D10:6SD16 - GroupNames

G = D₁₀:₆SD₁₆ order 320 = 2⁶·5

2^nd semidirect product of D₁₀ and SD₁₆ acting via SD₁₆/D₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₀:₆SD₁₆, D₂₀.₁₆D₄, (C₂xC₈):₁₇D₁₀, (C₂xQ₈):₃D₁₀, (C₅xD₄).₉D₄, C₄.₆₂(D₄xD₅), C₂₀.₄₇(C₂xD₄), (C₂xC₄₀):₃₃C₂², D₁₀:₃Q₈:₃C₂, (C₂xSD₁₆):₁₀D₅, D₄.₈(C₅:D₄), C₅:₄(C₂²:SD₁₆), D₂₀:₅C₄:₃₅C₂, D₁₀:₁C₈:₃₂C₂, C₂.₂₈(D₅xSD₁₆), (Q₈xC₁₀):₃C₂², C₁₀.₅₇C₂²wrC₂, (C₁₀xSD₁₆):₂₀C₂, (C₂xD₄).₁₄₆D₁₀, D₄:Dic₅:₃₃C₂, C₄:Dic₅:₂₀C₂², (C₂xDic₅).₇₉D₄, C₁₀.₄₅(C₂xSD₁₆), C₂².₂₆₆(D₄xD₅), C₂.₂₈(D₄₀:C₂), C₁₀.₇₈(C₈:C₂²), (C₂xC₂₀).₄₄₆C₂³, (C₂²xD₅).₁₂₈D₄, (D₄xC₁₀).₉₅C₂², C₂.₂₅(C₂³:D₁₀), (C₂xD₂₀).₁₂₄C₂², (C₂xD₄xD₅).₆C₂, (C₂xQ₈:D₅):₁₇C₂, C₄.₄₂(C₂xC₅:D₄), (C₂xC₅:₂C₈):₈C₂², (C₂xC₄xD₅).₅₁C₂², (C₂xC₁₀).₃₅₈(C₂xD₄), (C₂xC₄).₅₃₅(C₂²xD₅), SmallGroup(320,796)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₂₀ — D₁₀:₆SD₁₆

Chief series

C₁ — C₅ — C₁₀ — C₂xC₁₀ — C₂xC₂₀ — C₂xC₄xD₅ — C₂xD₄xD₅ — D₁₀:₆SD₁₆

Lower central

C₅ — C₁₀ — C₂xC₂₀ — D₁₀:₆SD₁₆

Upper central

C₁ — C₂² — C₂xC₄ — C₂xSD₁₆

Generators and relations for D₁₀:₆SD₁₆
G = < a,b,c,d | a¹⁰=b²=c⁸=d²=1, bab=a^-1, ac=ca, ad=da, cbc^-1=a⁵b, bd=db, dcd=c³ >

Subgroups: 990 in 188 conjugacy classes, 45 normal (37 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, C₂³, D₅, C₁₀, C₁₀, C₂²:C₄, C₄:C₄, C₂xC₈, C₂xC₈, SD₁₆, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₂⁴, Dic₅, C₂₀, C₂₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₂²:C₈, D₄:C₄, C₂²:Q₈, C₂xSD₁₆, C₂xSD₁₆, C₂²xD₄, C₅:₂C₈, C₄₀, C₄xD₅, D₂₀, D₂₀, C₂xDic₅, C₂xDic₅, C₅:D₄, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₅xD₄, C₅xQ₈, C₂²xD₅, C₂²xD₅, C₂²xC₁₀, C₂²:SD₁₆, C₂xC₅:₂C₈, C₁₀.D₄, C₄:Dic₅, D₁₀:C₄, Q₈:D₅, C₂xC₄₀, C₅xSD₁₆, C₂xC₄xD₅, C₂xD₂₀, D₄xD₅, C₂xC₅:D₄, D₄xC₁₀, Q₈xC₁₀, C₂³xD₅, D₁₀:₁C₈, D₂₀:₅C₄, D₄:Dic₅, C₂xQ₈:D₅, D₁₀:₃Q₈, C₁₀xSD₁₆, C₂xD₄xD₅, D₁₀:₆SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, SD₁₆, C₂xD₄, D₁₀, C₂²wrC₂, C₂xSD₁₆, C₈:C₂², C₅:D₄, C₂²xD₅, C₂²:SD₁₆, D₄xD₅, C₂xC₅:D₄, D₅xSD₁₆, D₄₀:C₂, C₂³:D₁₀, D₁₀:₆SD₁₆

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

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

(11 49)(12 50)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 78)(22 79)(23 80)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(51 69)(52 70)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)

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

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

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

47 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	5A	5B	8A	8B	8C	8D	10A	···	10F	10G	10H	10I	10J	20A	20B	20C	20D	20E	20F	20G	20H	40A	···	40H
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	5	5	8	8	8	8	10	···	10	10	10	10	10	20	20	20	20	20	20	20	20	40	···	40
size	1	1	1	1	4	4	10	10	20	20	2	2	8	20	40	2	2	4	4	20	20	2	···	2	8	8	8	8	4	4	4	4	8	8	8	8	4	···	4

47 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

SD₁₆

D₁₀

C₅:D₄

C₈:C₂²

D₄xD₅

D₅xSD₁₆

D₄₀:C₂

kernel

D₁₀:₆SD₁₆

D₁₀:₁C₈

D₂₀:₅C₄

D₄:Dic₅

C₂xQ₈:D₅

D₁₀:₃Q₈

C₁₀xSD₁₆

C₂xD₄xD₅

D₂₀

C₂xDic₅

C₅xD₄

C₂²xD₅

C₂xSD₁₆

D₁₀

C₂xC₈

C₂xD₄

C₂xQ₈

D₄

C₁₀

C₄

C₂²

C₂

# reps

Matrix representation of D₁₀:₆SD₁₆ ►in GL₆(F₄₁)

40	7	0	0	0	0
34	7	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	1	0
0	0	0	0	0	1

40	0	0	0	0	0
34	1	0	0	0	0
0	0	40	0	0	0
0	0	24	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

40	0	0	0	0	0
0	40	0	0	0	0
0	0	12	1	0	0
0	0	19	29	0	0
0	0	0	0	15	15
0	0	0	0	26	15

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	17	40	0	0
0	0	0	0	1	0
0	0	0	0	0	40

G:=sub<GL(6,GF(41))| [40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,34,0,0,0,0,0,1,0,0,0,0,0,0,40,24,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,12,19,0,0,0,0,1,29,0,0,0,0,0,0,15,26,0,0,0,0,15,15],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

D₁₀:₆SD₁₆ in GAP, Magma, Sage, TeX

D_{10}\rtimes_6{\rm SD}_{16}

% in TeX

G:=Group("D10:6SD16");

// GroupNames label

G:=SmallGroup(320,796);

// by ID

G=gap.SmallGroup(320,796);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,184,851,438,102,12550]);

// Polycyclic

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

// generators/relations

G = D10:6SD16 order 320 = 26·5

2nd semidirect product of D10 and SD16 acting via SD16/D4=C2

G = D₁₀:₆SD₁₆ order 320 = 2⁶·5

2^nd semidirect product of D₁₀ and SD₁₆ acting via SD₁₆/D₄=C₂