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G = D4.6D20order 320 = 26·5

1st non-split extension by D4 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.6D20, D20.8D4, D105SD16, C4⋊C42D10, (C2×C8)⋊16D10, (C5×D4).1D4, C4.85(D4×D5), C4.2(C2×D20), C20.1(C2×D4), 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, (D4×C10).37C22, (C22×D5).109D4, (C2×D20).54C22, 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 — D4.6D20
C1C5C10C20C2×C20C2×C4×D5C2×D4×D5 — D4.6D20
C5C10C2×C20 — D4.6D20
C1C22C2×C4D4⋊C4

Generators and relations for D4.6D20
 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, D4.6D20
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, D4.6D20

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

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

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

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

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
kernelD4.6D20D206C4D101C8C5×D4⋊C4D102Q8C2×C40⋊C2C2×D4.D5C2×D4×D5D20C2×Dic5C5×D4C22×D5D4⋊C4D10C4⋊C4C2×C8C2×D4D4C10C4C22C2C2
# reps11111111212124222812244

Matrix representation of D4.6D20 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] >;

D4.6D20 in GAP, Magma, Sage, TeX

D_4._6D_{20}
% in TeX

G:=Group("D4.6D20");
// 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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