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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.D4:3D5, C20.93(C2xD4), C4.148(D4xD5), C5:2C8.40D4, (C2xD4).15D10, C8.D10:6C2, C5:1(D4.3D4), (C2xC20).5C23, D20.2C4:5C2, C20.53D4:1C2, C4.12D20:6C2, C10.9(C4:D4), C4oD20.3C22, D4.D10.1C2, (D4xC10).15C22, C2.12(D10:D4), C4.Dic5.2C22, C22.13(C4oD20), (C2xDic10).49C22, (C5xM4(2)).10C22, (C2xD4.D5):1C2, (C5xC4.D4):1C2, (C2xC4).5(C22xD5), (C2xC5:2C8).1C22, (C2xC10).30(C4oD4), SmallGroup(320,375)

Series: Derived Chief Lower central Upper central

C1C2xC20 — D20.2D4
C1C5C10C20C2xC20C4oD20D20.2C4 — D20.2D4
C5C10C2xC20 — D20.2D4
C1C2C2xC4C4.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, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, D4, Q8, C23, D5, C10, C10, C2xC8, M4(2), M4(2), D8, SD16, Q16, C2xD4, C2xQ8, C4oD4, Dic5, C20, D10, C2xC10, C2xC10, C4.D4, C4.10D4, C8.C4, C8oD4, C2xSD16, C8:C22, C8.C22, C5:2C8, C5:2C8, C40, Dic10, Dic10, C4xD5, D20, C2xDic5, C5:D4, C2xC20, C5xD4, C22xC10, D4.3D4, C8xD5, C8:D5, C40:C2, Dic20, C2xC5:2C8, C4.Dic5, D4:D5, D4.D5, C5xM4(2), C2xDic10, C4oD20, D4xC10, C20.53D4, C4.12D20, C5xC4.D4, D20.2C4, C8.D10, D4.D10, C2xD4.D5, D20.2D4
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C4oD4, D10, C4:D4, C22xD5, D4.3D4, C4oD20, D4xD5, 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 76)(22 75)(23 74)(24 73)(25 72)(26 71)(27 70)(28 69)(29 68)(30 67)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 80)(38 79)(39 78)(40 77)
(1 27 6 22 11 37 16 32)(2 38 7 33 12 28 17 23)(3 29 8 24 13 39 18 34)(4 40 9 35 14 30 19 25)(5 31 10 26 15 21 20 36)(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 37 6 22 11 27 16 32)(2 38 7 23 12 28 17 33)(3 39 8 24 13 29 18 34)(4 40 9 25 14 30 19 35)(5 21 10 26 15 31 20 36)(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)(21,76)(22,75)(23,74)(24,73)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,80)(38,79)(39,78)(40,77), (1,27,6,22,11,37,16,32)(2,38,7,33,12,28,17,23)(3,29,8,24,13,39,18,34)(4,40,9,35,14,30,19,25)(5,31,10,26,15,21,20,36)(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), (1,37,6,22,11,27,16,32)(2,38,7,23,12,28,17,33)(3,39,8,24,13,29,18,34)(4,40,9,25,14,30,19,35)(5,21,10,26,15,31,20,36)(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)(21,76)(22,75)(23,74)(24,73)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,80)(38,79)(39,78)(40,77), (1,27,6,22,11,37,16,32)(2,38,7,33,12,28,17,23)(3,29,8,24,13,39,18,34)(4,40,9,35,14,30,19,25)(5,31,10,26,15,21,20,36)(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), (1,37,6,22,11,27,16,32)(2,38,7,23,12,28,17,33)(3,39,8,24,13,29,18,34)(4,40,9,25,14,30,19,35)(5,21,10,26,15,31,20,36)(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),(21,76),(22,75),(23,74),(24,73),(25,72),(26,71),(27,70),(28,69),(29,68),(30,67),(31,66),(32,65),(33,64),(34,63),(35,62),(36,61),(37,80),(38,79),(39,78),(40,77)], [(1,27,6,22,11,37,16,32),(2,38,7,33,12,28,17,23),(3,29,8,24,13,39,18,34),(4,40,9,35,14,30,19,25),(5,31,10,26,15,21,20,36),(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)], [(1,37,6,22,11,27,16,32),(2,38,7,23,12,28,17,33),(3,39,8,24,13,29,18,34),(4,40,9,25,14,30,19,35),(5,21,10,26,15,31,20,36),(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+++++++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D4D5C4oD4D10D10C4oD20D4.3D4D4xD5D20.2D4
kernelD20.2D4C20.53D4C4.12D20C5xC4.D4D20.2C4C8.D10D4.D10C2xD4.D5C5:2C8Dic10D20C4.D4C2xC10M4(2)C2xD4C22C5C4C1
# 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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