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G = D20.4D4order 320 = 26·5

4th non-split extension by D20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.4D4, Dic10.4D4, M4(2).3D10, (C2×C4).6D20, (C2×C20).8D4, C4.80(D4×D5), C20.97(C2×D4), (C2×Q8).5D10, D207C43C2, C8.D107C2, C4.10D41D5, (C2×C20).9C23, C10.17C22≀C2, Dic5⋊Q81C2, C51(D4.10D4), C4○D20.5C22, C22.12(C2×D20), (Q8×C10).7C22, C2.20(C22⋊D20), (C4×Dic5).7C22, Q8.10D10.1C2, (C5×M4(2)).2C22, (C2×Dic10).51C22, (C2×C10).22(C2×D4), (C2×C4).9(C22×D5), (C5×C4.10D4)⋊3C2, SmallGroup(320,379)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.4D4
C1C5C10C20C2×C20C4○D20Q8.10D10 — D20.4D4
C5C10C2×C20 — D20.4D4
C1C2C2×C4C4.10D4

Generators and relations for D20.4D4
 G = < a,b,c,d | a20=b2=1, c4=a10, d2=a15, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, dbd-1=a5b, dcd-1=a15c3 >

Subgroups: 622 in 142 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×7], C22, C22 [×2], C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×8], D4 [×6], Q8 [×8], D5 [×2], C10, C10, C42, C4⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×Q8, C2×Q8 [×3], C4○D4 [×6], Dic5 [×5], C20 [×2], C20 [×2], D10 [×2], C2×C10, C4.10D4, C4≀C2 [×2], C4⋊Q8, C8.C22 [×2], 2- 1+4, C40 [×2], Dic10 [×2], Dic10 [×4], C4×D5 [×6], D20 [×2], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], D4.10D4, C40⋊C2 [×2], Dic20 [×2], C4×Dic5, C10.D4 [×2], C5×M4(2) [×2], C2×Dic10, C4○D20 [×2], C4○D20 [×2], Q8×D5 [×2], Q82D5 [×2], Q8×C10, D207C4 [×2], C5×C4.10D4, C8.D10 [×2], Dic5⋊Q8, Q8.10D10, D20.4D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, D20 [×2], C22×D5, D4.10D4, C2×D20, D4×D5 [×2], C22⋊D20, D20.4D4

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I5A5B8A8B10A10B10C10D20A20B20C20D20E20F20G20H40A···40H
order12222444444444558810101010202020202020202040···40
size11220202244202020204022882244444488888···8

38 irreducible representations

dim1111112222222448
type+++++++++++++-+-
imageC1C2C2C2C2C2D4D4D4D5D10D10D20D4.10D4D4×D5D20.4D4
kernelD20.4D4D207C4C5×C4.10D4C8.D10Dic5⋊Q8Q8.10D10Dic10D20C2×C20C4.10D4M4(2)C2×Q8C2×C4C5C4C1
# reps1212112222428242

Matrix representation of D20.4D4 in GL6(𝔽41)

3410000
4000000
000100
0040000
00404014
000212040
,
4070000
010000
00114037
0000400
0004000
000212040
,
4000000
0400000
000010
00114037
000100
00210040
,
4000000
0400000
0011112938
00111037
00304000
006212029

G:=sub<GL(6,GF(41))| [34,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,40,0,0,0,1,0,40,21,0,0,0,0,1,20,0,0,0,0,4,40],[40,0,0,0,0,0,7,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,40,21,0,0,40,40,0,20,0,0,37,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,21,0,0,0,1,1,0,0,0,1,40,0,0,0,0,0,37,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,11,1,30,6,0,0,11,1,40,21,0,0,29,10,0,20,0,0,38,37,0,29] >;

D20.4D4 in GAP, Magma, Sage, TeX

D_{20}._4D_4
% in TeX

G:=Group("D20.4D4");
// GroupNames label

G:=SmallGroup(320,379);
// by ID

G=gap.SmallGroup(320,379);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,58,1123,570,136,1684,438,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^15,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^15*b,d*b*d^-1=a^5*b,d*c*d^-1=a^15*c^3>;
// generators/relations

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