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G = Dic5:D4  order 160 = 25·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic5:3D4, C23.9D10, (C2xD4):5D5, (C2xC10):3D4, (D4xC10):9C2, C5:5(C4:D4), C2.27(D4xD5), C10.51(C2xD4), (C2xC4).19D10, C22:1(C5:D4), C23.D5:12C2, D10:C4:15C2, C10.32(C4oD4), C10.D4:15C2, (C2xC10).54C23, (C2xC20).62C22, (C22xDic5):6C2, C2.18(D4:2D5), C22.61(C22xD5), (C22xC10).21C22, (C2xDic5).41C22, (C22xD5).11C22, (C2xC5:D4):6C2, C2.15(C2xC5:D4), SmallGroup(160,160)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC10 — Dic5:D4
Chief series
C1 — C5 — C10 — C2xC10 — C22xD5 — C2xC5:D4 — Dic5:D4
Lower central
C5 — C2xC10 — Dic5:D4
Upper central
C1 — C22 — C2xD4

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, C2xC4, C2xC4, D4, C23, C23, D5, C10, C10, C22:C4, C4:C4, C22xC4, C2xD4, C2xD4, Dic5, Dic5, C20, D10, C2xC10, C2xC10, C2xC10, C4:D4, C2xDic5, C2xDic5, C5:D4, C2xC20, C5xD4, C22xD5, C22xC10, C10.D4, D10:C4, C23.D5, C22xDic5, C2xC5:D4, D4xC10, Dic5:D4
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C4oD4, D10, C4:D4, C5:D4, C22xD5, D4xD5, D4:2D5, C2xC5: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)]])
 

Dic5:D4 is a maximal subgroup of
 C42.102D10  C42.104D10  C42:12D10  Dic10:23D4  C42:17D10  C42.118D10  C42.119D10  C24.56D10  C24:4D10  C24.33D10  C24.34D10  C24.35D10  C24.36D10  C20:(C4oD4)  C10.682- 1+4  Dic10:19D4  Dic10:20D4  C10.342+ 1+4  D5xC4:D4  C10.372+ 1+4  C10.392+ 1+4  C10.402+ 1+4  C10.422+ 1+4  C10.442+ 1+4  C10.452+ 1+4  C10.462+ 1+4  C10.1152+ 1+4  C10.472+ 1+4  C10.482+ 1+4  C10.742- 1+4  C4:C4.197D10  C10.1212+ 1+4  C10.822- 1+4  C10.642+ 1+4  C10.842- 1+4  C10.662+ 1+4  C10.852- 1+4  C10.692+ 1+4  C42.137D10  C42.138D10  C42:20D10  C42.145D10  C42:26D10  Dic10:11D4  C42.168D10  C42:28D10  D4xC5:D4  C24:8D10  C24.42D10  C10.1042- 1+4  C10.1452+ 1+4  (C2xC20):17D4  C10.1472+ 1+4  Dic5:D12  Dic15:2D4  Dic15:4D4  (S3xC10):D4  (C2xC10):4D12  Dic15:18D4  Dic15:12D4  Dic5:S4
Dic5:D4 is a maximal quotient of
 C24.44D10  C24.4D10  C24.6D10  C24.9D10  C24.13D10  C23.45D20  C24.14D10  C24.16D10  C10.96(C4xD4)  (C2xC20).54D4  C10.90(C4xD4)  (C2xC20).56D4  (C2xC10):D8  C4:D4:D5  C5:2C8:23D4  C4.(D4xD5)  C5:2C8:24D4  C22:Q8:D5  (C2xC10):Q16  C5:(C8.D4)  Dic5:D8  (C2xD8).D5  Dic5:3SD16  Dic5:5SD16  (C5xD4).D4  (C5xQ8).D4  Dic5:3Q16  (C2xQ16):D5  M4(2).D10  M4(2).13D10  M4(2).15D10  M4(2).16D10  C24.18D10  C24.20D10  C24.21D10  Dic5:D12  Dic15:2D4  Dic15:4D4  (S3xC10):D4  (C2xC10):4D12  Dic15:18D4  Dic15:12D4

34 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B10A···10F10G···10N20A20B20C20D
order122222224444445510···1010···1020202020
size11112242041010101020222···24···44444

34 irreducible representations

dim1111111222222244
type+++++++++++++-
imageC1C2C2C2C2C2C2D4D4D5C4oD4D10D10C5:D4D4xD5D4:2D5
kernelDic5:D4C10.D4D10:C4C23.D5C22xDic5C2xC5:D4D4xC10Dic5C2xC10C2xD4C10C2xC4C23C22C2C2
# reps1111121222224822

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

04000
1700
0010
0001
,
174000
32400
0010
0001
,
174000
12400
00040
0010
,
40000
04000
0001
0010
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
 

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