Q16:D10 - GroupNames

G = Q₁₆:D₁₀ order 320 = 2⁶·5

4^th semidirect product of Q₁₆ and D₁₀ acting via D₁₀/C₁₀=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₈:₁₀D₁₀, Q₁₆:₉D₁₀, SD₁₆:₁₀D₁₀, D₄₀:₂₂C₂², C₂₀.₁₅C₂⁴, C₄₀.₃₇C₂³, D₂₀.₁₀C₂³, Dic₂₀:₁₉C₂², Dic₁₀.₁₀C₂³, C₄oD₈:₃D₅, C₄oD₄:₈D₁₀, (C₂xC₈):₁₂D₁₀, D₈:D₅:₇C₂, D₄₀:C₂:₇C₂, (C₂xC₄₀):₅C₂², C₄.₂₂₂(D₄xD₅), (D₄xD₅):₇C₂², D₄₀:₇C₂:₇C₂, (Q₈xD₅):₈C₂², C₂².₅(D₄xD₅), D₄:D₅:₁₃C₂², Q₁₆:D₅:₇C₂, D₁₀.₈₇(C₂xD₄), (C₄xD₅).₁₀₇D₄, C₂₀.₃₈₁(C₂xD₄), SD₁₆:D₅:₇C₂, C₄oD₂₀:₆C₂², (C₅xD₈):₁₅C₂², Q₈:D₅:₁₂C₂², C₅:₂C₈.₆C₂³, D₄.₉(C₂²xD₅), (C₄xD₅).₈C₂³, (C₅xD₄).₉C₂³, C₄.₁₅(C₂³xD₅), C₈.₁₅(C₂²xD₅), D₄.₈D₁₀:₂C₂, Q₈.₉(C₂²xD₅), (C₅xQ₈).₉C₂³, C₄₀:C₂:₁₆C₂², C₈:D₅:₁₅C₂², C₅:₂(D₈:C₂²), D₄.D₅:₁₂C₂², (C₂xDic₅).₈₈D₄, Dic₅.₉₈(C₂xD₄), (C₅xQ₁₆):₁₃C₂², C₅:Q₁₆:₁₁C₂², (C₂²xD₅).₅₀D₄, (C₂xC₂₀).₅₃₂C₂³, (C₅xSD₁₆):₁₀C₂², D₄:₂D₅.₉C₂², C₁₀.₁₁₆(C₂²xD₄), Q₈:₂D₅.₉C₂², C₂.₈₉(C₂xD₄xD₅), (C₅xC₄oD₈):₃C₂, (D₅xC₄oD₄):₂C₂, (C₂xC₈:D₅):₁C₂, (C₂xC₁₀).₁₂(C₂xD₄), (C₅xC₄oD₄):₂C₂², (C₂xC₅:₂C₈):₁₆C₂², (C₂xC₄xD₅).₁₆₈C₂², (C₂xC₄).₆₁₉(C₂²xD₅), SmallGroup(320,1440)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — Q₁₆:D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₄xD₅ — C₂xC₄xD₅ — D₅xC₄oD₄ — Q₁₆:D₁₀

Lower central

C₅ — C₁₀ — C₂₀ — Q₁₆:D₁₀

Upper central

C₁ — C₄ — C₂xC₄ — C₄oD₈

Generators and relations for Q₁₆:D₁₀
G = < a,b,c,d | a⁸=c¹⁰=d²=1, b²=a⁴, bab^-1=cac^-1=a^-1, dad=a³, cbc^-1=a⁶b, dbd=a²b, dcd=c^-1 >

Subgroups: 1022 in 262 conjugacy classes, 99 normal (33 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, D₅, C₁₀, C₁₀, C₂xC₈, C₂xC₈, M₄(2), D₈, D₈, SD₁₆, SD₁₆, Q₁₆, Q₁₆, C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₄oD₄, Dic₅, Dic₅, C₂₀, C₂₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₂xM₄(2), C₄oD₈, C₄oD₈, C₈:C₂², C₈.C₂², C₂xC₄oD₄, C₅:₂C₈, C₄₀, Dic₁₀, Dic₁₀, C₄xD₅, C₄xD₅, D₂₀, D₂₀, C₂xDic₅, C₂xDic₅, C₅:D₄, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₅xD₄, C₅xQ₈, C₂²xD₅, C₂²xD₅, D₈:C₂², C₈:D₅, C₄₀:C₂, D₄₀, Dic₂₀, C₂xC₅:₂C₈, D₄:D₅, D₄.D₅, Q₈:D₅, C₅:Q₁₆, C₂xC₄₀, C₅xD₈, C₅xSD₁₆, C₅xQ₁₆, C₂xC₄xD₅, C₂xC₄xD₅, C₄oD₂₀, C₄oD₂₀, D₄xD₅, D₄xD₅, D₄:₂D₅, D₄:₂D₅, Q₈xD₅, Q₈:₂D₅, C₅xC₄oD₄, C₂xC₈:D₅, D₄₀:₇C₂, D₈:D₅, D₄₀:C₂, SD₁₆:D₅, Q₁₆:D₅, D₄.₈D₁₀, C₅xC₄oD₈, D₅xC₄oD₄, Q₁₆:D₁₀
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂xD₄, C₂⁴, D₁₀, C₂²xD₄, C₂²xD₅, D₈:C₂², D₄xD₅, C₂³xD₅, C₂xD₄xD₅, Q₁₆:D₁₀

Smallest permutation representation of Q₁₆:D₁₀
►On 80 points

Generators in S₈₀

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

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

D₁₀

D₈:C₂²

D₄xD₅

Q₁₆:D₁₀

kernel

Q₁₆:D₁₀

C₂xC₈:D₅

D₄₀:₇C₂

D₈:D₅

D₄₀:C₂

SD₁₆:D₅

Q₁₆:D₅

D₄.₈D₁₀

C₅xC₄oD₈

D₅xC₄oD₄

C₄xD₅

C₂xDic₅

C₂²xD₅

C₄oD₈

C₂xC₈

D₈

SD₁₆

Q₁₆

C₄oD₄

C₅

C₄

C₂²

C₁

# reps

Matrix representation of Q₁₆:D₁₀ ►in GL₆(F₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	40
0	0	0	40	0	0
0	0	40	0	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	32	0	0
0	0	32	0	0	0
0	0	0	0	9	0
0	0	0	0	0	32

40	35	0	0	0	0
6	35	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

1	0	0	0	0	0
35	40	0	0	0	0
0	0	0	0	0	32
0	0	0	0	9	0
0	0	0	32	0	0
0	0	9	0	0	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,32],[40,6,0,0,0,0,35,35,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,35,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,9,0,0,0,0,32,0,0,0,0,9,0,0,0,0,32,0,0,0] >;

Q₁₆:D₁₀ in GAP, Magma, Sage, TeX

Q_{16}\rtimes D_{10}

% in TeX

G:=Group("Q16:D10");

// GroupNames label

G:=SmallGroup(320,1440);

// by ID

G=gap.SmallGroup(320,1440);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,570,185,438,235,102,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^8=c^10=d^2=1,b^2=a^4,b*a*b^-1=c*a*c^-1=a^-1,d*a*d=a^3,c*b*c^-1=a^6*b,d*b*d=a^2*b,d*c*d=c^-1>;

// generators/relations

G = Q16:D10 order 320 = 26·5

4th semidirect product of Q16 and D10 acting via D10/C10=C2

G = Q₁₆:D₁₀ order 320 = 2⁶·5

4^th semidirect product of Q₁₆ and D₁₀ acting via D₁₀/C₁₀=C₂