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G = D4.11D20order 320 = 26·5

1st non-split extension by D4 of D20 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.11D20, Q8.11D20, D4011C22, C20.61C24, C40.10C23, M4(2)⋊20D10, D20.24C23, Dic2010C22, Dic10.24C23, C8○D43D5, (C2×C8)⋊6D10, (C2×C40)⋊9C22, (C5×D4).23D4, C20.73(C2×D4), C4.27(C2×D20), C51(D4○SD16), (C5×Q8).23D4, D48D103C2, C8⋊D1011C2, C4○D4.38D10, C4○D201C22, D407C211C2, C8.55(C22×D5), C4.58(C23×D5), C22.3(C2×D20), C8.D1011C2, C40⋊C211C22, C2.30(C22×D20), C10.28(C22×D4), D4.10D103C2, (C2×C20).515C23, (C2×Dic10)⋊35C22, (C2×D20).181C22, (C5×M4(2))⋊22C22, (C5×C8○D4)⋊3C2, (C2×C40⋊C2)⋊6C2, (C2×C10).8(C2×D4), (C5×C4○D4).45C22, (C2×C4).226(C22×D5), SmallGroup(320,1423)

Series: Derived Chief Lower central Upper central

C1C20 — D4.11D20
C1C5C10C20D20C2×D20D48D10 — D4.11D20
C5C10C20 — D4.11D20
C1C2C4○D4C8○D4

Generators and relations for D4.11D20
 G = < a,b,c,d | a4=b2=1, c20=d2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c19 >

Subgroups: 1094 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×7], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], C5, C8, C8 [×3], C2×C4 [×3], C2×C4 [×9], D4 [×3], D4 [×13], Q8, Q8 [×7], C23 [×3], D5 [×4], C10, C10 [×3], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×6], C2×Q8 [×4], C4○D4, C4○D4 [×10], Dic5 [×4], C20, C20 [×3], D10 [×7], C2×C10 [×3], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ 1+4, 2- 1+4, C40, C40 [×3], Dic10, Dic10 [×3], Dic10 [×3], C4×D5 [×6], D20, D20 [×3], D20 [×3], C2×Dic5 [×3], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×3], D4○SD16, C40⋊C2, C40⋊C2 [×9], D40 [×3], Dic20 [×3], C2×C40 [×3], C5×M4(2) [×3], C2×Dic10 [×3], C2×D20 [×3], C4○D20 [×6], D4×D5 [×3], D42D5 [×3], Q8×D5, Q82D5, C5×C4○D4, C2×C40⋊C2 [×3], D407C2 [×3], C8⋊D10 [×3], C8.D10 [×3], C5×C8○D4, D48D10, D4.10D10, D4.11D20
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, D20 [×4], C22×D5 [×7], D4○SD16, C2×D20 [×6], C23×D5, C22×D20, D4.11D20

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E10A10B10C···10H20A20B20C20D20E···20J40A···40H40I···40T
order122222222444444445588888101010···102020202020···2040···4040···40
size11222202020202222202020202222444224···422224···42···24···4

62 irreducible representations

dim111111112222222244
type++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D5D10D10D10D20D20D4○SD16D4.11D20
kernelD4.11D20C2×C40⋊C2D407C2C8⋊D10C8.D10C5×C8○D4D48D10D4.10D10C5×D4C5×Q8C8○D4C2×C8M4(2)C4○D4D4Q8C5C1
# reps1333311131266212428

Matrix representation of D4.11D20 in GL4(𝔽41) generated by

0010
0001
40000
04000
,
0010
0001
1000
0100
,
231600
251200
002316
002512
,
121400
162900
001214
001629
G:=sub<GL(4,GF(41))| [0,0,40,0,0,0,0,40,1,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[23,25,0,0,16,12,0,0,0,0,23,25,0,0,16,12],[12,16,0,0,14,29,0,0,0,0,12,16,0,0,14,29] >;

D4.11D20 in GAP, Magma, Sage, TeX

D_4._{11}D_{20}
% in TeX

G:=Group("D4.11D20");
// GroupNames label

G:=SmallGroup(320,1423);
// by ID

G=gap.SmallGroup(320,1423);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,80,1684,102,12550]);
// Polycyclic

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

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