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G = D20.3D4order 320 = 26·5

3rd non-split extension by D20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.3D4, Dic10.3D4, M4(2).2D10, C8⋊D106C2, C4.D44D5, C4.149(D4×D5), C20.94(C2×D4), C52C8.41D4, (C2×D4).16D10, C51(D4.4D4), (C2×C20).6C23, D20.2C46C2, D4.D101C2, C20.53D42C2, C20.46D46C2, C4○D20.4C22, C10.10(C4⋊D4), (D4×C10).16C22, (C2×D20).43C22, C2.13(D10⋊D4), C4.Dic5.3C22, C22.14(C4○D20), (C5×M4(2)).11C22, (C2×D4⋊D5)⋊1C2, (C5×C4.D4)⋊2C2, (C2×C4).6(C22×D5), (C2×C52C8).2C22, (C2×C10).31(C4○D4), SmallGroup(320,376)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.3D4
C1C5C10C20C2×C20C4○D20D20.2C4 — D20.3D4
C5C10C2×C20 — D20.3D4
C1C2C2×C4C4.D4

Generators and relations for D20.3D4
 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=a5c3 >

Subgroups: 510 in 108 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C5, C8 [×5], C2×C4, C2×C4, D4 [×6], Q8, C23 [×2], D5 [×2], C10, C10 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×2], D8 [×4], SD16 [×2], C2×D4, C2×D4, C4○D4, Dic5, C20 [×2], D10 [×3], C2×C10, C2×C10 [×2], C4.D4, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22 [×2], C52C8 [×2], C52C8, C40 [×2], Dic10, C4×D5, D20, D20 [×2], C5⋊D4, C2×C20, C5×D4 [×2], C22×D5, C22×C10, D4.4D4, C8×D5, C8⋊D5, C40⋊C2, D40, C2×C52C8, C4.Dic5, D4⋊D5 [×3], D4.D5, C5×M4(2) [×2], C2×D20, C4○D20, D4×C10, C20.53D4, C20.46D4, C5×C4.D4, D20.2C4, C8⋊D10, C2×D4⋊D5, D4.D10, D20.3D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C22×D5, D4.4D4, C4○D20, D4×D5 [×2], D10⋊D4, D20.3D4

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C5A5B8A8B8C8D8E8F8G10A10B10C10D10E10F10G10H20A20B20C20D40A···40H
order12222244455888888810101010101010102020202040···40
size11282040222022448101020402244888844448···8

38 irreducible representations

dim1111111122222222448
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10C4○D20D4.4D4D4×D5D20.3D4
kernelD20.3D4C20.53D4C20.46D4C5×C4.D4D20.2C4C8⋊D10C2×D4⋊D5D4.D10C52C8Dic10D20C4.D4C2×C10M4(2)C2×D4C22C5C4C1
# reps1111111121122428242

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

3410000
4000000
000100
0040000
0004040
0037010
,
1340000
0400000
0029291717
0015262417
0035401512
0013352612
,
100000
010000
00436390
00364039
0000375
0010537
,
100000
010000
0015262417
0029291717
0013352612
0035401512

G:=sub<GL(6,GF(41))| [34,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,37,0,0,1,0,4,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,0,0,0,0,0,34,40,0,0,0,0,0,0,29,15,35,13,0,0,29,26,40,35,0,0,17,24,15,26,0,0,17,17,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,36,0,1,0,0,36,4,0,0,0,0,39,0,37,5,0,0,0,39,5,37],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,29,13,35,0,0,26,29,35,40,0,0,24,17,26,15,0,0,17,17,12,12] >;

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

D_{20}._3D_4
% in TeX

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

G:=SmallGroup(320,376);
// by ID

G=gap.SmallGroup(320,376);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,297,136,1684,851,12550]);
// 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^5*c^3>;
// generators/relations

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