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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 [×3], C2 [×6], C4 [×2], C4 [×3], C22, C22 [×20], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], D4 [×8], Q8 [×2], C23 [×11], D5 [×4], C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×D4 [×6], C2×Q8, C24, Dic5 [×2], C20 [×2], C20, D10 [×2], D10 [×14], C2×C10, C2×C10 [×4], C22⋊C8, D4⋊C4, D4⋊C4, C22⋊Q8, C2×SD16 [×2], C22×D4, C52C8, C40, Dic10 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×D5, C22×D5 [×9], C22×C10, C22⋊SD16, C40⋊C2 [×2], C2×C52C8, C4⋊Dic5, D10⋊C4, D4.D5 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, D4×D5 [×4], 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 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×SD16, C8⋊C22, D20 [×2], C22×D5, C22⋊SD16, C2×D20, D4×D5 [×2], C22⋊D20, D8⋊D5, D5×SD16, D20.8D4

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

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

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

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

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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