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G = D20.8D4order 320 = 26·5

8th non-split extension by D20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.8D4, D4.6D20, D105SD16, C4⋊C42D10, (C2×C8)⋊16D10, (C5×D4).1D4, C20.1(C2×D4), C4.85(D4×D5), C4.2(C2×D20), D206C47C2, D101C89C2, D4⋊C410D5, (C2×C40)⋊15C22, D102Q81C2, C52(C22⋊SD16), C2.11(D5×SD16), C10.20C22≀C2, (C2×D4).135D10, (C2×Dic5).28D4, C10.23(C2×SD16), C22.174(D4×D5), C2.13(D8⋊D5), C10.31(C8⋊C22), (C2×C20).216C23, (C22×D5).109D4, (C2×D20).54C22, (D4×C10).37C22, C2.23(C22⋊D20), (C2×Dic10)⋊13C22, (C2×D4×D5).5C2, (C5×C4⋊C4)⋊4C22, (C2×D4.D5)⋊3C2, (C2×C40⋊C2)⋊14C2, (C2×C52C8)⋊3C22, (C5×D4⋊C4)⋊10C2, (C2×C4×D5).13C22, (C2×C10).229(C2×D4), (C2×C4).323(C22×D5), SmallGroup(320,403)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.8D4
C1C5C10C20C2×C20C2×C4×D5C2×D4×D5 — D20.8D4
C5C10C2×C20 — D20.8D4
C1C22C2×C4D4⋊C4

Generators and relations for D20.8D4
 G = < a,b,c,d | a4=b2=c20=1, d2=a2, bab=cac-1=dad-1=a-1, cbc-1=dbd-1=a-1b, dcd-1=a2c-1 >

Subgroups: 1022 in 188 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C52C8, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×D5, C22×C10, C22⋊SD16, C40⋊C2, C2×C52C8, C4⋊Dic5, D10⋊C4, D4.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, D206C4, D101C8, C5×D4⋊C4, D102Q8, C2×C40⋊C2, C2×D4.D5, C2×D4×D5, D20.8D4
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C22≀C2, C2×SD16, C8⋊C22, D20, C22×D5, C22⋊SD16, C2×D20, D4×D5, C22⋊D20, D8⋊D5, D5×SD16, D20.8D4

Smallest permutation representation of D20.8D4
On 80 points
Generators in S80
(1 62 39 53)(2 54 40 63)(3 64 21 55)(4 56 22 65)(5 66 23 57)(6 58 24 67)(7 68 25 59)(8 60 26 69)(9 70 27 41)(10 42 28 71)(11 72 29 43)(12 44 30 73)(13 74 31 45)(14 46 32 75)(15 76 33 47)(16 48 34 77)(17 78 35 49)(18 50 36 79)(19 80 37 51)(20 52 38 61)
(1 72)(2 30)(3 74)(4 32)(5 76)(6 34)(7 78)(8 36)(9 80)(10 38)(11 62)(12 40)(13 64)(14 22)(15 66)(16 24)(17 68)(18 26)(19 70)(20 28)(21 45)(23 47)(25 49)(27 51)(29 53)(31 55)(33 57)(35 59)(37 41)(39 43)(42 52)(44 54)(46 56)(48 58)(50 60)(61 71)(63 73)(65 75)(67 77)(69 79)
(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 38 39 20)(2 19 40 37)(3 36 21 18)(4 17 22 35)(5 34 23 16)(6 15 24 33)(7 32 25 14)(8 13 26 31)(9 30 27 12)(10 11 28 29)(41 73 70 44)(42 43 71 72)(45 69 74 60)(46 59 75 68)(47 67 76 58)(48 57 77 66)(49 65 78 56)(50 55 79 64)(51 63 80 54)(52 53 61 62)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222222224444455888810···1010101010202020202020202040···40
size111144101020202282040224420202···28888444488884···4

47 irreducible representations

dim11111111222222222244444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D5SD16D10D10D10D20C8⋊C22D4×D5D4×D5D8⋊D5D5×SD16
kernelD20.8D4D206C4D101C8C5×D4⋊C4D102Q8C2×C40⋊C2C2×D4.D5C2×D4×D5D20C2×Dic5C5×D4C22×D5D4⋊C4D10C4⋊C4C2×C8C2×D4D4C10C4C22C2C2
# reps11111111212124222812244

Matrix representation of D20.8D4 in GL6(𝔽41)

100000
010000
001000
000100
000019
00001840
,
4000000
0400000
001000
000100
000019
0000040
,
15370000
36260000
00354000
001000
00001129
00001730
,
15370000
15260000
00403500
000100
00001129
00001730

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,18,0,0,0,0,9,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,9,40],[15,36,0,0,0,0,37,26,0,0,0,0,0,0,35,1,0,0,0,0,40,0,0,0,0,0,0,0,11,17,0,0,0,0,29,30],[15,15,0,0,0,0,37,26,0,0,0,0,0,0,40,0,0,0,0,0,35,1,0,0,0,0,0,0,11,17,0,0,0,0,29,30] >;

D20.8D4 in GAP, Magma, Sage, TeX

D_{20}._8D_4
% in TeX

G:=Group("D20.8D4");
// GroupNames label

G:=SmallGroup(320,403);
// by ID

G=gap.SmallGroup(320,403);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,135,268,570,297,136,12550]);
// Polycyclic

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

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