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G = D4.3D20order 320 = 26·5

3rd non-split extension by D4 of D20 acting via D20/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.3D20, C40.87D4, Q8.3D20, M4(2).33D10, C8○D41D5, (C5×D4).20D4, C4.19(C2×D20), (C2×C8).80D10, C20.42(C2×D4), (C5×Q8).20D4, C4○D4.31D10, C54(D4.3D4), C8.44(C5⋊D4), C40.6C414C2, D4⋊D10.1C2, D4.9D103C2, (C2×C40).66C22, C20.46D413C2, C4.12D2013C2, C2.24(C207D4), C10.76(C4⋊D4), (C2×C20).421C23, C22.8(C4○D20), (C2×D20).115C22, C4.Dic5.16C22, (C5×M4(2)).36C22, (C2×Dic10).121C22, (C5×C8○D4)⋊1C2, (C2×C40⋊C2)⋊3C2, C4.117(C2×C5⋊D4), (C2×C10).6(C4○D4), (C5×C4○D4).36C22, (C2×C4).123(C22×D5), SmallGroup(320,768)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D4.3D20
C1C5C10C20C2×C20C2×D20C2×C40⋊C2 — D4.3D20
C5C10C2×C20 — D4.3D20
C1C2C2×C4C8○D4

Generators and relations for D4.3D20
 G = < a,b,c,d | a4=b2=1, c20=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c19 >

Subgroups: 446 in 104 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×3], C5, C8 [×2], C8 [×3], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8 [×2], C23, D5, C10, C10 [×2], C2×C8, C2×C8, M4(2), M4(2) [×3], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, Dic5, C20 [×2], C20, D10 [×2], C2×C10, C2×C10, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C52C8 [×2], C40 [×2], C40, Dic10 [×2], D20 [×2], C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, D4.3D4, C40⋊C2 [×2], C4.Dic5 [×2], D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C2×Dic10, C2×D20, C5×C4○D4, C40.6C4, C20.46D4, C4.12D20, C2×C40⋊C2, D4⋊D10, D4.9D10, C5×C8○D4, D4.3D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, D20 [×2], C5⋊D4 [×2], C22×D5, D4.3D4, C2×D20, C4○D20, C2×C5⋊D4, C207D4, D4.3D20

Smallest permutation representation of D4.3D20
On 80 points
Generators in S80
(1 31 21 11)(2 32 22 12)(3 33 23 13)(4 34 24 14)(5 35 25 15)(6 36 26 16)(7 37 27 17)(8 38 28 18)(9 39 29 19)(10 40 30 20)(41 51 61 71)(42 52 62 72)(43 53 63 73)(44 54 64 74)(45 55 65 75)(46 56 66 76)(47 57 67 77)(48 58 68 78)(49 59 69 79)(50 60 70 80)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 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 20 21 40)(2 39 22 19)(3 18 23 38)(4 37 24 17)(5 16 25 36)(6 35 26 15)(7 14 27 34)(8 33 28 13)(9 12 29 32)(10 31 30 11)(41 50 61 70)(42 69 62 49)(43 48 63 68)(44 67 64 47)(45 46 65 66)(51 80 71 60)(52 59 72 79)(53 78 73 58)(54 57 74 77)(55 76 75 56)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D4A4B4C4D5A5B8A8B8C8D8E8F8G10A10B10C···10H20A20B20C20D20E···20J40A···40H40I···40T
order122224444558888888101010···102020202020···2040···4040···40
size1124402244022224444040224···422224···42···24···4

56 irreducible representations

dim1111111122222222222244
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C5⋊D4D20D20C4○D20D4.3D4D4.3D20
kernelD4.3D20C40.6C4C20.46D4C4.12D20C2×C40⋊C2D4⋊D10D4.9D10C5×C8○D4C40C5×D4C5×Q8C8○D4C2×C10C2×C8M4(2)C4○D4C8D4Q8C22C5C1
# reps1111111121122222844828

Matrix representation of D4.3D20 in GL6(𝔽41)

100000
010000
001100
00394000
00004040
000021
,
100000
010000
00004040
000021
001100
00394000
,
19270000
27190000
00112600
0030000
00001126
0000300
,
9300000
11320000
0002600
0011000
00003015
00001111

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,39,0,0,0,0,1,40,0,0,0,0,0,0,40,2,0,0,0,0,40,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,39,0,0,0,0,1,40,0,0,40,2,0,0,0,0,40,1,0,0],[19,27,0,0,0,0,27,19,0,0,0,0,0,0,11,30,0,0,0,0,26,0,0,0,0,0,0,0,11,30,0,0,0,0,26,0],[9,11,0,0,0,0,30,32,0,0,0,0,0,0,0,11,0,0,0,0,26,0,0,0,0,0,0,0,30,11,0,0,0,0,15,11] >;

D4.3D20 in GAP, Magma, Sage, TeX

D_4._3D_{20}
% in TeX

G:=Group("D4.3D20");
// GroupNames label

G:=SmallGroup(320,768);
// by ID

G=gap.SmallGroup(320,768);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,1123,297,136,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^20=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^19>;
// generators/relations

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