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G = D20.2D4order 320 = 26·5

2nd non-split extension by D20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.2D4, Dic10.2D4, M4(2).1D10, C4.D43D5, C20.93(C2×D4), C4.148(D4×D5), C52C8.40D4, (C2×D4).15D10, C8.D106C2, C51(D4.3D4), (C2×C20).5C23, D20.2C45C2, C20.53D41C2, C4.12D206C2, C10.9(C4⋊D4), C4○D20.3C22, D4.D10.1C2, (D4×C10).15C22, C2.12(D10⋊D4), C4.Dic5.2C22, C22.13(C4○D20), (C2×Dic10).49C22, (C5×M4(2)).10C22, (C2×D4.D5)⋊1C2, (C5×C4.D4)⋊1C2, (C2×C4).5(C22×D5), (C2×C52C8).1C22, (C2×C10).30(C4○D4), SmallGroup(320,375)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.2D4
C1C5C10C20C2×C20C4○D20D20.2C4 — D20.2D4
C5C10C2×C20 — D20.2D4
C1C2C2×C4C4.D4

Generators and relations for D20.2D4
 G = < a,b,c,d | a20=b2=1, c4=a10, d2=a5, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, dbd-1=a15b, dcd-1=a15c3 >

Subgroups: 414 in 104 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×3], C5, C8 [×5], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×3], C23, D5, C10, C10 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×2], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, Dic5 [×2], C20 [×2], D10, C2×C10, C2×C10 [×2], C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C52C8 [×2], C52C8, C40 [×2], Dic10, Dic10 [×2], C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4 [×2], C22×C10, D4.3D4, C8×D5, C8⋊D5, C40⋊C2, Dic20, C2×C52C8, C4.Dic5, D4⋊D5, D4.D5 [×3], C5×M4(2) [×2], C2×Dic10, C4○D20, D4×C10, C20.53D4, C4.12D20, C5×C4.D4, D20.2C4, C8.D10, D4.D10, C2×D4.D5, D20.2D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C22×D5, D4.3D4, C4○D20, D4×D5 [×2], D10⋊D4, D20.2D4

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D5A5B8A8B8C8D8E8F8G10A10B10C10D10E10F10G10H20A20B20C20D40A···40H
order12222444455888888810101010101010102020202040···40
size11282022204022448101020402244888844448···8

38 irreducible representations

dim1111111122222222448
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10C4○D20D4.3D4D4×D5D20.2D4
kernelD20.2D4C20.53D4C4.12D20C5×C4.D4D20.2C4C8.D10D4.D10C2×D4.D5C52C8Dic10D20C4.D4C2×C10M4(2)C2×D4C22C5C4C1
# reps1111111121122428242

Matrix representation of D20.2D4 in GL6(𝔽41)

4010000
3370000
001900
00184000
0054040
0036010
,
32250000
590000
000151130
00172707
0037402612
003462629
,
100000
010000
0015390
00092318
0000364
0010436
,
4000000
0400000
0015390
0018402323
004163637
00362145

G:=sub<GL(6,GF(41))| [40,33,0,0,0,0,1,7,0,0,0,0,0,0,1,18,5,36,0,0,9,40,4,0,0,0,0,0,0,1,0,0,0,0,40,0],[32,5,0,0,0,0,25,9,0,0,0,0,0,0,0,17,37,34,0,0,15,27,40,6,0,0,11,0,26,26,0,0,30,7,12,29],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,5,9,0,0,0,0,39,23,36,4,0,0,0,18,4,36],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,18,4,36,0,0,5,40,16,21,0,0,39,23,36,4,0,0,0,23,37,5] >;

D20.2D4 in GAP, Magma, Sage, TeX

D_{20}._2D_4
% in TeX

G:=Group("D20.2D4");
// GroupNames label

G:=SmallGroup(320,375);
// by ID

G=gap.SmallGroup(320,375);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,297,136,1684,851,12550]);
// Polycyclic

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

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