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## G = SD16⋊3F5order 320 = 26·5

### The semidirect product of SD16 and F5 acting through Inn(SD16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — SD16⋊3F5
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — D5⋊C8 — D4.F5 — SD16⋊3F5
 Lower central C5 — C10 — C20 — SD16⋊3F5
 Upper central C1 — C2 — C4 — SD16

Generators and relations for SD163F5
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a3, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 402 in 106 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×3], C4, C4 [×5], C22 [×3], C5, C8, C8 [×5], C2×C4 [×4], D4, D4 [×3], Q8, Q8, D5 [×2], C10, C10, C42, C2×C8 [×4], M4(2) [×4], D8, SD16, SD16, Q16, C4○D4 [×2], Dic5, Dic5, C20, C20, F5 [×2], D10, D10, C2×C10, C4×C8, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C52C8, C40, C5⋊C8 [×2], C5⋊C8 [×2], Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C2×F5, C8○D8, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D5⋊C8, D5⋊C8, C4.F5 [×2], C4.F5, C4×F5, C2×C5⋊C8, C22.F5, D42D5, Q82D5, C8×F5, C40.C4, D4⋊F5, Q82F5, SD163D5, D4.F5, Q8.F5, SD163F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×F5 [×3], C8○D8, C22×F5, D4×F5, SD163F5

Smallest permutation representation of SD163F5
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 74)(2 77)(3 80)(4 75)(5 78)(6 73)(7 76)(8 79)(9 48)(10 43)(11 46)(12 41)(13 44)(14 47)(15 42)(16 45)(17 36)(18 39)(19 34)(20 37)(21 40)(22 35)(23 38)(24 33)(25 58)(26 61)(27 64)(28 59)(29 62)(30 57)(31 60)(32 63)(49 69)(50 72)(51 67)(52 70)(53 65)(54 68)(55 71)(56 66)
(1 62 33 14 71)(2 63 34 15 72)(3 64 35 16 65)(4 57 36 9 66)(5 58 37 10 67)(6 59 38 11 68)(7 60 39 12 69)(8 61 40 13 70)(17 48 56 75 30)(18 41 49 76 31)(19 42 50 77 32)(20 43 51 78 25)(21 44 52 79 26)(22 45 53 80 27)(23 46 54 73 28)(24 47 55 74 29)
(9 57 36 66)(10 58 37 67)(11 59 38 68)(12 60 39 69)(13 61 40 70)(14 62 33 71)(15 63 34 72)(16 64 35 65)(17 50 44 28)(18 51 45 29)(19 52 46 30)(20 53 47 31)(21 54 48 32)(22 55 41 25)(23 56 42 26)(24 49 43 27)(73 75 77 79)(74 76 78 80)

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 5 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 8M 8N 10A 10B 20A 20B 40A 40B order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 5 8 8 8 8 8 8 8 8 8 8 8 8 8 8 10 10 20 20 40 40 size 1 1 4 10 20 2 4 5 5 10 10 10 10 20 4 2 2 5 5 5 5 10 10 10 10 20 20 20 20 4 16 8 16 8 8

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 8 8 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 C8○D8 F5 C2×F5 C2×F5 C2×F5 D4×F5 SD16⋊3F5 kernel SD16⋊3F5 C8×F5 C40.C4 D4⋊F5 Q8⋊2F5 SD16⋊3D5 D4.F5 Q8.F5 C40⋊C2 D4⋊D5 C5⋊Q16 C5×SD16 C5⋊C8 D10 C5 SD16 C8 D4 Q8 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 8 1 1 1 1 1 2

Matrix representation of SD163F5 in GL6(𝔽41)

 3 0 0 0 0 0 0 27 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 1
,
 0 38 0 0 0 0 27 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 1 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(41))| [3,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,27,0,0,0,0,38,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[1,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

SD163F5 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_3F_5
% in TeX

G:=Group("SD16:3F5");
// GroupNames label

G:=SmallGroup(320,1074);
// by ID

G=gap.SmallGroup(320,1074);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,184,136,851,438,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^3,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations

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