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G = D8⋊11D10order 320 = 26·5

5th semidirect product of D8 and D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D8⋊11D10
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C4○D20 — D4⋊8D10 — D8⋊11D10
 Lower central C5 — C10 — C20 — D8⋊11D10
 Upper central C1 — C2 — C2×C4 — C4○D8

Generators and relations for D811D10
G = < a,b,c,d | a8=b2=c10=d2=1, bab=a-1, ac=ca, dad=a3, cbc-1=a4b, dbd=a6b, dcd=c-1 >

Subgroups: 1046 in 258 conjugacy classes, 99 normal (53 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), D8, D8, SD16, SD16, Q16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C8○D4, C2×SD16, C4○D8, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C52C8, C40, Dic10, Dic10, Dic10, C4×D5, C4×D5, D20, D20, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, D4○SD16, C8×D5, C8⋊D5, C40⋊C2, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×C40, C5×D8, C5×SD16, C5×Q16, C2×Dic10, C2×Dic10, C2×D20, C2×D20, C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, D20.3C4, C2×C40⋊C2, D8⋊D5, D5×SD16, SD163D5, Q16⋊D5, D4⋊D10, D4.9D10, C5×C4○D8, D48D10, D4.10D10, D811D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, D4○SD16, D4×D5, C23×D5, C2×D4×D5, D811D10

Smallest permutation representation of D811D10
On 80 points
Generators in S80
```(1 41 26 78 63 16 40 51)(2 42 27 79 64 17 31 52)(3 43 28 80 65 18 32 53)(4 44 29 71 66 19 33 54)(5 45 30 72 67 20 34 55)(6 46 21 73 68 11 35 56)(7 47 22 74 69 12 36 57)(8 48 23 75 70 13 37 58)(9 49 24 76 61 14 38 59)(10 50 25 77 62 15 39 60)
(1 51)(2 79)(3 53)(4 71)(5 55)(6 73)(7 57)(8 75)(9 59)(10 77)(11 35)(12 22)(13 37)(14 24)(15 39)(16 26)(17 31)(18 28)(19 33)(20 30)(21 46)(23 48)(25 50)(27 42)(29 44)(32 43)(34 45)(36 47)(38 49)(40 41)(52 64)(54 66)(56 68)(58 70)(60 62)(61 76)(63 78)(65 80)(67 72)(69 74)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 70)(38 69)(39 68)(40 67)(51 55)(52 54)(56 60)(57 59)(71 79)(72 78)(73 77)(74 76)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 8E 10A 10B 10C 10D 10E 10F 10G 10H 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 40A ··· 40H order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 8 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 2 4 4 10 10 20 20 2 2 4 4 10 10 20 20 2 2 2 2 4 20 20 2 2 4 4 8 8 8 8 2 2 2 2 4 4 8 8 8 8 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 D10 D10 D4○SD16 D4×D5 D4×D5 D8⋊11D10 kernel D8⋊11D10 D20.3C4 C2×C40⋊C2 D8⋊D5 D5×SD16 SD16⋊3D5 Q16⋊D5 D4⋊D10 D4.9D10 C5×C4○D8 D4⋊8D10 D4.10D10 Dic10 D20 C5⋊D4 C4○D8 C2×C8 D8 SD16 Q16 C4○D4 C5 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 1 2 2 2 2 4 2 4 2 2 2 8

Matrix representation of D811D10 in GL4(𝔽41) generated by

 0 0 40 29 0 0 12 1 2 24 39 17 17 39 24 2
,
 39 17 1 12 24 2 29 40 39 17 2 24 24 2 17 39
,
 14 27 27 14 14 30 27 11 28 13 27 14 28 19 27 11
,
 1 7 0 0 0 40 0 0 2 14 40 34 0 39 0 1
`G:=sub<GL(4,GF(41))| [0,0,2,17,0,0,24,39,40,12,39,24,29,1,17,2],[39,24,39,24,17,2,17,2,1,29,2,17,12,40,24,39],[14,14,28,28,27,30,13,19,27,27,27,27,14,11,14,11],[1,0,2,0,7,40,14,39,0,0,40,0,0,0,34,1] >;`

D811D10 in GAP, Magma, Sage, TeX

`D_8\rtimes_{11}D_{10}`
`% in TeX`

`G:=Group("D8:11D10");`
`// GroupNames label`

`G:=SmallGroup(320,1442);`
`// by ID`

`G=gap.SmallGroup(320,1442);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,570,185,136,438,235,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^3,c*b*c^-1=a^4*b,d*b*d=a^6*b,d*c*d=c^-1>;`
`// generators/relations`

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