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## G = Dic5⋊4D4order 160 = 25·5

### 1st semidirect product of Dic5 and D4 acting through Inn(Dic5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Dic5⋊4D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — Dic5⋊4D4
 Lower central C5 — C10 — Dic5⋊4D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for Dic54D4
G = < a,b,c,d | a10=c4=d2=1, b2=a5, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 280 in 94 conjugacy classes, 43 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C4×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, Dic54D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C4×D5, C22×D5, C2×C4×D5, D4×D5, D42D5, Dic54D4

Smallest permutation representation of Dic54D4
On 80 points
Generators in S80
(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)
(1 38 6 33)(2 37 7 32)(3 36 8 31)(4 35 9 40)(5 34 10 39)(11 66 16 61)(12 65 17 70)(13 64 18 69)(14 63 19 68)(15 62 20 67)(21 45 26 50)(22 44 27 49)(23 43 28 48)(24 42 29 47)(25 41 30 46)(51 75 56 80)(52 74 57 79)(53 73 58 78)(54 72 59 77)(55 71 60 76)
(1 63 28 53)(2 62 29 52)(3 61 30 51)(4 70 21 60)(5 69 22 59)(6 68 23 58)(7 67 24 57)(8 66 25 56)(9 65 26 55)(10 64 27 54)(11 46 75 36)(12 45 76 35)(13 44 77 34)(14 43 78 33)(15 42 79 32)(16 41 80 31)(17 50 71 40)(18 49 72 39)(19 48 73 38)(20 47 74 37)
(1 58)(2 59)(3 60)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 41)(18 42)(19 43)(20 44)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 size 1 1 1 1 2 2 10 10 2 2 2 2 5 5 5 5 10 10 10 10 2 2 2 ··· 2 4 4 4 4 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D5 C4○D4 D10 D10 C4×D5 D4×D5 D4⋊2D5 kernel Dic5⋊4D4 C4×Dic5 C10.D4 D10⋊C4 C5×C22⋊C4 C2×C4×D5 C22×Dic5 C2×C5⋊D4 C5⋊D4 Dic5 C22⋊C4 C10 C2×C4 C23 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 2 2 4 2 8 2 2

Matrix representation of Dic54D4 in GL4(𝔽41) generated by

 0 40 0 0 1 35 0 0 0 0 40 0 0 0 0 40
,
 9 0 0 0 13 32 0 0 0 0 32 0 0 0 0 32
,
 1 0 0 0 6 40 0 0 0 0 1 4 0 0 20 40
,
 40 0 0 0 0 40 0 0 0 0 40 37 0 0 0 1
G:=sub<GL(4,GF(41))| [0,1,0,0,40,35,0,0,0,0,40,0,0,0,0,40],[9,13,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[1,6,0,0,0,40,0,0,0,0,1,20,0,0,4,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,37,1] >;

Dic54D4 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_4D_4
% in TeX

G:=Group("Dic5:4D4");
// GroupNames label

G:=SmallGroup(160,102);
// by ID

G=gap.SmallGroup(160,102);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,188,50,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^4=d^2=1,b^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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