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G = D10⋊D4order 160 = 25·5

1st semidirect product of D10 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D101D4, Dic52D4, C23.5D10, C2.9(D4×D5), (C2×D20)⋊3C2, C51(C4⋊D4), C22⋊C44D5, (C2×C4).7D10, C10.20(C2×D4), C10.9(C4○D4), C10.D45C2, D10⋊C411C2, C2.11(C4○D20), (C2×C20).53C22, (C2×C10).25C23, (C2×Dic5).7C22, C22.43(C22×D5), (C22×C10).14C22, (C22×D5).21C22, (C2×C4×D5)⋊11C2, (C2×C5⋊D4)⋊2C2, (C5×C22⋊C4)⋊6C2, SmallGroup(160,105)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D10⋊D4
C1C5C10C2×C10C22×D5C2×C4×D5 — D10⋊D4
C5C2×C10 — D10⋊D4
C1C22C22⋊C4

Generators and relations for D10⋊D4
 G = < a,b,c,d | a10=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=a8b, dbd=a3b, dcd=c-1 >

Subgroups: 352 in 94 conjugacy classes, 33 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×5], C22, C22 [×10], C5, C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], D5 [×3], C10 [×3], C10, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×2], Dic5, C20 [×2], D10 [×2], D10 [×5], C2×C10, C2×C10 [×3], C4⋊D4, C4×D5 [×2], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C22×D5 [×2], C22×C10, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4 [×2], D10⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C22×D5, C4○D20, D4×D5 [×2], D10⋊D4

Smallest permutation representation of D10⋊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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 80)(9 79)(10 78)(11 22)(12 21)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 60)(41 63)(42 62)(43 61)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)
(1 38 28 50)(2 37 29 49)(3 36 30 48)(4 35 21 47)(5 34 22 46)(6 33 23 45)(7 32 24 44)(8 31 25 43)(9 40 26 42)(10 39 27 41)(11 70 73 58)(12 69 74 57)(13 68 75 56)(14 67 76 55)(15 66 77 54)(16 65 78 53)(17 64 79 52)(18 63 80 51)(19 62 71 60)(20 61 72 59)
(1 23)(2 22)(3 21)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 73)(12 72)(13 71)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(31 40)(32 39)(33 38)(34 37)(35 36)(41 44)(42 43)(45 50)(46 49)(47 48)(51 55)(52 54)(56 60)(57 59)(61 69)(62 68)(63 67)(64 66)

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

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

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

D10⋊D4 is a maximal subgroup of
C24.27D10  C24.30D10  C42.93D10  C42.95D10  C42.97D10  C42.100D10  C42.104D10  C4212D10  C42.228D10  D2023D4  Dic1024D4  C42.113D10  C4217D10  C42.116D10  C243D10  C244D10  C24.34D10  C24.36D10  C10.682- 1+4  Dic1020D4  D5×C4⋊D4  C10.392+ 1+4  D2019D4  C10.402+ 1+4  C10.442+ 1+4  C10.482+ 1+4  C10.172- 1+4  D2022D4  Dic1022D4  C10.202- 1+4  C10.222- 1+4  C10.262- 1+4  C10.1202+ 1+4  C10.1212+ 1+4  C10.822- 1+4  C4⋊C428D10  C10.612+ 1+4  C10.642+ 1+4  C10.662+ 1+4  C10.682+ 1+4  C10.692+ 1+4  C42.233D10  C42.138D10  C4218D10  D2010D4  Dic1010D4  C4220D10  C42.145D10  C4223D10  C42.189D10  C42.161D10  C42.163D10  C42.164D10  C4225D10  Dic15⋊D4  D30⋊D4  D10⋊D12  D3012D4  D306D4  Dic153D4  D309D4
D10⋊D4 is a maximal quotient of
C10.49(C4×D4)  C2.(C20⋊Q8)  (C2×C4)⋊9D20  D102(C4⋊C4)  C10.54(C4×D4)  (C2×Dic5)⋊3D4  C10.(C4⋊D4)  (C22×D5).Q8  D20.2D4  D20.3D4  D20.6D4  D20.7D4  Dic102D4  Dic10.D4  D10⋊D8  D10⋊SD16  C52C8⋊D4  C5⋊(C82D4)  D203D4  D20.D4  Dic5⋊Q16  Dic10.11D4  D102SD16  C5⋊(C8⋊D4)  D10⋊Q16  C52C8.D4  Dic5⋊SD16  D20.12D4  C24.6D10  C24.8D10  C24.12D10  C24.13D10  C24.14D10  C232D20  Dic15⋊D4  D30⋊D4  D10⋊D12  D3012D4  D306D4  Dic153D4  D309D4

34 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B10A···10F10G10H10I10J20A···20H
order122222224444445510···101010101020···20
size11114101020224101020222···244444···4

34 irreducible representations

dim111111122222224
type+++++++++++++
imageC1C2C2C2C2C2C2D4D4D5C4○D4D10D10C4○D20D4×D5
kernelD10⋊D4C10.D4D10⋊C4C5×C22⋊C4C2×C4×D5C2×D20C2×C5⋊D4Dic5D10C22⋊C4C10C2×C4C23C2C2
# reps111111222224284

Matrix representation of D10⋊D4 in GL4(𝔽41) generated by

343400
7100
00400
00040
,
142700
112700
00329
00239
,
174000
32400
00400
00040
,
1000
344000
00400
00391
G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,40,0,0,0,0,40],[14,11,0,0,27,27,0,0,0,0,32,23,0,0,9,9],[17,3,0,0,40,24,0,0,0,0,40,0,0,0,0,40],[1,34,0,0,0,40,0,0,0,0,40,39,0,0,0,1] >;

D10⋊D4 in GAP, Magma, Sage, TeX

D_{10}\rtimes D_4
% in TeX

G:=Group("D10:D4");
// GroupNames label

G:=SmallGroup(160,105);
// by ID

G=gap.SmallGroup(160,105);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,55,506,188,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^8*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations

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