D8:13D10 - GroupNames

G = D₈:₁₃D₁₀ order 320 = 2⁶·5

2^nd semidirect product of D₈ and D₁₀ acting through Inn(D₈)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₈:₁₃D₁₀, D₂₀.₂₈D₄, C₂₀.₄C₂⁴, D₄₀:₁₆C₂², C₄₀.₄₀C₂³, D₂₀.₂C₂³, Dic₁₀.₂₈D₄, Dic₂₀:₁₄C₂², Dic₁₀.₂C₂³, (D₅xD₈):₆C₂, (C₂xC₈):₉D₁₀, C₅:₂(D₄oD₈), (C₁₀xD₈):₃C₂, (C₂xD₈):₁₂D₅, C₄.₇₅(D₄xD₅), C₅:D₄.₈D₄, (C₂xD₄):₁₄D₁₀, D₈:₃D₅:₆C₂, D₈:D₅:₅C₂, (C₂xC₄₀):₃C₂², D₄:D₅:₁C₂², C₂₀.₇₉(C₂xD₄), (C₈xD₅):₇C₂², (D₄xD₅):₁C₂², D₄₀:₇C₂:₃C₂, D₄:₆D₁₀:₅C₂, C₄.₄(C₂³xD₅), D₁₀.₄₉(C₂xD₄), C₄oD₂₀:₃C₂², (C₅xD₈):₁₁C₂², C₅:₂C₈.₁C₂³, D₄.D₅:₁C₂², (C₅xD₄).₂C₂³, D₄.₂(C₂²xD₅), (C₄xD₅).₂C₂³, C₂².₂₀(D₄xD₅), C₈.₁₀(C₂²xD₅), D₂₀.₃C₄:₂C₂, D₄.D₁₀:₇C₂, D₄:₂D₅:₁C₂², (D₄xC₁₀):₂₀C₂², C₄₀:C₂:₁₄C₂², C₈:D₅:₁₃C₂², Dic₅.₅₅(C₂xD₄), (C₂xC₂₀).₅₂₁C₂³, C₁₀.₁₀₅(C₂²xD₄), C₄.Dic₅:₂₈C₂², C₂.₇₈(C₂xD₄xD₅), (C₂xC₁₀).₃₉₄(C₂xD₄), (C₂xC₄).₂₂₉(C₂²xD₅), SmallGroup(320,1429)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — D₈:₁₃D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₄xD₅ — C₄oD₂₀ — D₄:₆D₁₀ — D₈:₁₃D₁₀

Lower central

C₅ — C₁₀ — C₂₀ — D₈:₁₃D₁₀

Upper central

C₁ — C₂ — C₂xC₄ — C₂xD₈

Generators and relations for D₈:₁₃D₁₀
G = < a,b,c,d | a⁸=b²=c¹⁰=d²=1, bab=cac^-1=a^-1, ad=da, cbc^-1=a⁶b, dbd=a⁴b, dcd=c^-1 >

Subgroups: 1142 in 268 conjugacy classes, 99 normal (29 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, C₂³, D₅, C₁₀, C₁₀, C₂xC₈, C₂xC₈, M₄(2), D₈, D₈, SD₁₆, Q₁₆, C₂xD₄, C₂xD₄, C₄oD₄, Dic₅, Dic₅, C₂₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₈oD₄, C₂xD₈, C₂xD₈, C₄oD₈, C₈:C₂², 2₊¹⁺⁴, C₅:₂C₈, C₄₀, Dic₁₀, Dic₁₀, C₄xD₅, C₄xD₅, D₂₀, D₂₀, C₂xDic₅, C₅:D₄, C₅:D₄, C₂xC₂₀, C₅xD₄, C₅xD₄, C₂²xD₅, C₂²xC₁₀, D₄oD₈, C₈xD₅, C₈:D₅, C₄₀:C₂, D₄₀, Dic₂₀, C₄.Dic₅, D₄:D₅, D₄.D₅, C₂xC₄₀, C₅xD₈, C₄oD₂₀, C₄oD₂₀, D₄xD₅, D₄xD₅, D₄:₂D₅, D₄:₂D₅, C₂xC₅:D₄, D₄xC₁₀, D₂₀.₃C₄, D₄₀:₇C₂, D₅xD₈, D₈:D₅, D₈:₃D₅, D₄.D₁₀, C₁₀xD₈, D₄:₆D₁₀, D₈:₁₃D₁₀
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂xD₄, C₂⁴, D₁₀, C₂²xD₄, C₂²xD₅, D₄oD₈, D₄xD₅, C₂³xD₅, C₂xD₄xD₅, D₈:₁₃D₁₀

Smallest permutation representation of D₈:₁₃D₁₀
►On 80 points

Generators in S₈₀

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

(2 29)(4 21)(6 23)(8 25)(10 27)(11 49)(12 55)(13 41)(14 57)(15 43)(16 59)(17 45)(18 51)(19 47)(20 53)(22 70)(24 62)(26 64)(28 66)(30 68)(32 50)(34 42)(36 44)(38 46)(40 48)(61 78)(63 80)(65 72)(67 74)(69 76)

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

D₁₀

D₄oD₈

D₄xD₅

D₈:₁₃D₁₀

kernel

D₈:₁₃D₁₀

D₂₀.₃C₄

D₄₀:₇C₂

D₅xD₈

D₈:D₅

D₈:₃D₅

D₄.D₁₀

C₁₀xD₈

D₄:₆D₁₀

Dic₁₀

D₂₀

C₅:D₄

C₂xD₈

C₂xC₈

D₈

C₂xD₄

C₅

C₄

C₂²

C₁

# reps

Matrix representation of D₈:₁₃D₁₀ ►in GL₆(F₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	24	24	0	0
0	0	29	0	0	0
0	0	0	12	12	29
0	0	29	29	12	12

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

35	35	0	0	0	0
6	40	0	0	0	0
0	0	0	0	17	17
0	0	29	0	24	0
0	0	0	12	29	29
0	0	29	29	12	12

35	35	0	0	0	0
40	6	0	0	0	0
0	0	40	0	0	39
0	0	1	0	40	1
0	0	40	40	0	40
0	0	0	0	0	1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,24,29,0,29,0,0,24,0,12,29,0,0,0,0,12,12,0,0,0,0,29,12],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,40,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[35,6,0,0,0,0,35,40,0,0,0,0,0,0,0,29,0,29,0,0,0,0,12,29,0,0,17,24,29,12,0,0,17,0,29,12],[35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,1,40,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,39,1,40,1] >;

D₈:₁₃D₁₀ in GAP, Magma, Sage, TeX

D_8\rtimes_{13}D_{10}

% in TeX

G:=Group("D8:13D10");

// GroupNames label

G:=SmallGroup(320,1429);

// by ID

G=gap.SmallGroup(320,1429);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,185,438,235,102,12550]);

// Polycyclic

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

// generators/relations

G = D8:13D10 order 320 = 26·5

2nd semidirect product of D8 and D10 acting through Inn(D8)

G = D₈:₁₃D₁₀ order 320 = 2⁶·5

2^nd semidirect product of D₈ and D₁₀ acting through Inn(D₈)