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G = C202D4order 160 = 25·5

2nd semidirect product of C20 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C202D4, D103D4, C23.8D10, (C2×D4)⋊4D5, (D4×C10)⋊3C2, C42(C5⋊D4), C54(C4⋊D4), C2.26(D4×D5), C4⋊Dic514C2, (C2×C4).51D10, C10.50(C2×D4), C23.D511C2, C10.31(C4○D4), (C2×C10).53C23, (C2×C20).34C22, C2.17(D42D5), C22.60(C22×D5), (C22×C10).20C22, (C2×Dic5).19C22, (C22×D5).29C22, (C2×C4×D5)⋊2C2, (C2×C5⋊D4)⋊5C2, C2.14(C2×C5⋊D4), SmallGroup(160,159)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C202D4
C1C5C10C2×C10C22×D5C2×C4×D5 — C202D4
C5C2×C10 — C202D4
C1C22C2×D4

Generators and relations for C202D4
 G = < a,b,c | a20=b4=c2=1, bab-1=a-1, cac=a9, cbc=b-1 >

Subgroups: 304 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×10], C5, C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, D5 [×2], C10 [×3], C10 [×2], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], Dic5 [×3], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×6], C4⋊D4, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], C4⋊Dic5, C23.D5 [×2], C2×C4×D5, C2×C5⋊D4 [×2], D4×C10, C202D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C5⋊D4 [×2], C22×D5, D4×D5, D42D5, C2×C5⋊D4, C202D4

Smallest permutation representation of C202D4
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 76 56 23)(2 75 57 22)(3 74 58 21)(4 73 59 40)(5 72 60 39)(6 71 41 38)(7 70 42 37)(8 69 43 36)(9 68 44 35)(10 67 45 34)(11 66 46 33)(12 65 47 32)(13 64 48 31)(14 63 49 30)(15 62 50 29)(16 61 51 28)(17 80 52 27)(18 79 53 26)(19 78 54 25)(20 77 55 24)
(1 11)(2 20)(3 9)(4 18)(5 7)(6 16)(8 14)(10 12)(13 19)(15 17)(21 68)(22 77)(23 66)(24 75)(25 64)(26 73)(27 62)(28 71)(29 80)(30 69)(31 78)(32 67)(33 76)(34 65)(35 74)(36 63)(37 72)(38 61)(39 70)(40 79)(41 51)(42 60)(43 49)(44 58)(45 47)(46 56)(48 54)(50 52)(53 59)(55 57)

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

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

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

C202D4 is a maximal subgroup of
D10.SD16  D10.12D8  D10⋊D8  D10.16SD16  D10⋊SD16  C406C4⋊C2  C52C8⋊D4  C5⋊(C82D4)  C405C4⋊C2  D20⋊D4  C406D4  Dic10⋊D4  C4012D4  D108SD16  C4014D4  D207D4  C408D4  C42.228D10  D2024D4  C42.229D10  C42.113D10  C42.115D10  C42.116D10  C42.117D10  C243D10  C24.33D10  C24.35D10  C24.36D10  D5×C4⋊D4  C4⋊C421D10  C10.382+ 1+4  C10.392+ 1+4  C10.732- 1+4  D2020D4  C10.422+ 1+4  C10.432+ 1+4  C10.442+ 1+4  C10.452+ 1+4  C10.1152+ 1+4  C10.472+ 1+4  C10.482+ 1+4  C10.742- 1+4  C10.612+ 1+4  C10.632+ 1+4  C10.642+ 1+4  C10.692+ 1+4  D2010D4  Dic1010D4  C42.234D10  C42.144D10  C42.238D10  D2011D4  Dic1011D4  C42.168D10  D4×C5⋊D4  C24.41D10  C24.42D10  (C2×C20)⋊15D4  C10.1072- 1+4  C10.1472+ 1+4  C10.1482+ 1+4  C60⋊D4  C202D12  (C6×D5)⋊D4  D307D4  C602D4
C202D4 is a maximal quotient of
C24.3D10  C24.6D10  C24.12D10  C24.16D10  C204(C4⋊C4)  (C2×C20).287D4  D104(C4⋊C4)  (C2×C20).290D4  C42.61D10  D20.23D4  C202D8  Dic109D4  C205SD16  C20⋊Q16  C406D4  C4012D4  C40.23D4  C4014D4  C408D4  C40.44D4  D103Q16  C40.36D4  C40.29D4  C24.19D10  C24.20D10  C24.21D10  C60⋊D4  C202D12  (C6×D5)⋊D4  D307D4  C602D4

34 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B10A···10F10G···10N20A20B20C20D
order122222224444445510···1010···1020202020
size11114410102210102020222···24···44444

34 irreducible representations

dim111111222222244
type++++++++++++-
imageC1C2C2C2C2C2D4D4D5C4○D4D10D10C5⋊D4D4×D5D42D5
kernelC202D4C4⋊Dic5C23.D5C2×C4×D5C2×C5⋊D4D4×C10C20D10C2×D4C10C2×C4C23C4C2C2
# reps112121222224822

Matrix representation of C202D4 in GL4(𝔽41) generated by

1100
5600
00814
001033
,
20300
32100
00123
00040
,
6700
363500
00400
00040
G:=sub<GL(4,GF(41))| [1,5,0,0,1,6,0,0,0,0,8,10,0,0,14,33],[20,3,0,0,3,21,0,0,0,0,1,0,0,0,23,40],[6,36,0,0,7,35,0,0,0,0,40,0,0,0,0,40] >;

C202D4 in GAP, Magma, Sage, TeX

C_{20}\rtimes_2D_4
% in TeX

G:=Group("C20:2D4");
// GroupNames label

G:=SmallGroup(160,159);
// by ID

G=gap.SmallGroup(160,159);
# by ID

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

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

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