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G = D5.D16order 320 = 26·5

The non-split extension by D5 of D16 acting via D16/D8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D81F5, D404C4, D5.2D16, D10.19D8, D5.2SD32, Dic5.4SD16, C5⋊(C2.D16), (C5×D8)⋊4C4, D5⋊C161C2, C40.8(C2×C4), D5.D81C2, (D5×D8).4C2, C8.10(C2×F5), (C4×D5).21D4, C52C8.12D4, C4.2(C22⋊F5), C20.2(C22⋊C4), C2.7(D20⋊C4), (C8×D5).17C22, C10.6(D4⋊C4), SmallGroup(320,242)

Series: Derived Chief Lower central Upper central

C1C40 — D5.D16
C1C5C10C20C4×D5C8×D5D5.D8 — D5.D16
C5C10C20C40 — D5.D16
C1C2C4C8D8

Generators and relations for D5.D16
 G = < a,b,c,d | a5=b2=c16=1, d2=a-1b, bab=a-1, cac-1=dad-1=a3, cbc-1=dbd-1=a2b, dcd-1=a-1bc-1 >

Subgroups: 458 in 66 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2 [×4], C4, C4 [×2], C22 [×5], C5, C8, C8, C2×C4 [×2], D4 [×3], C23, D5 [×2], D5, C10, C10, C16, C4⋊C4, C2×C8, D8, D8 [×2], C2×D4, Dic5, C20, F5, D10, D10 [×3], C2×C10, C2.D8, C2×C16, C2×D8, C52C8, C40, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C22×D5, C2.D16, C5⋊C16, C8×D5, D40, D4⋊D5, C5×D8, C4⋊F5, D4×D5, D5⋊C16, D5.D8, D5×D8, D5.D16
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, F5, D4⋊C4, D16, SD32, C2×F5, C2.D16, C22⋊F5, D20⋊C4, D5.D16

Character table of D5.D16

 class 12A2B2C2D2E4A4B4C4D58A8B8C8D10A10B10C16A16B16C16D16E16F16G16H2040A40B
 size 115584021040404221010416161010101010101010888
ρ111111111111111111111111111111    trivial
ρ21111-1-11111111111-1-1-1-1-1-1-1-1-1-1111    linear of order 2
ρ31111-1-111-1-1111111-1-111111111111    linear of order 2
ρ411111111-1-111111111-1-1-1-1-1-1-1-1111    linear of order 2
ρ511-1-11-11-1-ii111-1-1111-iiiii-i-i-i111    linear of order 4
ρ611-1-11-11-1i-i111-1-1111i-i-i-i-iiii111    linear of order 4
ρ711-1-1-111-1-ii111-1-11-1-1i-i-i-i-iiii111    linear of order 4
ρ811-1-1-111-1i-i111-1-11-1-1-iiiii-i-i-i111    linear of order 4
ρ922-2-2002-2002-2-222200000000002-2-2    orthogonal lifted from D4
ρ1022220022002-2-2-2-2200000000002-2-2    orthogonal lifted from D4
ρ11222200-2-20020000200-222-2-222-2-200    orthogonal lifted from D8
ρ12222200-2-200200002002-2-222-2-22-200    orthogonal lifted from D8
ρ132-22-20000002-22-22-200ζ16516316716ζ16716165163ζ16516316716ζ167161651630-22    orthogonal lifted from D16
ρ142-22-200000022-22-2-200ζ16716ζ16516316516316716ζ16716ζ1651631651631671602-2    orthogonal lifted from D16
ρ152-22-20000002-22-22-200165163ζ1671616716ζ165163165163ζ1671616716ζ1651630-22    orthogonal lifted from D16
ρ162-22-200000022-22-2-20016716165163ζ165163ζ1671616716165163ζ165163ζ1671602-2    orthogonal lifted from D16
ρ1722-2-200-220020000200--2--2--2-2-2-2-2--2-200    complex lifted from SD16
ρ182-2-220000002-222-2-200ζ16716ζ165163ζ16131611ζ16716ζ1615169ζ16131611ζ165163ζ16151690-22    complex lifted from SD32
ρ1922-2-200-220020000200-2-2-2--2--2--2--2-2-200    complex lifted from SD16
ρ202-2-2200000022-2-22-200ζ165163ζ1615169ζ16716ζ165163ζ16131611ζ16716ζ1615169ζ1613161102-2    complex lifted from SD32
ρ212-2-2200000022-2-22-200ζ16131611ζ16716ζ1615169ζ16131611ζ165163ζ1615169ζ16716ζ16516302-2    complex lifted from SD32
ρ222-2-220000002-222-2-200ζ1615169ζ16131611ζ165163ζ1615169ζ16716ζ165163ζ16131611ζ167160-22    complex lifted from SD32
ρ234400-404000-14400-11100000000-1-1-1    orthogonal lifted from C2×F5
ρ244400404000-14400-1-1-100000000-1-1-1    orthogonal lifted from F5
ρ254400004000-1-4-400-1-5500000000-111    orthogonal lifted from C22⋊F5
ρ264400004000-1-4-400-15-500000000-111    orthogonal lifted from C22⋊F5
ρ27880000-8000-20000-20000000000200    orthogonal lifted from D20⋊C4, Schur index 2
ρ288-800000000-2-4242002000000000002-2    orthogonal faithful, Schur index 2
ρ298-800000000-242-4200200000000000-22    orthogonal faithful, Schur index 2

Smallest permutation representation of D5.D16
On 80 points
Generators in S80
(1 78 33 22 54)(2 23 79 55 34)(3 56 24 35 80)(4 36 57 65 25)(5 66 37 26 58)(6 27 67 59 38)(7 60 28 39 68)(8 40 61 69 29)(9 70 41 30 62)(10 31 71 63 42)(11 64 32 43 72)(12 44 49 73 17)(13 74 45 18 50)(14 19 75 51 46)(15 52 20 47 76)(16 48 53 77 21)
(1 54)(2 34)(3 80)(4 25)(5 58)(6 38)(7 68)(8 29)(9 62)(10 42)(11 72)(12 17)(13 50)(14 46)(15 76)(16 21)(18 74)(19 51)(22 78)(23 55)(26 66)(27 59)(30 70)(31 63)(35 56)(36 65)(39 60)(40 69)(43 64)(44 73)(47 52)(48 77)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 26 44 37)(18 36 45 25)(19 24 46 35)(20 34 47 23)(21 22 48 33)(27 32 38 43)(28 42 39 31)(29 30 40 41)(49 58 73 66)(50 65 74 57)(51 56 75 80)(52 79 76 55)(53 54 77 78)(59 64 67 72)(60 71 68 63)(61 62 69 70)

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

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

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

Matrix representation of D5.D16 in GL6(𝔽241)

100000
010000
000100
0024018900
0000240189
00005252
,
24000000
02400000
000100
001000
0000240189
000001
,
1441610000
36620000
000010
000001
00240000
0052100
,
1441610000
208970000
000010
000001
001000
0018924000

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,1,189,0,0,0,0,0,0,240,52,0,0,0,0,189,52],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,189,1],[144,36,0,0,0,0,161,62,0,0,0,0,0,0,0,0,240,52,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[144,208,0,0,0,0,161,97,0,0,0,0,0,0,0,0,1,189,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0] >;

D5.D16 in GAP, Magma, Sage, TeX

D_5.D_{16}
% in TeX

G:=Group("D5.D16");
// GroupNames label

G:=SmallGroup(320,242);
// by ID

G=gap.SmallGroup(320,242);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,675,346,192,1684,851,102,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^16=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^-1*b*c^-1>;
// generators/relations

Export

Character table of D5.D16 in TeX

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