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## G = SD16×F5order 320 = 26·5

### Direct product of SD16 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — SD16×F5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4×F5 — D4×F5 — SD16×F5
 Lower central C5 — C10 — C20 — SD16×F5
 Upper central C1 — C2 — C4 — SD16

Generators and relations for SD16×F5
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a3, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 546 in 122 conjugacy classes, 44 normal (42 characteristic)
C1, C2, C2 [×4], C4, C4 [×8], C22 [×5], C5, C8, C8 [×2], C2×C4 [×8], D4, D4 [×2], Q8, Q8 [×2], C23, D5 [×2], D5, C10, C10, C42 [×2], C22⋊C4, C4⋊C4 [×3], C2×C8 [×2], SD16, SD16 [×3], C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, F5 [×2], F5 [×3], D10, D10 [×3], C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C52C8, C40, C5⋊C8, Dic10, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C2×F5 [×2], C2×F5 [×4], C22×D5, C4×SD16, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D5⋊C8, C4×F5, C4×F5, C4⋊F5 [×2], C4⋊F5, C22⋊F5, D4×D5, Q8×D5, C22×F5, C8×F5, C40⋊C4, D20⋊C4, Q8⋊F5, D5×SD16, D4×F5, Q8×F5, SD16×F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, SD16 [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×SD16, C4○D8, C2×F5 [×3], C4×SD16, C22×F5, D4×F5, SD16×F5

Smallest permutation representation of SD16×F5
On 40 points
Generators in S40
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 21)(18 24)(20 22)(25 27)(26 30)(29 31)(33 37)(34 40)(36 38)
(1 14 23 32 35)(2 15 24 25 36)(3 16 17 26 37)(4 9 18 27 38)(5 10 19 28 39)(6 11 20 29 40)(7 12 21 30 33)(8 13 22 31 34)
(1 5)(2 6)(3 7)(4 8)(9 22 38 31)(10 23 39 32)(11 24 40 25)(12 17 33 26)(13 18 34 27)(14 19 35 28)(15 20 36 29)(16 21 37 30)

G:=sub<Sym(40)| (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,21)(18,24)(20,22)(25,27)(26,30)(29,31)(33,37)(34,40)(36,38), (1,14,23,32,35)(2,15,24,25,36)(3,16,17,26,37)(4,9,18,27,38)(5,10,19,28,39)(6,11,20,29,40)(7,12,21,30,33)(8,13,22,31,34), (1,5)(2,6)(3,7)(4,8)(9,22,38,31)(10,23,39,32)(11,24,40,25)(12,17,33,26)(13,18,34,27)(14,19,35,28)(15,20,36,29)(16,21,37,30)>;

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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,21)(18,24)(20,22)(25,27)(26,30)(29,31)(33,37)(34,40)(36,38), (1,14,23,32,35)(2,15,24,25,36)(3,16,17,26,37)(4,9,18,27,38)(5,10,19,28,39)(6,11,20,29,40)(7,12,21,30,33)(8,13,22,31,34), (1,5)(2,6)(3,7)(4,8)(9,22,38,31)(10,23,39,32)(11,24,40,25)(12,17,33,26)(13,18,34,27)(14,19,35,28)(15,20,36,29)(16,21,37,30) );

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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,21),(18,24),(20,22),(25,27),(26,30),(29,31),(33,37),(34,40),(36,38)], [(1,14,23,32,35),(2,15,24,25,36),(3,16,17,26,37),(4,9,18,27,38),(5,10,19,28,39),(6,11,20,29,40),(7,12,21,30,33),(8,13,22,31,34)], [(1,5),(2,6),(3,7),(4,8),(9,22,38,31),(10,23,39,32),(11,24,40,25),(12,17,33,26),(13,18,34,27),(14,19,35,28),(15,20,36,29),(16,21,37,30)])

35 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J ··· 4N 5 8A 8B 8C ··· 8H 10A 10B 20A 20B 40A 40B order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 ··· 4 5 8 8 8 ··· 8 10 10 20 20 40 40 size 1 1 4 5 5 20 2 4 5 5 5 5 10 10 10 20 ··· 20 4 2 2 10 ··· 10 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 2 4 4 4 4 8 8 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 SD16 C4○D8 F5 C2×F5 C2×F5 C2×F5 D4×F5 SD16×F5 kernel SD16×F5 C8×F5 C40⋊C4 D20⋊C4 Q8⋊F5 D5×SD16 D4×F5 Q8×F5 C40⋊C2 D4.D5 Q8⋊D5 C5×SD16 C2×F5 Dic5 F5 D5 SD16 C8 D4 Q8 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 1 1 1 1 1 2

Matrix representation of SD16×F5 in GL6(𝔽41)

 15 15 0 0 0 0 26 15 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
,
 40 0 0 0 0 0 0 1 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
,
 32 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))| [15,26,0,0,0,0,15,15,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

SD16×F5 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times F_5
% in TeX

G:=Group("SD16xF5");
// GroupNames label

G:=SmallGroup(320,1072);
// by ID

G=gap.SmallGroup(320,1072);
# by ID

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

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