Copied to
clipboard

G = D20.40D4order 320 = 26·5

10th non-split extension by D20 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.40D4, Dic10.40D4, M4(2).17D10, C4○D4.8D10, (C5×D4).17D4, C4.107(D4×D5), (C5×Q8).17D4, C8.C223D5, C20.201(C2×D4), (C2×Q8).71D10, (C2×Dic5).6D4, D207C412C2, C22.38(D4×D5), C10.66C22≀C2, Dic5⋊Q87C2, D42Dic59C2, C20.C236C2, D4.12(C5⋊D4), C54(D4.10D4), (C2×C20).20C23, Q8.12(C5⋊D4), C4.12D2012C2, C4○D20.26C22, (Q8×C10).98C22, C2.34(C23⋊D10), D4.10D10.2C2, (C4×Dic5).65C22, C4.Dic5.29C22, (C5×M4(2)).27C22, (C2×Dic10).141C22, C4.57(C2×C5⋊D4), (C2×C10).37(C2×D4), (C5×C8.C22)⋊7C2, (C2×C4).20(C22×D5), (C5×C4○D4).18C22, SmallGroup(320,832)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.40D4
C1C5C10C20C2×C20C4○D20D4.10D10 — D20.40D4
C5C10C2×C20 — D20.40D4
C1C2C2×C4C8.C22

Generators and relations for D20.40D4
 G = < a,b,c,d | a20=b2=1, c4=d2=a10, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd-1=a10c3 >

Subgroups: 558 in 142 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×7], C22, C22 [×2], C5, C8 [×2], C2×C4, C2×C4 [×10], D4, D4 [×5], Q8, Q8 [×7], D5, C10, C10 [×2], C42, C4⋊C4 [×2], M4(2), M4(2), SD16 [×2], Q16 [×2], C2×Q8, C2×Q8 [×3], C4○D4, C4○D4 [×5], Dic5 [×5], C20 [×2], C20 [×2], D10, C2×C10, C2×C10, C4.10D4, C4≀C2 [×2], C4⋊Q8, C8.C22, C8.C22, 2- 1+4, C52C8, C40, Dic10, Dic10 [×4], C4×D5 [×3], D20, C2×Dic5 [×2], C2×Dic5 [×3], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×2], D4.10D4, C4.Dic5, C4×Dic5, C10.D4 [×2], Q8⋊D5, C5⋊Q16, C5×M4(2), C5×SD16, C5×Q16, C2×Dic10, C2×Dic10, C4○D20, C4○D20, D42D5 [×3], Q8×D5, Q8×C10, C5×C4○D4, C4.12D20, D207C4, D42Dic5, C20.C23, Dic5⋊Q8, C5×C8.C22, D4.10D10, D20.40D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×2], C22×D5, D4.10D4, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, D20.40D4

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E···4I5A5B8A8B10A10B10C10D10E10F20A20B20C20D20E···20J40A40B40C40D
order1222244444···455881010101010102020202020···2040404040
size112420224820···202284022448844448···88888

38 irreducible representations

dim11111111222222222224448
type+++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2D4D4D4D4D4D5D10D10D10C5⋊D4C5⋊D4D4.10D4D4×D5D4×D5D20.40D4
kernelD20.40D4C4.12D20D207C4D42Dic5C20.C23Dic5⋊Q8C5×C8.C22D4.10D10Dic10D20C2×Dic5C5×D4C5×Q8C8.C22M4(2)C2×Q8C4○D4D4Q8C5C4C22C1
# reps11111111112112222442222

Matrix representation of D20.40D4 in GL8(𝔽41)

040000000
17000000
000400000
00170000
000040400
000020100
000000404
000000201
,
40172420000
24139170000
17391240000
22417400000
000000417
0000004037
0000372400
00001400
,
17402240000
12417390000
39172410000
24240170000
000000833
000000333
000041700
0000403700
,
39172410000
24240170000
17402240000
12417390000
00000010
00000001
000040000
000004000

G:=sub<GL(8,GF(41))| [0,1,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,40,20,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,40,20,0,0,0,0,0,0,4,1],[40,24,17,2,0,0,0,0,17,1,39,24,0,0,0,0,24,39,1,17,0,0,0,0,2,17,24,40,0,0,0,0,0,0,0,0,0,0,37,1,0,0,0,0,0,0,24,4,0,0,0,0,4,40,0,0,0,0,0,0,17,37,0,0],[17,1,39,24,0,0,0,0,40,24,17,2,0,0,0,0,2,17,24,40,0,0,0,0,24,39,1,17,0,0,0,0,0,0,0,0,0,0,4,40,0,0,0,0,0,0,17,37,0,0,0,0,8,3,0,0,0,0,0,0,33,33,0,0],[39,24,17,1,0,0,0,0,17,2,40,24,0,0,0,0,24,40,2,17,0,0,0,0,1,17,24,39,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

D20.40D4 in GAP, Magma, Sage, TeX

D_{20}._{40}D_4
% in TeX

G:=Group("D20.40D4");
// GroupNames label

G:=SmallGroup(320,832);
// by ID

G=gap.SmallGroup(320,832);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,184,1123,297,136,851,438,102,12550]);
// Polycyclic

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

׿
×
𝔽