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G = D20.6D4order 320 = 26·5

6th non-split extension by D20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.6D4, Dic10.6D4, M4(2).5D10, C20.99(C2xD4), C4.152(D4xD5), C5:2C8.42D4, (C2xQ8).7D10, C8:D10.1C2, C4.10D4:3D5, C5:2(D4.3D4), D20.2C4:7C2, C20.46D4:8C2, C20.C23:1C2, C20.53D4:3C2, (C2xC20).11C23, C4oD20.7C22, (Q8xC10).9C22, C10.11(C4:D4), (C2xD20).46C22, C2.14(D10:D4), C4.Dic5.6C22, C22.15(C4oD20), (C5xM4(2)).14C22, (C2xQ8:D5):1C2, (C5xC4.10D4):1C2, (C2xC5:2C8).3C22, (C2xC4).11(C22xD5), (C2xC10).32(C4oD4), SmallGroup(320,381)

Series: Derived Chief Lower central Upper central

C1C2xC20 — D20.6D4
C1C5C10C20C2xC20C4oD20D20.2C4 — D20.6D4
C5C10C2xC20 — D20.6D4
C1C2C2xC4C4.10D4

Generators and relations for D20.6D4
 G = < a,b,c,d | a20=b2=1, c4=a10, d2=a15, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, dbd-1=a15b, dcd-1=a15c3 >

Subgroups: 478 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, C20, D10, C2xC10, C4.D4, C4.10D4, C8.C4, C8oD4, C2xSD16, C8:C22, C8.C22, C5:2C8, C5:2C8, C40, Dic10, C4xD5, D20, D20, C5:D4, C2xC20, C2xC20, C5xQ8, C22xD5, D4.3D4, C8xD5, C8:D5, C40:C2, D40, C2xC5:2C8, C4.Dic5, Q8:D5, C5:Q16, C5xM4(2), C2xD20, C4oD20, Q8xC10, C20.53D4, C20.46D4, C5xC4.10D4, D20.2C4, C8:D10, C2xQ8:D5, C20.C23, D20.6D4
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C4oD4, D10, C4:D4, C22xD5, D4.3D4, C4oD20, D4xD5, D10:D4, D20.6D4

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D5A5B8A8B8C8D8E8F8G10A10B10C10D20A20B20C20D20E20F20G20H40A···40H
order12222444455888888810101010202020202020202040···40
size11220402282022448101020402244444488888···8

38 irreducible representations

dim1111111122222222448
type++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5C4oD4D10D10C4oD20D4.3D4D4xD5D20.6D4
kernelD20.6D4C20.53D4C20.46D4C5xC4.10D4D20.2C4C8:D10C2xQ8:D5C20.C23C5:2C8Dic10D20C4.10D4C2xC10M4(2)C2xQ8C22C5C4C1
# reps1111111121122428242

Matrix representation of D20.6D4 in GL6(F41)

4010000
3370000
000100
0040000
0004040
0037010
,
4000000
3310000
0012292417
0015152424
006402612
001362629
,
100000
010000
00537180
00375018
0000364
0090436
,
100000
010000
0015152424
0012292417
0025352629
0035282612

G:=sub<GL(6,GF(41))| [40,33,0,0,0,0,1,7,0,0,0,0,0,0,0,40,0,37,0,0,1,0,4,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,33,0,0,0,0,0,1,0,0,0,0,0,0,12,15,6,13,0,0,29,15,40,6,0,0,24,24,26,26,0,0,17,24,12,29],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,37,0,9,0,0,37,5,0,0,0,0,18,0,36,4,0,0,0,18,4,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,12,25,35,0,0,15,29,35,28,0,0,24,24,26,26,0,0,24,17,29,12] >;

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

D_{20}._6D_4
% in TeX

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

G:=SmallGroup(320,381);
// by ID

G=gap.SmallGroup(320,381);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,184,297,136,1684,851,12550]);
// 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^10*b,d*b*d^-1=a^15*b,d*c*d^-1=a^15*c^3>;
// generators/relations

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