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G = SD16⋊2F5  order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: SD16⋊2F5, C5⋊C8.2D4, D4⋊D5⋊4C4, C8.7(C2×F5), C40⋊C2⋊2C4, C8⋊F5⋊5C2, C5⋊Q16⋊2C4, D4.F5⋊4C2, Q8.F5⋊2C2, D4.6(C2×F5), C2.22(D4×F5), Q8.2(C2×F5), C5⋊2(C8.26D4), C40.15(C2×C4), D4⋊F5⋊4C2, Q8⋊2F5⋊2C2, (C5×SD16)⋊2C4, D20.4(C2×C4), C10.21(C4×D4), D10.Q8⋊5C2, C4.8(C22×F5), C20.8(C22×C4), D5⋊C8.4C22, (C4×F5).4C22, D10.4(C4○D4), C4.F5.4C22, Dic10.4(C2×C4), Dic5.75(C2×D4), (C8×D5).26C22, (C4×D5).30C23, SD16⋊3D5.1C2, D4⋊2D5.7C22, Q8⋊2D5.5C22, (C5×D4).6(C2×C4), (C5×Q8).2(C2×C4), C5⋊2C8.10(C2×C4), SmallGroup(320,1075)

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

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, C5⋊2C8, 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, D4⋊2D5, Q8⋊2D5, C8⋊F5, D10.Q8, D4⋊F5, Q8⋊2F5, SD16⋊3D5, D4.F5, Q8.F5, SD16⋊2F5

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, SD16⋊2F5

Generators and relations
 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 >

Smallest permutation representation
►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 48)(10 43)(11 46)(12 41)(13 44)(14 47)(15 42)(16 45)(17 38)(18 33)(19 36)(20 39)(21 34)(22 37)(23 40)(24 35)(25 73)(26 76)(27 79)(28 74)(29 77)(30 80)(31 75)(32 78)(49 69)(50 72)(51 67)(52 70)(53 65)(54 68)(55 71)(56 66)

(1 77 35 14 71)(2 78 36 15 72)(3 79 37 16 65)(4 80 38 9 66)(5 73 39 10 67)(6 74 40 11 68)(7 75 33 12 69)(8 76 34 13 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 76 38 70)(10 73 39 67)(11 78 40 72)(12 75 33 69)(13 80 34 66)(14 77 35 71)(15 74 36 68)(16 79 37 65)(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)(9,48)(10,43)(11,46)(12,41)(13,44)(14,47)(15,42)(16,45)(17,38)(18,33)(19,36)(20,39)(21,34)(22,37)(23,40)(24,35)(25,73)(26,76)(27,79)(28,74)(29,77)(30,80)(31,75)(32,78)(49,69)(50,72)(51,67)(52,70)(53,65)(54,68)(55,71)(56,66), (1,77,35,14,71)(2,78,36,15,72)(3,79,37,16,65)(4,80,38,9,66)(5,73,39,10,67)(6,74,40,11,68)(7,75,33,12,69)(8,76,34,13,70)(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)(9,76,38,70)(10,73,39,67)(11,78,40,72)(12,75,33,69)(13,80,34,66)(14,77,35,71)(15,74,36,68)(16,79,37,65)(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)(9,48)(10,43)(11,46)(12,41)(13,44)(14,47)(15,42)(16,45)(17,38)(18,33)(19,36)(20,39)(21,34)(22,37)(23,40)(24,35)(25,73)(26,76)(27,79)(28,74)(29,77)(30,80)(31,75)(32,78)(49,69)(50,72)(51,67)(52,70)(53,65)(54,68)(55,71)(56,66), (1,77,35,14,71)(2,78,36,15,72)(3,79,37,16,65)(4,80,38,9,66)(5,73,39,10,67)(6,74,40,11,68)(7,75,33,12,69)(8,76,34,13,70)(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)(9,76,38,70)(10,73,39,67)(11,78,40,72)(12,75,33,69)(13,80,34,66)(14,77,35,71)(15,74,36,68)(16,79,37,65)(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),(9,48),(10,43),(11,46),(12,41),(13,44),(14,47),(15,42),(16,45),(17,38),(18,33),(19,36),(20,39),(21,34),(22,37),(23,40),(24,35),(25,73),(26,76),(27,79),(28,74),(29,77),(30,80),(31,75),(32,78),(49,69),(50,72),(51,67),(52,70),(53,65),(54,68),(55,71),(56,66)], [(1,77,35,14,71),(2,78,36,15,72),(3,79,37,16,65),(4,80,38,9,66),(5,73,39,10,67),(6,74,40,11,68),(7,75,33,12,69),(8,76,34,13,70),(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),(9,76,38,70),(10,73,39,67),(11,78,40,72),(12,75,33,69),(13,80,34,66),(14,77,35,71),(15,74,36,68),(16,79,37,65),(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 ►G ⊆ GL8(𝔽41)

400000000
040000000
004000000
000400000
000020037
00003803220
00002106
000090039
,
10000000
01000000
00100000
00010000
000028050
000030161
0000320130
0000191230
,
000400000
100400000
010400000
001400000
00001000
00000100
00000010
00000001
,
00100000
10000000
00010000
01000000
00001000
000040900
0000260320
000010040

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] >;
 

Character table of SD16⋊2F5

 class 12A2B2C2D4A4B4C4D4E4F4G58A8B8C8D8E8F8G8H8I8J10A10B20A20B40A40B
 size 114102024552020204410101010202020202041681688
ρ111111111111111111111111111111    trivial
ρ21111-11-1111111-1-1-1-1-1-111-1-1111-1-1-1    linear of order 2
ρ311-11-11-11111-11111111-1-1-1-11-11-111    linear of order 2
ρ411-111111111-11-1-1-1-1-1-1-1-1111-111-1-1    linear of order 2
ρ511-1111111-1-1-11-11111-111-1-11-111-1-1    linear of order 2
ρ611-11-11-111-1-1-111-1-1-1-1111111-11-111    linear of order 2
ρ71111-11-111-1-111-11111-1-1-111111-1-1-1    linear of order 2
ρ8111111111-1-1111-1-1-1-11-1-1-1-1111111    linear of order 2
ρ9111-1-111-1-1i-i-111-iii-i-1-ii-ii111111    linear of order 4
ρ10111-111-1-1-1i-i-11-1i-i-ii1-iii-i111-1-1-1    linear of order 4
ρ11111-1-111-1-1-ii-111i-i-ii-1i-ii-i111111    linear of order 4
ρ12111-111-1-1-1-ii-11-1-iii-i1i-i-ii111-1-1-1    linear of order 4
ρ1311-1-111-1-1-1-ii111i-i-ii-1-ii-ii1-11-111    linear of order 4
ρ1411-1-1-111-1-1-ii11-1-iii-i1-iii-i1-111-1-1    linear of order 4
ρ1511-1-111-1-1-1i-i111-iii-i-1i-ii-i1-11-111    linear of order 4
ρ1611-1-1-111-1-1i-i11-1i-i-ii1i-i-ii1-111-1-1    linear of order 4
ρ17220-20-202200020-22-220000020-2000    orthogonal lifted from D4
ρ18220-20-2022000202-22-20000020-2000    orthogonal lifted from D4
ρ1922020-20-2-2000202i2i2i2i0000020-2000    complex lifted from C4○D4
ρ2022020-20-2-2000202i2i2i2i0000020-2000    complex lifted from C4○D4
ρ2144-4004400000-1-4000000000-11-1-111    orthogonal lifted from C2×F5
ρ2244-4004-400000-14000000000-11-11-1-1    orthogonal lifted from C2×F5
ρ23444004400000-14000000000-1-1-1-1-1-1    orthogonal lifted from F5
ρ24444004-400000-1-4000000000-1-1-1111    orthogonal lifted from C2×F5
ρ254-4000004i4i00040000000000-400000    complex lifted from C8.26D4
ρ264-4000004i4i00040000000000-400000    complex lifted from C8.26D4
ρ2788000-8000000-20000000000-202000    orthogonal lifted from D4×F5
ρ288-80000000000-200000000002000√-10√-10    complex faithful
ρ298-80000000000-200000000002000√-10√-10    complex faithful

In GAP, Magma, Sage, TeX 

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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