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G = D8⋊D10order 320 = 26·5

2nd semidirect product of D8 and D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D88D10, Q167D10, C20.51D8, C40.32D4, C40.29C23, D40.11C22, C4○D82D5, C5⋊D166C2, (C2×C10).9D8, (C2×D40)⋊22C2, C55(C16⋊C22), C10.68(C2×D8), (C2×C8).98D10, C5⋊SD326C2, (C5×D8)⋊8C22, C8.7(C5⋊D4), C20.4C86C2, C4.24(D4⋊D5), C52C164C22, C20.191(C2×D4), (C2×C20).185D4, (C5×Q16)⋊7C22, C8.35(C22×D5), C22.5(D4⋊D5), (C2×C40).104C22, (C5×C4○D8)⋊4C2, C2.23(C2×D4⋊D5), C4.17(C2×C5⋊D4), (C2×C4).81(C5⋊D4), SmallGroup(320,820)

Series: Derived Chief Lower central Upper central

C1C40 — D8⋊D10
C1C5C10C20C40D40C2×D40 — D8⋊D10
C5C10C20C40 — D8⋊D10
C1C2C2×C4C2×C8C4○D8

Generators and relations for D8⋊D10
 G = < a,b,c,d | a8=b2=c10=d2=1, bab=dad=a-1, ac=ca, cbc-1=a4b, dbd=ab, dcd=c-1 >

Subgroups: 494 in 90 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C5, C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, D5 [×2], C10, C10 [×2], C16 [×2], C2×C8, D8, D8 [×3], SD16, Q16, C2×D4, C4○D4, C20 [×2], C20, D10 [×4], C2×C10, C2×C10, M5(2), D16 [×2], SD32 [×2], C2×D8, C4○D8, C40 [×2], D20 [×3], C2×C20, C2×C20, C5×D4 [×2], C5×Q8, C22×D5, C16⋊C22, C52C16 [×2], D40 [×2], D40, C2×C40, C5×D8, C5×SD16, C5×Q16, C2×D20, C5×C4○D4, C20.4C8, C5⋊D16 [×2], C5⋊SD32 [×2], C2×D40, C5×C4○D8, D8⋊D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], C2×D8, C5⋊D4 [×2], C22×D5, C16⋊C22, D4⋊D5 [×2], C2×C5⋊D4, C2×D4⋊D5, D8⋊D10

Smallest permutation representation of D8⋊D10
On 80 points
Generators in S80
(1 33 13 27 8 38 18 23)(2 34 14 28 9 39 19 24)(3 35 15 29 10 40 20 25)(4 31 11 30 6 36 16 21)(5 32 12 26 7 37 17 22)(41 64 58 78 46 69 53 73)(42 65 59 79 47 70 54 74)(43 66 60 80 48 61 55 75)(44 67 51 71 49 62 56 76)(45 68 52 72 50 63 57 77)
(1 73)(2 79)(3 75)(4 71)(5 77)(6 76)(7 72)(8 78)(9 74)(10 80)(11 67)(12 63)(13 69)(14 65)(15 61)(16 62)(17 68)(18 64)(19 70)(20 66)(21 49)(22 45)(23 41)(24 47)(25 43)(26 50)(27 46)(28 42)(29 48)(30 44)(31 51)(32 57)(33 53)(34 59)(35 55)(36 56)(37 52)(38 58)(39 54)(40 60)
(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 5)(2 4)(6 9)(7 8)(11 19)(12 18)(13 17)(14 16)(15 20)(21 34)(22 33)(23 32)(24 31)(25 35)(26 38)(27 37)(28 36)(29 40)(30 39)(41 63)(42 62)(43 61)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 80)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C5A5B8A8B8C10A10B10C10D10E10F10G10H16A16B16C16D20A20B20C20D20E20F20G20H20I20J40A···40H
order122222444558881010101010101010161616162020202020202020202040···40
size1128404022822224224488882020202022224488884···4

44 irreducible representations

dim11111122222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2D4D4D5D8D8D10D10D10C5⋊D4C5⋊D4C16⋊C22D4⋊D5D4⋊D5D8⋊D10
kernelD8⋊D10C20.4C8C5⋊D16C5⋊SD32C2×D40C5×C4○D8C40C2×C20C4○D8C20C2×C10C2×C8D8Q16C8C2×C4C5C4C22C1
# reps11221111222222442228

Matrix representation of D8⋊D10 in GL4(𝔽241) generated by

2021200
2919900
021222829
292084232
,
2929198170
202022192
02913212
2033194179
,
2405200
1895200
175227190189
131184510
,
1000
5224000
9821420
152046027
G:=sub<GL(4,GF(241))| [20,29,0,29,212,199,212,208,0,0,228,4,0,0,29,232],[29,20,0,20,29,20,29,33,198,22,13,194,170,192,212,179],[240,189,175,131,52,52,227,184,0,0,190,51,0,0,189,0],[1,52,9,15,0,240,8,204,0,0,214,60,0,0,20,27] >;

D8⋊D10 in GAP, Magma, Sage, TeX

D_8\rtimes D_{10}
% in TeX

G:=Group("D8:D10");
// GroupNames label

G:=SmallGroup(320,820);
// by ID

G=gap.SmallGroup(320,820);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,387,675,185,192,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations

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