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G = C20⋊D4order 160 = 25·5

3rd semidirect product of C20 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C203D4, Dic51D4, C23.10D10, (C2×D4)⋊6D5, (D4×C10)⋊4C2, (C2×D20)⋊9C2, C41(C5⋊D4), C52(C41D4), C2.28(D4×D5), (C4×Dic5)⋊6C2, C10.52(C2×D4), (C2×C4).52D10, (C2×C10).55C23, (C2×C20).35C22, C22.62(C22×D5), (C22×C10).22C22, (C2×Dic5).42C22, (C22×D5).12C22, (C2×C5⋊D4)⋊7C2, C2.16(C2×C5⋊D4), SmallGroup(160,161)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20⋊D4
C1C5C10C2×C10C22×D5C2×D20 — C20⋊D4
C5C2×C10 — C20⋊D4
C1C22C2×D4

Generators and relations for C20⋊D4
 G = < a,b,c | a20=b4=c2=1, bab-1=a9, cac=a-1, cbc=b-1 >

Subgroups: 400 in 108 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×12], C5, C2×C4, C2×C4 [×2], D4 [×12], C23 [×2], C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C42, C2×D4, C2×D4 [×5], Dic5 [×4], C20 [×2], D10 [×6], C2×C10, C2×C10 [×6], C41D4, D20 [×2], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20, C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], C4×Dic5, C2×D20, C2×C5⋊D4 [×4], D4×C10, C20⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C41D4, C5⋊D4 [×2], C22×D5, D4×D5 [×2], C2×C5⋊D4, C20⋊D4

Smallest permutation representation of C20⋊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 51 66 35)(2 60 67 24)(3 49 68 33)(4 58 69 22)(5 47 70 31)(6 56 71 40)(7 45 72 29)(8 54 73 38)(9 43 74 27)(10 52 75 36)(11 41 76 25)(12 50 77 34)(13 59 78 23)(14 48 79 32)(15 57 80 21)(16 46 61 30)(17 55 62 39)(18 44 63 28)(19 53 64 37)(20 42 65 26)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 45)(22 44)(23 43)(24 42)(25 41)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)

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

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

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

C20⋊D4 is a maximal subgroup of
C23.2D20  (C2×D4)⋊F5  Dic5.SD16  D201D4  Dic5.5D8  Dic102D4  C4⋊C4.D10  D203D4  Dic5⋊D8  C405D4  C4011D4  Dic55SD16  C4015D4  C409D4  D2018D4  2+ 1+4⋊D5  C42.228D10  Dic1024D4  C42.114D10  C42.116D10  C244D10  C24.34D10  C24.36D10  C20⋊(C4○D4)  Dic1020D4  C10.382+ 1+4  D2019D4  C10.442+ 1+4  C10.472+ 1+4  C10.482+ 1+4  C10.662+ 1+4  C10.672+ 1+4  C10.682+ 1+4  C42.233D10  C4218D10  C42.143D10  D5×C41D4  C4226D10  Dic1011D4  D4×C5⋊D4  C24.41D10  C10.1462+ 1+4  (C2×C20)⋊17D4  C10.1482+ 1+4  C20⋊D12  C6010D4  Dic155D4  C603D4
C20⋊D4 is a maximal quotient of
C24.7D10  C24.13D10  C232D20  C205(C4⋊C4)  (C2×D20)⋊22C4  (C2×C20).289D4  C42.64D10  C42.214D10  C42.65D10  C20⋊D8  C42.74D10  C204SD16  C206SD16  C42.80D10  C203Q16  C405D4  C4011D4  C40.22D4  C40.31D4  C40.43D4  C4015D4  C409D4  C40.26D4  C40.37D4  C40.28D4  C24.19D10  C24.21D10  C20⋊D12  C6010D4  Dic155D4  C603D4

34 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B10A···10F10G···10N20A20B20C20D
order122222224444445510···1010···1020202020
size11114420202210101010222···24···44444

34 irreducible representations

dim111112222224
type+++++++++++
imageC1C2C2C2C2D4D4D5D10D10C5⋊D4D4×D5
kernelC20⋊D4C4×Dic5C2×D20C2×C5⋊D4D4×C10Dic5C20C2×D4C2×C4C23C4C2
# reps111414222484

Matrix representation of C20⋊D4 in GL4(𝔽41) generated by

7100
40000
0001
00400
,
174000
32400
00040
0010
,
1000
344000
0010
00040
G:=sub<GL(4,GF(41))| [7,40,0,0,1,0,0,0,0,0,0,40,0,0,1,0],[17,3,0,0,40,24,0,0,0,0,0,1,0,0,40,0],[1,34,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;

C20⋊D4 in GAP, Magma, Sage, TeX

C_{20}\rtimes D_4
% in TeX

G:=Group("C20:D4");
// GroupNames label

G:=SmallGroup(160,161);
// by ID

G=gap.SmallGroup(160,161);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,218,188,4613]);
// Polycyclic

G:=Group<a,b,c|a^20=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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