Copied to
clipboard

## G = Dic5⋊D4order 160 = 25·5

### 2nd semidirect product of Dic5 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic5⋊D4
G = < a,b,c,d | a10=c4=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=c-1 >

Subgroups: 304 in 94 conjugacy classes, 35 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C4⋊D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C10.D4, D10⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4, D4×C10, Dic5⋊D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C5⋊D4, C22×D5, D4×D5, D42D5, C2×C5⋊D4, Dic5⋊D4

Smallest permutation representation of Dic5⋊D4
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 48 6 43)(2 47 7 42)(3 46 8 41)(4 45 9 50)(5 44 10 49)(11 54 16 59)(12 53 17 58)(13 52 18 57)(14 51 19 56)(15 60 20 55)(21 33 26 38)(22 32 27 37)(23 31 28 36)(24 40 29 35)(25 39 30 34)(61 73 66 78)(62 72 67 77)(63 71 68 76)(64 80 69 75)(65 79 70 74)
(1 14 28 76)(2 15 29 77)(3 16 30 78)(4 17 21 79)(5 18 22 80)(6 19 23 71)(7 20 24 72)(8 11 25 73)(9 12 26 74)(10 13 27 75)(31 63 43 51)(32 64 44 52)(33 65 45 53)(34 66 46 54)(35 67 47 55)(36 68 48 56)(37 69 49 57)(38 70 50 58)(39 61 41 59)(40 62 42 60)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 51)(7 52)(8 53)(9 54)(10 55)(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), (1,48,6,43)(2,47,7,42)(3,46,8,41)(4,45,9,50)(5,44,10,49)(11,54,16,59)(12,53,17,58)(13,52,18,57)(14,51,19,56)(15,60,20,55)(21,33,26,38)(22,32,27,37)(23,31,28,36)(24,40,29,35)(25,39,30,34)(61,73,66,78)(62,72,67,77)(63,71,68,76)(64,80,69,75)(65,79,70,74), (1,14,28,76)(2,15,29,77)(3,16,30,78)(4,17,21,79)(5,18,22,80)(6,19,23,71)(7,20,24,72)(8,11,25,73)(9,12,26,74)(10,13,27,75)(31,63,43,51)(32,64,44,52)(33,65,45,53)(34,66,46,54)(35,67,47,55)(36,68,48,56)(37,69,49,57)(38,70,50,58)(39,61,41,59)(40,62,42,60), (1,56)(2,57)(3,58)(4,59)(5,60)(6,51)(7,52)(8,53)(9,54)(10,55)(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), (1,48,6,43)(2,47,7,42)(3,46,8,41)(4,45,9,50)(5,44,10,49)(11,54,16,59)(12,53,17,58)(13,52,18,57)(14,51,19,56)(15,60,20,55)(21,33,26,38)(22,32,27,37)(23,31,28,36)(24,40,29,35)(25,39,30,34)(61,73,66,78)(62,72,67,77)(63,71,68,76)(64,80,69,75)(65,79,70,74), (1,14,28,76)(2,15,29,77)(3,16,30,78)(4,17,21,79)(5,18,22,80)(6,19,23,71)(7,20,24,72)(8,11,25,73)(9,12,26,74)(10,13,27,75)(31,63,43,51)(32,64,44,52)(33,65,45,53)(34,66,46,54)(35,67,47,55)(36,68,48,56)(37,69,49,57)(38,70,50,58)(39,61,41,59)(40,62,42,60), (1,56)(2,57)(3,58)(4,59)(5,60)(6,51)(7,52)(8,53)(9,54)(10,55)(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)], [(1,48,6,43),(2,47,7,42),(3,46,8,41),(4,45,9,50),(5,44,10,49),(11,54,16,59),(12,53,17,58),(13,52,18,57),(14,51,19,56),(15,60,20,55),(21,33,26,38),(22,32,27,37),(23,31,28,36),(24,40,29,35),(25,39,30,34),(61,73,66,78),(62,72,67,77),(63,71,68,76),(64,80,69,75),(65,79,70,74)], [(1,14,28,76),(2,15,29,77),(3,16,30,78),(4,17,21,79),(5,18,22,80),(6,19,23,71),(7,20,24,72),(8,11,25,73),(9,12,26,74),(10,13,27,75),(31,63,43,51),(32,64,44,52),(33,65,45,53),(34,66,46,54),(35,67,47,55),(36,68,48,56),(37,69,49,57),(38,70,50,58),(39,61,41,59),(40,62,42,60)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,51),(7,52),(8,53),(9,54),(10,55),(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)]])

34 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 5A 5B 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 20 20 20 size 1 1 1 1 2 2 4 20 4 10 10 10 10 20 2 2 2 ··· 2 4 ··· 4 4 4 4 4

34 irreducible representations

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

Matrix representation of Dic5⋊D4 in GL4(𝔽41) generated by

 0 40 0 0 1 7 0 0 0 0 1 0 0 0 0 1
,
 17 40 0 0 3 24 0 0 0 0 1 0 0 0 0 1
,
 17 40 0 0 1 24 0 0 0 0 0 40 0 0 1 0
,
 40 0 0 0 0 40 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(41))| [0,1,0,0,40,7,0,0,0,0,1,0,0,0,0,1],[17,3,0,0,40,24,0,0,0,0,1,0,0,0,0,1],[17,1,0,0,40,24,0,0,0,0,0,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;

Dic5⋊D4 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes D_4
% in TeX

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

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

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

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,218,188,4613]);
// Polycyclic

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

׿
×
𝔽