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G = D45D20order 320 = 26·5

1st semidirect product of D4 and D20 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D45D20, C4215D10, C10.172+ 1+4, C4⋊C448D10, (C5×D4)⋊10D4, (C4×D4)⋊16D5, (C4×D20)⋊30C2, (D4×C20)⋊18C2, C207D49C2, C53(D45D4), C20.54(C2×D4), C4.22(C2×D20), D107(C4○D4), C22⋊D206C2, C4⋊D2015C2, (C4×C20)⋊20C22, C22⋊C447D10, (C2×D20)⋊6C22, (C22×C4)⋊13D10, C4⋊Dic58C22, C22.1(C2×D20), D102Q814C2, (C2×D4).248D10, C4.D2018C2, (C2×C10).98C24, C10.16(C22×D4), C2.18(C22×D20), (C2×C20).159C23, (C22×C20)⋊10C22, C22.D205C2, C2.18(D46D10), D10⋊C452C22, (C2×Dic10)⋊17C22, (D4×C10).259C22, (C2×Dic5).42C23, (C22×Dic5)⋊9C22, (C22×D5).33C23, (C23×D5).40C22, C22.123(C23×D5), C23.172(C22×D5), (C22×C10).168C23, (C2×D4×D5)⋊4C2, (C2×C4×D5)⋊3C22, C2.22(D5×C4○D4), (C2×C10).1(C2×D4), (C2×D42D5)⋊3C2, (C5×C4⋊C4)⋊60C22, (C2×C5⋊D4)⋊4C22, C10.139(C2×C4○D4), (C2×D10⋊C4)⋊21C2, (C5×C22⋊C4)⋊50C22, (C2×C4).160(C22×D5), SmallGroup(320,1226)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D45D20
C1C5C10C2×C10C22×D5C23×D5C2×D4×D5 — D45D20
C5C2×C10 — D45D20
C1C22C4×D4

Generators and relations for D45D20
 G = < a,b,c,d | a4=b2=c20=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 1462 in 334 conjugacy classes, 113 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×4], C22 [×25], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×14], D5 [×5], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×4], C20 [×2], C20 [×4], D10 [×2], D10 [×19], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C22⋊C4 [×2], C4×D4, C4×D4, C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×6], C2×Dic5 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5 [×2], C22×D5 [×2], C22×D5 [×10], C22×C10 [×2], D45D4, C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×8], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×2], C2×D20 [×2], C2×D20 [×2], D4×D5 [×4], D42D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C23×D5 [×2], C4×D20, C4.D20, C22⋊D20 [×2], C22.D20 [×2], C4⋊D20, D102Q8, C2×D10⋊C4 [×2], C207D4 [×2], D4×C20, C2×D4×D5, C2×D42D5, D45D20
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ 1+4, D20 [×4], C22×D5 [×7], D45D4, C2×D20 [×6], C23×D5, C22×D20, D46D10, D5×C4○D4, D45D20

Smallest permutation representation of D45D20
On 80 points
Generators in S80
(1 68 32 57)(2 58 33 69)(3 70 34 59)(4 60 35 71)(5 72 36 41)(6 42 37 73)(7 74 38 43)(8 44 39 75)(9 76 40 45)(10 46 21 77)(11 78 22 47)(12 48 23 79)(13 80 24 49)(14 50 25 61)(15 62 26 51)(16 52 27 63)(17 64 28 53)(18 54 29 65)(19 66 30 55)(20 56 31 67)
(1 57)(2 69)(3 59)(4 71)(5 41)(6 73)(7 43)(8 75)(9 45)(10 77)(11 47)(12 79)(13 49)(14 61)(15 51)(16 63)(17 53)(18 65)(19 55)(20 67)(21 46)(22 78)(23 48)(24 80)(25 50)(26 62)(27 52)(28 64)(29 54)(30 66)(31 56)(32 68)(33 58)(34 70)(35 60)(36 72)(37 42)(38 74)(39 44)(40 76)
(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 60)(18 59)(19 58)(20 57)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 70)(30 69)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 80)(40 79)

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L4A4B4C4D4E4F4G4H4I4J4K4L5A5B10A···10F10G···10N20A···20H20I···20X
order12222222222224444444444445510···1010···1020···2020···20
size11112222101020202022224441010202020222···24···42···24···4

65 irreducible representations

dim111111111111222222222444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D5C4○D4D10D10D10D10D10D202+ 1+4D46D10D5×C4○D4
kernelD45D20C4×D20C4.D20C22⋊D20C22.D20C4⋊D20D102Q8C2×D10⋊C4C207D4D4×C20C2×D4×D5C2×D42D5C5×D4C4×D4D10C42C22⋊C4C4⋊C4C22×C4C2×D4D4C10C2C2
# reps1112211221114242424216144

Matrix representation of D45D20 in GL6(𝔽41)

100000
010000
001000
000100
00004039
000011
,
100000
010000
001000
000100
00004039
000001
,
1400000
8340000
0013200
00234000
0000320
000099
,
100000
8400000
0013200
0004000
00003223
000099

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1],[1,8,0,0,0,0,40,34,0,0,0,0,0,0,1,23,0,0,0,0,32,40,0,0,0,0,0,0,32,9,0,0,0,0,0,9],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,32,40,0,0,0,0,0,0,32,9,0,0,0,0,23,9] >;

D45D20 in GAP, Magma, Sage, TeX

D_4\rtimes_5D_{20}
% in TeX

G:=Group("D4:5D20");
// GroupNames label

G:=SmallGroup(320,1226);
// by ID

G=gap.SmallGroup(320,1226);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,192,12550]);
// Polycyclic

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

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