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## G = SD16⋊2F5order 320 = 26·5

### 2nd semidirect product of SD16 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — SD16⋊2F5
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — D5⋊C8 — D4.F5 — SD16⋊2F5
 Lower central C5 — C10 — C20 — SD16⋊2F5
 Upper central C1 — C2 — C4 — SD16

Generators and relations for SD162F5
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a3, ac=ca, dad-1=a5, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 402 in 104 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×3], C4, C4 [×4], 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, D10, D10, C2×C10, C8⋊C4, 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, C8.26D4, 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, D10.Q8, D4⋊F5, Q82F5, SD163D5, D4.F5, Q8.F5, SD162F5
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], C8.26D4, C22×F5, D4×F5, SD162F5

Character table of SD162F5

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 5 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A 10B 20A 20B 40A 40B size 1 1 4 10 20 2 4 5 5 20 20 20 4 4 10 10 10 10 20 20 20 20 20 4 16 8 16 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 1 -1 1 -1 1 1 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 linear of order 2 ρ4 1 1 -1 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 linear of order 2 ρ5 1 1 -1 1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ6 1 1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 1 -1 1 1 linear of order 2 ρ7 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ9 1 1 1 -1 -1 1 1 -1 -1 i -i -1 1 1 -i i i -i -1 -i i -i i 1 1 1 1 1 1 linear of order 4 ρ10 1 1 1 -1 1 1 -1 -1 -1 i -i -1 1 -1 i -i -i i 1 -i i i -i 1 1 1 -1 -1 -1 linear of order 4 ρ11 1 1 1 -1 -1 1 1 -1 -1 -i i -1 1 1 i -i -i i -1 i -i i -i 1 1 1 1 1 1 linear of order 4 ρ12 1 1 1 -1 1 1 -1 -1 -1 -i i -1 1 -1 -i i i -i 1 i -i -i i 1 1 1 -1 -1 -1 linear of order 4 ρ13 1 1 -1 -1 1 1 -1 -1 -1 -i i 1 1 1 i -i -i i -1 -i i -i i 1 -1 1 -1 1 1 linear of order 4 ρ14 1 1 -1 -1 -1 1 1 -1 -1 -i i 1 1 -1 -i i i -i 1 -i i i -i 1 -1 1 1 -1 -1 linear of order 4 ρ15 1 1 -1 -1 1 1 -1 -1 -1 i -i 1 1 1 -i i i -i -1 i -i i -i 1 -1 1 -1 1 1 linear of order 4 ρ16 1 1 -1 -1 -1 1 1 -1 -1 i -i 1 1 -1 i -i -i i 1 i -i -i i 1 -1 1 1 -1 -1 linear of order 4 ρ17 2 2 0 -2 0 -2 0 2 2 0 0 0 2 0 -2 2 -2 2 0 0 0 0 0 2 0 -2 0 0 0 orthogonal lifted from D4 ρ18 2 2 0 -2 0 -2 0 2 2 0 0 0 2 0 2 -2 2 -2 0 0 0 0 0 2 0 -2 0 0 0 orthogonal lifted from D4 ρ19 2 2 0 2 0 -2 0 -2 -2 0 0 0 2 0 -2i -2i 2i 2i 0 0 0 0 0 2 0 -2 0 0 0 complex lifted from C4○D4 ρ20 2 2 0 2 0 -2 0 -2 -2 0 0 0 2 0 2i 2i -2i -2i 0 0 0 0 0 2 0 -2 0 0 0 complex lifted from C4○D4 ρ21 4 4 -4 0 0 4 4 0 0 0 0 0 -1 -4 0 0 0 0 0 0 0 0 0 -1 1 -1 -1 1 1 orthogonal lifted from C2×F5 ρ22 4 4 -4 0 0 4 -4 0 0 0 0 0 -1 4 0 0 0 0 0 0 0 0 0 -1 1 -1 1 -1 -1 orthogonal lifted from C2×F5 ρ23 4 4 4 0 0 4 4 0 0 0 0 0 -1 4 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ24 4 4 4 0 0 4 -4 0 0 0 0 0 -1 -4 0 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 orthogonal lifted from C2×F5 ρ25 4 -4 0 0 0 0 0 -4i 4i 0 0 0 4 0 0 0 0 0 0 0 0 0 0 -4 0 0 0 0 0 complex lifted from C8.26D4 ρ26 4 -4 0 0 0 0 0 4i -4i 0 0 0 4 0 0 0 0 0 0 0 0 0 0 -4 0 0 0 0 0 complex lifted from C8.26D4 ρ27 8 8 0 0 0 -8 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 -2 0 2 0 0 0 orthogonal lifted from D4×F5 ρ28 8 -8 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 -√-10 √-10 complex faithful ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 √-10 -√-10 complex faithful

Smallest permutation representation of SD162F5
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 59)(2 62)(3 57)(4 60)(5 63)(6 58)(7 61)(8 64)(9 28)(10 31)(11 26)(12 29)(13 32)(14 27)(15 30)(16 25)(17 36)(18 39)(19 34)(20 37)(21 40)(22 35)(23 38)(24 33)(41 77)(42 80)(43 75)(44 78)(45 73)(46 76)(47 79)(48 74)(49 69)(50 72)(51 67)(52 70)(53 65)(54 68)(55 71)(56 66)
(1 12 33 79 71)(2 13 34 80 72)(3 14 35 73 65)(4 15 36 74 66)(5 16 37 75 67)(6 9 38 76 68)(7 10 39 77 69)(8 11 40 78 70)(17 48 56 60 30)(18 41 49 61 31)(19 42 50 62 32)(20 43 51 63 25)(21 44 52 64 26)(22 45 53 57 27)(23 46 54 58 28)(24 47 55 59 29)
(2 6)(4 8)(9 34 68 80)(10 39 69 77)(11 36 70 74)(12 33 71 79)(13 38 72 76)(14 35 65 73)(15 40 66 78)(16 37 67 75)(17 54 44 32)(18 51 45 29)(19 56 46 26)(20 53 47 31)(21 50 48 28)(22 55 41 25)(23 52 42 30)(24 49 43 27)(57 59 61 63)(58 64 62 60)

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

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

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

Matrix representation of SD162F5 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 2 0 0 37 0 0 0 0 38 0 32 20 0 0 0 0 2 1 0 6 0 0 0 0 9 0 0 39
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 5 0 0 0 0 0 3 0 16 1 0 0 0 0 32 0 13 0 0 0 0 0 19 1 23 0
,
 0 0 0 40 0 0 0 0 1 0 0 40 0 0 0 0 0 1 0 40 0 0 0 0 0 0 1 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 9 0 0 0 0 0 0 26 0 32 0 0 0 0 0 1 0 0 40

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,2,38,2,9,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,37,20,6,39],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,3,32,19,0,0,0,0,0,0,0,1,0,0,0,0,5,16,13,23,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,40,26,1,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40] >;

SD162F5 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_2F_5
% in TeX

G:=Group("SD16:2F5");
// GroupNames label

G:=SmallGroup(320,1075);
// by ID

G=gap.SmallGroup(320,1075);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,184,136,851,438,102,6278,1595]);
// 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,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations

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