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G = D2016D4order 320 = 26·5

4th semidirect product of D20 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2016D4, C4⋊C43D10, (C2×C10)⋊2D8, (C2×D4)⋊1D10, C4⋊D41D5, C4.98(D4×D5), C53(C22⋊D8), C10.54(C2×D8), (C2×C20).71D4, C222(D4⋊D5), C20.145(C2×D4), (D4×C10)⋊1C22, D206C433C2, C10.44C22≀C2, (C22×D20)⋊13C2, (C22×C10).82D4, C20.55D410C2, (C2×C20).355C23, (C22×C4).119D10, C23.58(C5⋊D4), C2.12(D4⋊D10), C2.12(C23⋊D10), C10.114(C8⋊C22), (C2×D20).248C22, (C22×C20).159C22, (C2×D4⋊D5)⋊8C2, C2.9(C2×D4⋊D5), (C5×C4⋊D4)⋊1C2, (C5×C4⋊C4)⋊5C22, (C2×C52C8)⋊5C22, (C2×C10).486(C2×D4), (C2×C4).49(C5⋊D4), (C2×C4).455(C22×D5), C22.161(C2×C5⋊D4), SmallGroup(320,663)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D2016D4
C1C5C10C20C2×C20C2×D20C22×D20 — D2016D4
C5C10C2×C20 — D2016D4
C1C22C22×C4C4⋊D4

Generators and relations for D2016D4
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd=c-1 >

Subgroups: 1054 in 198 conjugacy classes, 47 normal (27 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×21], C5, C8 [×2], C2×C4 [×2], C2×C4 [×3], D4 [×14], C23, C23 [×11], D5 [×4], C10 [×3], C10 [×3], C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4, C2×D4 [×8], C24, C20 [×2], C20 [×2], D10 [×16], C2×C10, C2×C10 [×2], C2×C10 [×5], C22⋊C8, D4⋊C4 [×2], C4⋊D4, C2×D8 [×2], C22×D4, C52C8 [×2], D20 [×4], D20 [×6], C2×C20 [×2], C2×C20 [×3], C5×D4 [×4], C22×D5 [×10], C22×C10, C22×C10, C22⋊D8, C2×C52C8 [×2], D4⋊D5 [×4], C5×C22⋊C4, C5×C4⋊C4, C2×D20 [×2], C2×D20 [×5], C22×C20, D4×C10, D4×C10, C23×D5, D206C4 [×2], C20.55D4, C2×D4⋊D5 [×2], C5×C4⋊D4, C22×D20, D2016D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, D8 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×D8, C8⋊C22, C5⋊D4 [×2], C22×D5, C22⋊D8, D4⋊D5 [×2], D4×D5 [×2], C2×C5⋊D4, C2×D4⋊D5, C23⋊D10, D4⋊D10, D2016D4

Smallest permutation representation of D2016D4
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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 40)(17 39)(18 38)(19 37)(20 36)(41 73)(42 72)(43 71)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)(51 63)(52 62)(53 61)(54 80)(55 79)(56 78)(57 77)(58 76)(59 75)(60 74)
(1 57 36 73)(2 48 37 64)(3 59 38 75)(4 50 39 66)(5 41 40 77)(6 52 21 68)(7 43 22 79)(8 54 23 70)(9 45 24 61)(10 56 25 72)(11 47 26 63)(12 58 27 74)(13 49 28 65)(14 60 29 76)(15 51 30 67)(16 42 31 78)(17 53 32 69)(18 44 33 80)(19 55 34 71)(20 46 35 62)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D5A5B8A8B8C8D10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order12222222222444455888810···10101010101010101020···2020202020
size111122820202020224822202020202···2444488884···48888

47 irreducible representations

dim11111122222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5D8D10D10D10C5⋊D4C5⋊D4C8⋊C22D4×D5D4⋊D5D4⋊D10
kernelD2016D4D206C4C20.55D4C2×D4⋊D5C5×C4⋊D4C22×D20D20C2×C20C22×C10C4⋊D4C2×C10C4⋊C4C22×C4C2×D4C2×C4C23C10C4C22C2
# reps12121141124222441444

Matrix representation of D2016D4 in GL6(𝔽41)

100000
010000
0040100
0033700
00004039
000011
,
4000000
0400000
0040000
0033100
00004039
000001
,
0400000
100000
0040000
0004000
0000024
0000120
,
100000
0400000
001000
000100
0000400
0000040

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,1,7,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,33,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,12,0,0,0,0,24,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

D2016D4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{16}D_4
% in TeX

G:=Group("D20:16D4");
// GroupNames label

G:=SmallGroup(320,663);
// by ID

G=gap.SmallGroup(320,663);
# by ID

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

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

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