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G = D4×Dic5order 160 = 25·5

Direct product of D4 and Dic5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4×Dic5, C23.17D10, C56(C4×D4), C206(C2×C4), (C5×D4)⋊6C4, C2.5(D4×D5), (C2×D4).7D5, C41(C2×Dic5), C4⋊Dic513C2, (C4×Dic5)⋊4C2, (D4×C10).4C2, (C2×C4).49D10, C10.37(C2×D4), C23.D57C2, C221(C2×Dic5), C10.28(C4○D4), C2.5(D42D5), (C2×C20).32C22, C10.38(C22×C4), (C2×C10).49C23, (C22×Dic5)⋊4C2, C2.6(C22×Dic5), C22.25(C22×D5), (C22×C10).17C22, (C2×Dic5).40C22, (C2×C10)⋊6(C2×C4), SmallGroup(160,155)

Series: Derived Chief Lower central Upper central

C1C10 — D4×Dic5
C1C5C10C2×C10C2×Dic5C22×Dic5 — D4×Dic5
C5C10 — D4×Dic5
C1C22C2×D4

Generators and relations for D4×Dic5
 G = < a,b,c,d | a4=b2=c10=1, d2=c5, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 232 in 94 conjugacy classes, 51 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×4], C22 [×4], C5, C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×D4, C2×Dic5 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×4], C22×C10 [×2], C4×Dic5, C4⋊Dic5, C23.D5 [×2], C22×Dic5 [×2], D4×C10, D4×Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C2×Dic5 [×6], C22×D5, D4×D5, D42D5, C22×Dic5, D4×Dic5

Smallest permutation representation of D4×Dic5
On 80 points
Generators in S80
(1 45 28 38)(2 46 29 39)(3 47 30 40)(4 48 21 31)(5 49 22 32)(6 50 23 33)(7 41 24 34)(8 42 25 35)(9 43 26 36)(10 44 27 37)(11 62 77 57)(12 63 78 58)(13 64 79 59)(14 65 80 60)(15 66 71 51)(16 67 72 52)(17 68 73 53)(18 69 74 54)(19 70 75 55)(20 61 76 56)
(1 38)(2 39)(3 40)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 61)(21 48)(22 49)(23 50)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 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 57 6 52)(2 56 7 51)(3 55 8 60)(4 54 9 59)(5 53 10 58)(11 50 16 45)(12 49 17 44)(13 48 18 43)(14 47 19 42)(15 46 20 41)(21 69 26 64)(22 68 27 63)(23 67 28 62)(24 66 29 61)(25 65 30 70)(31 74 36 79)(32 73 37 78)(33 72 38 77)(34 71 39 76)(35 80 40 75)

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

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

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

D4×Dic5 is a maximal subgroup of
Dic5.23D8  Dic54D8  D4.D55C4  Dic56SD16  Dic5.14D8  D4⋊Dic10  D4.Dic10  D4.2Dic10  D4⋊D56C4  Dic5⋊D8  D8⋊Dic5  (C2×D8).D5  Dic53SD16  SD16⋊Dic5  (C5×D4).D4  C5⋊C87D4  C202M4(2)  C4×D42D5  D45Dic10  D46Dic10  C4×D4×D5  C4211D10  C42.108D10  C24.56D10  C24.32D10  C24.33D10  C24.35D10  C20⋊(C4○D4)  Dic1019D4  C4⋊C4.178D10  C10.342+ 1+4  C10.352+ 1+4  C10.362+ 1+4  C4⋊C421D10  C10.732- 1+4  C10.432+ 1+4  C10.452+ 1+4  C10.462+ 1+4  C10.1152+ 1+4  C10.472+ 1+4  C4⋊C4.197D10  C10.802- 1+4  C10.1222+ 1+4  C10.852- 1+4  C42.139D10  C42.234D10  C42.143D10  C42.144D10  C42.166D10  C42.238D10  Dic1011D4  C42.168D10  C24.38D10  C24.42D10  C10.1042- 1+4  C10.1062- 1+4  C10.1452+ 1+4  Dic158D4  Dic1517D4
D4×Dic5 is a maximal quotient of
C24.47D10  C24.8D10  C4⋊C45Dic5  C206(C4⋊C4)  C42.47D10  C207M4(2)  D8⋊Dic5  SD16⋊Dic5  Q16⋊Dic5  D85Dic5  D84Dic5  C24.18D10  C24.19D10  Dic158D4  Dic1517D4

40 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4L5A5B10A···10F10G···10N20A20B20C20D
order122222224444444···45510···1010···1020202020
size1111222222555510···10222···24···44444

40 irreducible representations

dim111111122222244
type+++++++++-++-
imageC1C2C2C2C2C2C4D4D5C4○D4D10Dic5D10D4×D5D42D5
kernelD4×Dic5C4×Dic5C4⋊Dic5C23.D5C22×Dic5D4×C10C5×D4Dic5C2×D4C10C2×C4D4C23C2C2
# reps111221822228422

Matrix representation of D4×Dic5 in GL5(𝔽41)

10000
00100
040000
00010
00001
,
400000
00100
01000
00010
00001
,
400000
01000
00100
000401
000535
,
320000
01000
00100
00007
00060

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,5,0,0,0,1,35],[32,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,6,0,0,0,7,0] >;

D4×Dic5 in GAP, Magma, Sage, TeX

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

G:=Group("D4xDic5");
// GroupNames label

G:=SmallGroup(160,155);
// by ID

G=gap.SmallGroup(160,155);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,188,4613]);
// Polycyclic

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

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