Copied to
clipboard

## G = D10.SD16order 320 = 26·5

### 7th non-split extension by D10 of SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D10.SD16
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.3Q8 — D10.SD16
 Lower central C5 — C2×C10 — C2×C20 — D10.SD16
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for D10.SD16
G = < a,b,c,d | a10=b2=c8=d2=1, bab=a-1, cac-1=a3, ad=da, cbc-1=a7b, dbd=a5b, dcd=a4bc3 >

Subgroups: 490 in 90 conjugacy classes, 24 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×D4, Dic5, C20, F5, D10, D10, C2×C10, C2×C10, C2.C42, C22⋊C8, C4⋊D4, C5⋊C8, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C2×F5, C22×D5, C22×C10, C22.SD16, C4⋊Dic5, C23.D5, C2×C5⋊C8, C2×C4×D5, C2×C5⋊D4, D4×C10, C22×F5, D10⋊C8, D10.3Q8, C202D4, D10.SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, D8, SD16, F5, C23⋊C4, D4⋊C4, C4≀C2, C2×F5, C22.SD16, C22⋊F5, D20⋊C4, D4⋊F5, C23⋊F5, D10.SD16

Character table of D10.SD16

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 5 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 20A 20B size 1 1 1 1 8 10 10 4 10 10 20 20 20 20 40 4 20 20 20 20 4 4 4 8 8 8 8 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 -i i i -i -1 1 -i i i -i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 1 1 1 -1 -1 1 -1 -1 i -i -i i -1 1 i -i -i i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ7 1 1 1 1 -1 -1 -1 1 -1 -1 -i i i -i 1 1 i -i -i i 1 1 1 -1 -1 -1 -1 1 1 linear of order 4 ρ8 1 1 1 1 -1 -1 -1 1 -1 -1 i -i -i i 1 1 -i i i -i 1 1 1 -1 -1 -1 -1 1 1 linear of order 4 ρ9 2 2 2 2 0 -2 -2 -2 2 2 0 0 0 0 0 2 0 0 0 0 2 2 2 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ10 2 2 2 2 0 2 2 -2 -2 -2 0 0 0 0 0 2 0 0 0 0 2 2 2 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 2 -2 0 0 0 0 0 0 0 0 2 √2 -√2 √2 -√2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D8 ρ12 2 -2 2 -2 0 2 -2 0 0 0 0 0 0 0 0 2 -√2 √2 -√2 √2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D8 ρ13 2 -2 -2 2 0 0 0 0 -2i 2i -1-i 1-i -1+i 1+i 0 2 0 0 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4≀C2 ρ14 2 -2 -2 2 0 0 0 0 2i -2i -1+i 1+i -1-i 1-i 0 2 0 0 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4≀C2 ρ15 2 -2 2 -2 0 -2 2 0 0 0 0 0 0 0 0 2 -√-2 -√-2 √-2 √-2 2 -2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ16 2 -2 2 -2 0 -2 2 0 0 0 0 0 0 0 0 2 √-2 √-2 -√-2 -√-2 2 -2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ17 2 -2 -2 2 0 0 0 0 2i -2i 1-i -1-i 1+i -1+i 0 2 0 0 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4≀C2 ρ18 2 -2 -2 2 0 0 0 0 -2i 2i 1+i -1+i 1-i -1-i 0 2 0 0 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4≀C2 ρ19 4 4 4 4 4 0 0 4 0 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ20 4 4 4 4 -4 0 0 4 0 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from C2×F5 ρ21 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 -4 4 -4 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 4 4 4 0 0 0 -4 0 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 -√5 -√5 √5 √5 1 1 orthogonal lifted from C22⋊F5 ρ23 4 4 4 4 0 0 0 -4 0 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 √5 √5 -√5 -√5 1 1 orthogonal lifted from C22⋊F5 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 1 2ζ52+2ζ5+1 2ζ54+2ζ53+1 2ζ53+2ζ5+1 2ζ54+2ζ52+1 -√5 √5 complex lifted from C23⋊F5 ρ25 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 1 2ζ54+2ζ53+1 2ζ52+2ζ5+1 2ζ54+2ζ52+1 2ζ53+2ζ5+1 -√5 √5 complex lifted from C23⋊F5 ρ26 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 1 2ζ54+2ζ52+1 2ζ53+2ζ5+1 2ζ52+2ζ5+1 2ζ54+2ζ53+1 √5 -√5 complex lifted from C23⋊F5 ρ27 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 1 2ζ53+2ζ5+1 2ζ54+2ζ52+1 2ζ54+2ζ53+1 2ζ52+2ζ5+1 √5 -√5 complex lifted from C23⋊F5 ρ28 8 -8 8 -8 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 -2 2 2 0 0 0 0 0 0 orthogonal lifted from D20⋊C4, Schur index 2 ρ29 8 -8 -8 8 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 2 2 -2 0 0 0 0 0 0 symplectic lifted from D4⋊F5, Schur index 2

Smallest permutation representation of D10.SD16
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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 71)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 65)(52 64)(53 63)(54 62)(55 61)(56 70)(57 69)(58 68)(59 67)(60 66)
(1 52 32 11 30 70 50 77)(2 59 31 14 21 67 49 80)(3 56 40 17 22 64 48 73)(4 53 39 20 23 61 47 76)(5 60 38 13 24 68 46 79)(6 57 37 16 25 65 45 72)(7 54 36 19 26 62 44 75)(8 51 35 12 27 69 43 78)(9 58 34 15 28 66 42 71)(10 55 33 18 29 63 41 74)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 41)(18 42)(19 43)(20 44)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,71)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,65)(52,64)(53,63)(54,62)(55,61)(56,70)(57,69)(58,68)(59,67)(60,66), (1,52,32,11,30,70,50,77)(2,59,31,14,21,67,49,80)(3,56,40,17,22,64,48,73)(4,53,39,20,23,61,47,76)(5,60,38,13,24,68,46,79)(6,57,37,16,25,65,45,72)(7,54,36,19,26,62,44,75)(8,51,35,12,27,69,43,78)(9,58,34,15,28,66,42,71)(10,55,33,18,29,63,41,74), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,51)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,41)(18,42)(19,43)(20,44)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)>;

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

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

Matrix representation of D10.SD16 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 40 0 0 0 0 1 0 0 0 40 0 1 0 0 0 0 40 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 1 0 0 0 40 0 1 0 0 0 0 0 1 0 0 0 0 0 1 40
,
 14 26 0 0 0 0 0 38 0 0 0 0 0 0 9 9 27 10 0 0 36 19 13 19 0 0 22 28 22 5 0 0 31 14 32 32
,
 27 29 0 0 0 0 6 14 0 0 0 0 0 0 14 32 31 32 0 0 28 5 22 22 0 0 19 19 36 13 0 0 9 10 9 27

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,1,1,1,0,0,40,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,1,1,1,1,0,0,0,0,0,40],[14,0,0,0,0,0,26,38,0,0,0,0,0,0,9,36,22,31,0,0,9,19,28,14,0,0,27,13,22,32,0,0,10,19,5,32],[27,6,0,0,0,0,29,14,0,0,0,0,0,0,14,28,19,9,0,0,32,5,19,10,0,0,31,22,36,9,0,0,32,22,13,27] >;

D10.SD16 in GAP, Magma, Sage, TeX

D_{10}.{\rm SD}_{16}
% in TeX

G:=Group("D10.SD16");
// GroupNames label

G:=SmallGroup(320,258);
// by ID

G=gap.SmallGroup(320,258);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,1571,570,136,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^7*b,d*b*d=a^5*b,d*c*d=a^4*b*c^3>;
// generators/relations

Export

׿
×
𝔽