D10.10D8 - GroupNames

G = D₁₀.₁₀D₈ order 320 = 2⁶·5

6^th non-split extension by D₁₀ of D₈ acting via D₈/C₈=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₀.₁₀D₈, C₂₀.₉C₄², D₁₀.₅Q₁₆, D₁₀.₉SD₁₆, C₄⋊F₅⋊₅C₄, (C₂×C₈)⋊₄F₅, (C₂×C₄₀)⋊₂C₄, C₄.₂(C₄×F₅), (C₄×D₅).₁₀₈D₄, C₂.₃(D₅.D₈), C₂.₃(C₄₀⋊C₄), C₁₀.₅(C₄.Q₈), C₁₀.₇(C₂.D₈), C₅⋊₂(C₂².₄Q₁₆), (C₂×Dic₅).₃₁Q₈, (C₂²×D₅).₉₆D₄, C₄.₂₄(C₂²⋊F₅), C₂².₁₉(C₄⋊F₅), D₅.₂(D₄⋊C₄), C₂₀.₂₄(C₂²⋊C₄), Dic₅.₂₄(C₄⋊C₄), D₅.₂(Q₈⋊C₄), D₁₀.₃₁(C₂²⋊C₄), C₂.₉(D₁₀.₃Q₈), C₁₀.₈(C₂.C₄²), (D₅×C₂×C₈).₁₂C₂, (C₂×C₅⋊₂C₈)⋊₁₆C₄, (C₂×C₄⋊F₅).₈C₂, (C₄×D₅).₆₂(C₂×C₄), (C₂×C₄).₁₂₄(C₂×F₅), (C₂×C₁₀).₁₂(C₄⋊C₄), (C₂×C₂₀).₁₁₈(C₂×C₄), (C₂×C₄×D₅).₃₈₇C₂², SmallGroup(320,231)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — D₁₀.₁₀D₈

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₄×D₅ — C₂×C₄×D₅ — C₂×C₄⋊F₅ — D₁₀.₁₀D₈

Lower central

C₅ — C₁₀ — C₂₀ — D₁₀.₁₀D₈

Upper central

C₁ — C₂² — C₂×C₄ — C₂×C₈

Generators and relations for D₁₀.₁₀D₈
G = < a,b,c,d | a¹⁰=b²=c⁸=1, d²=a⁴b, bab=a^-1, ac=ca, dad^-1=a³, bc=cb, dbd^-1=a²b, dcd^-1=a⁵c^-1 >

Subgroups: 514 in 114 conjugacy classes, 46 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂³, D₅, C₁₀, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, Dic₅, C₂₀, F₅, D₁₀, D₁₀, C₂×C₁₀, C₂×C₄⋊C₄, C₂²×C₈, C₅⋊₂C₈, C₄₀, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂×F₅, C₂²×D₅, C₂².₄Q₁₆, C₈×D₅, C₂×C₅⋊₂C₈, C₂×C₄₀, C₄⋊F₅, C₄⋊F₅, C₂×C₄×D₅, C₂²×F₅, D₅×C₂×C₈, C₂×C₄⋊F₅, D₁₀.₁₀D₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₄², C₂²⋊C₄, C₄⋊C₄, D₈, SD₁₆, Q₁₆, F₅, C₂.C₄², D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₂×F₅, C₂².₄Q₁₆, C₄×F₅, C₄⋊F₅, C₂²⋊F₅, C₄₀⋊C₄, D₅.D₈, D₁₀.₃Q₈, D₁₀.₁₀D₈

Smallest permutation representation of D₁₀.₁₀D₈
►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 20)(12 19)(13 18)(14 17)(15 16)(21 28)(22 27)(23 26)(24 25)(29 30)(31 34)(32 33)(35 40)(36 39)(37 38)(41 48)(42 47)(43 46)(44 45)(49 50)(51 54)(52 53)(55 60)(56 59)(57 58)(61 68)(62 67)(63 66)(64 65)(69 70)(71 74)(72 73)(75 80)(76 79)(77 78)

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

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4	4	4
type	+	+	+				+	-	+	+		-	+	+		+
image	C₁	C₂	C₂	C₄	C₄	C₄	D₄	Q₈	D₄	D₈	SD₁₆	Q₁₆	F₅	C₂×F₅	C₄×F₅	C₂²⋊F₅	C₄⋊F₅	C₄₀⋊C₄	D₅.D₈
kernel	D₁₀.₁₀D₈	D₅×C₂×C₈	C₂×C₄⋊F₅	C₂×C₅⋊₂C₈	C₂×C₄₀	C₄⋊F₅	C₄×D₅	C₂×Dic₅	C₂²×D₅	D₁₀	D₁₀	D₁₀	C₂×C₈	C₂×C₄	C₄	C₄	C₂²	C₂	C₂
# reps	1	1	2	2	2	8	2	1	1	2	4	2	1	1	2	2	2	4	4

Matrix representation of D₁₀.₁₀D₈ ►in GL₆(𝔽₄₁)

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

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	0	40	0	0
0	0	40	0	0	0

22	22	0	0	0	0
15	2	0	0	0	0
0	0	32	0	0	0
0	0	0	32	0	0
0	0	0	0	32	0
0	0	0	0	0	32

34	34	0	0	0	0
13	7	0	0	0	0
0	0	32	0	0	0
0	0	0	0	0	32
0	0	0	32	0	0
0	0	9	9	9	9

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,40,0,0,0,0,1,0,40,0,0,0,1,0,0,0,0,40,1,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0],[22,15,0,0,0,0,22,2,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[34,13,0,0,0,0,34,7,0,0,0,0,0,0,32,0,0,9,0,0,0,0,32,9,0,0,0,0,0,9,0,0,0,32,0,9] >;

D₁₀.₁₀D₈ in GAP, Magma, Sage, TeX

D_{10}._{10}D_8

% in TeX

G:=Group("D10.10D8");

// GroupNames label

G:=SmallGroup(320,231);

// by ID

G=gap.SmallGroup(320,231);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,1123,136,6278,3156]);

// Polycyclic

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

// generators/relations

G = D10.10D8 order 320 = 26·5

6th non-split extension by D10 of D8 acting via D8/C8=C2

G = D₁₀.₁₀D₈ order 320 = 2⁶·5

6^th non-split extension by D₁₀ of D₈ acting via D₈/C₈=C₂