80 26 61)(2 81 27 62)(3 82 28 63)(4 83 29 64)(5 84 30 65)(6 85 31 66)(7 86 32 67)(8 87 33 68)(9 88 34 69)(10 89 35 70)(11 90 36 71)(12 91 37 72)(13 92 38 73)(14 93 39 74)(15 94 40 75)(16 95 41 51)(17 96 42 52)(18 97 43 53)(19 98 44 54)(20 99 45 55)(21 100 46 56)(22 76 47 57)(23 77 48 58)(24 78 49 59)(25 79 50 60)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 80)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 76)(48 77)(49 78)(50 79)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(71 75)(72 74)(76 83)(77 82)(78 81)(79 80)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 10A 10B 10C 10D 10E 10F 20A 20B 25A ··· 25J 50A ··· 50J 50K ··· 50AD 100A ··· 100J order 1 2 2 2 2 2 2 2 4 4 5 5 10 10 10 10 10 10 20 20 25 ··· 25 50 ··· 50 50 ··· 50 100 ··· 100 size 1 1 2 2 25 25 50 50 2 50 2 2 2 2 4 4 4 4 4 4 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D5 D10 D10 D25 D50 D50 D4×D5 D4×D25 kernel D4×D25 C4×D25 D100 C25⋊D4 D4×C25 C22×D25 D25 C5×D4 C20 C2×C10 D4 C4 C22 C5 C1 # reps 1 1 1 2 1 2 2 2 2 4 10 10 20 2 10

Matrix representation of D4×D25 in GL4(𝔽101) generated by

 100 0 0 0 0 100 0 0 0 0 0 1 0 0 100 0
,
 100 0 0 0 0 100 0 0 0 0 0 1 0 0 1 0
,
 21 11 0 0 39 83 0 0 0 0 1 0 0 0 0 1
,
 80 32 0 0 62 21 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(101))| [100,0,0,0,0,100,0,0,0,0,0,100,0,0,1,0],[100,0,0,0,0,100,0,0,0,0,0,1,0,0,1,0],[21,39,0,0,11,83,0,0,0,0,1,0,0,0,0,1],[80,62,0,0,32,21,0,0,0,0,1,0,0,0,0,1] >;

D4×D25 in GAP, Magma, Sage, TeX

D_4\times D_{25}
% in TeX

G:=Group("D4xD25");
// GroupNames label

G:=SmallGroup(400,39);
// by ID

G=gap.SmallGroup(400,39);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,116,4324,628,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^25=d^2=1,b*a*b=a^-1,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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