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## G = D4×C5⋊D5order 400 = 24·52

### Direct product of D4 and C5⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — D4×C5⋊D5
 Chief series C1 — C5 — C52 — C5×C10 — C2×C5⋊D5 — C22×C5⋊D5 — D4×C5⋊D5
 Lower central C52 — C5×C10 — D4×C5⋊D5
 Upper central C1 — C2 — D4

Generators and relations for D4×C5⋊D5
G = < a,b,c,d,e | a4=b2=c5=d5=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1480 in 216 conjugacy classes, 61 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, D4, D4, C23, D5, C10, C10, C2×D4, Dic5, C20, D10, C2×C10, C52, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, C5⋊D5, C5⋊D5, C5×C10, C5×C10, D4×D5, C526C4, C5×C20, C2×C5⋊D5, C2×C5⋊D5, C2×C5⋊D5, C102, C4×C5⋊D5, C20⋊D5, C527D4, D4×C52, C22×C5⋊D5, D4×C5⋊D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22×D5, C5⋊D5, D4×D5, C2×C5⋊D5, C22×C5⋊D5, D4×C5⋊D5

Smallest permutation representation of D4×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 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 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), (1,54)(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), (1,2,3,4,5)(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), (1,54)(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), (1,2,3,4,5)(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)], [(1,54),(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)], [(1,2,3,4,5),(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)]])

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A ··· 5L 10A ··· 10L 10M ··· 10AJ 20A ··· 20L order 1 2 2 2 2 2 2 2 4 4 5 ··· 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 2 2 25 25 50 50 2 50 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D5 D10 D10 D4×D5 kernel D4×C5⋊D5 C4×C5⋊D5 C20⋊D5 C52⋊7D4 D4×C52 C22×C5⋊D5 C5⋊D5 C5×D4 C20 C2×C10 C5 # reps 1 1 1 2 1 2 2 12 12 24 12

Matrix representation of D4×C5⋊D5 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 7 1 0 0 0 0 33 40 0 0 0 0 0 0 40 1 0 0 0 0 5 35 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 6 0 0 0 0 34 34 0 0 0 0 0 0 40 1 0 0 0 0 5 35 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 33 40 0 0 0 0 0 0 40 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[7,33,0,0,0,0,1,40,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,34,0,0,0,0,6,34,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,33,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D4×C5⋊D5 in GAP, Magma, Sage, TeX

D_4\times C_5\rtimes D_5
% in TeX

G:=Group("D4xC5:D5");
// GroupNames label

G:=SmallGroup(400,195);
// by ID

G=gap.SmallGroup(400,195);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,116,1924,11525]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=c^5=d^5=e^2=1,b*a*b=a^-1,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

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