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G = C4.D20order 160 = 25·5

5th non-split extension by C4 of D20 acting via D20/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4.5D20, C426D5, C20.28D4, (C4×C20)⋊5C2, C2.6(C2×D20), C10.4(C2×D4), (C2×D20).3C2, (C2×C4).77D10, C51(C4.4D4), D10⋊C41C2, (C2×Dic10)⋊1C2, C2.7(C4○D20), C10.5(C4○D4), (C2×C20).74C22, (C2×C10).16C23, (C2×Dic5).3C22, (C22×D5).2C22, C22.37(C22×D5), SmallGroup(160,96)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4.D20
C1C5C10C2×C10C22×D5D10⋊C4 — C4.D20
C5C2×C10 — C4.D20
C1C22C42

Generators and relations for C4.D20
 G = < a,b,c | a4=b20=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >

Subgroups: 288 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C4.4D4, Dic10 [×2], D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], D10⋊C4 [×4], C4×C20, C2×Dic10, C2×D20, C4.D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, D20 [×2], C22×D5, C2×D20, C4○D20 [×2], C4.D20

Smallest permutation representation of C4.D20
On 80 points
Generators in S80
(1 40 42 63)(2 21 43 64)(3 22 44 65)(4 23 45 66)(5 24 46 67)(6 25 47 68)(7 26 48 69)(8 27 49 70)(9 28 50 71)(10 29 51 72)(11 30 52 73)(12 31 53 74)(13 32 54 75)(14 33 55 76)(15 34 56 77)(16 35 57 78)(17 36 58 79)(18 37 59 80)(19 38 60 61)(20 39 41 62)
(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 41 42 20)(2 19 43 60)(3 59 44 18)(4 17 45 58)(5 57 46 16)(6 15 47 56)(7 55 48 14)(8 13 49 54)(9 53 50 12)(10 11 51 52)(21 61 64 38)(22 37 65 80)(23 79 66 36)(24 35 67 78)(25 77 68 34)(26 33 69 76)(27 75 70 32)(28 31 71 74)(29 73 72 30)(39 63 62 40)

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

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

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

C4.D20 is a maximal subgroup of
C42.D10  C422F5  C42.2F5  C8.8D20  C42.264D10  C8⋊D20  C42.20D10  C8.D20  M4(2)⋊D10  D4.10D20  D20.19D4  C42.36D10  D4.1D20  Q8.1D20  C42.214D10  C42.216D10  C42.74D10  C42.80D10  C42.82D10  C42.276D10  C42.277D10  C429D10  C42.92D10  C4210D10  C42.97D10  C42.99D10  D2023D4  Dic1023D4  D45D20  C42.114D10  C4217D10  C42.122D10  Q85D20  C42.133D10  C42.135D10  C42.136D10  C42.233D10  D5×C4.4D4  C4222D10  C42.145D10  C42.237D10  C42.157D10  C42.158D10  C4223D10  C42.164D10  C4226D10  C42.171D10  C42.178D10  Dic3.D20  C60.44D4  C60.47D4  C427D15
C4.D20 is a maximal quotient of
(C2×C20).28D4  (C2×Dic5)⋊3D4  (C2×C4).20D20  (C2×C4).21D20  C20.14Q16  C4.5D40  C42.264D10  C42.14D10  C42.19D10  C42.20D10  (C2×C20)⋊10Q8  C429Dic5  (C2×C4)⋊6D20  (C2×C42)⋊D5  Dic3.D20  C60.44D4  C60.47D4  C427D15

46 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H5A5B10A···10F20A···20X
order1222224···4445510···1020···20
size111120202···22020222···22···2

46 irreducible representations

dim11111222222
type+++++++++
imageC1C2C2C2C2D4D5C4○D4D10D20C4○D20
kernelC4.D20D10⋊C4C4×C20C2×Dic10C2×D20C20C42C10C2×C4C4C2
# reps141112246816

Matrix representation of C4.D20 in GL4(𝔽41) generated by

21300
283900
00400
00040
,
62300
182100
00040
0010
,
62300
183500
0001
0010
G:=sub<GL(4,GF(41))| [2,28,0,0,13,39,0,0,0,0,40,0,0,0,0,40],[6,18,0,0,23,21,0,0,0,0,0,1,0,0,40,0],[6,18,0,0,23,35,0,0,0,0,0,1,0,0,1,0] >;

C4.D20 in GAP, Magma, Sage, TeX

C_4.D_{20}
% in TeX

G:=Group("C4.D20");
// GroupNames label

G:=SmallGroup(160,96);
// by ID

G=gap.SmallGroup(160,96);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,55,218,86,4613]);
// Polycyclic

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

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