D10.18D8 - GroupNames

G = D₁₀.₁₈D₈ order 320 = 2⁶·5

7^th non-split extension by D₁₀ of D₈ acting via D₈/D₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₀.₁₈D₈, C₂₀.₁C₄², D₁₀.₉Q₁₆, D₁₀.₁₈SD₁₆, C₄⋊C₄⋊₃F₅, C₄⋊F₅⋊₁C₄, D₅⋊C₈⋊₂C₄, C₄.₆(C₄×F₅), C₄⋊Dic₅⋊₈C₄, C₂₀.₁(C₄⋊C₄), (C₄×D₅).₃Q₈, D₅.(C₂.D₈), D₅.(C₄.Q₈), (C₄×D₅).₁₈D₄, C₄.₁₆(C₄⋊F₅), D₁₀.₂₀(C₄⋊C₄), Dic₅.₃(C₄⋊C₄), C₂.₂(D₂₀⋊C₄), C₅⋊₁(C₂².₄Q₁₆), C₂.₂(Q₈⋊F₅), (C₂×Dic₅).₉₈D₄, D₅.₁(D₄⋊C₄), C₁₀.₄(D₄⋊C₄), D₅.₁(Q₈⋊C₄), C₁₀.₄(Q₈⋊C₄), (C₂²×D₅).₁₄₂D₄, D₁₀.₂₉(C₂²⋊C₄), Dic₅.₂(C₂²⋊C₄), C₂.₇(D₁₀.₃Q₈), C₂².₃₃(C₂²⋊F₅), C₁₀.₅(C₂.C₄²), (C₅×C₄⋊C₄)⋊₃C₄, (C₂×C₄⋊F₅).₁C₂, (C₂×D₅⋊C₈).₁C₂, (D₅×C₄⋊C₄).₁₄C₂, (C₂×C₄).₆₆(C₂×F₅), (C₂×C₂₀).₃₂(C₂×C₄), (C₄×D₅).₁₂(C₂×C₄), (C₂×C₄×D₅).₁₈₄C₂², (C₂×C₁₀).₂₄(C₂²⋊C₄), SmallGroup(320,212)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — D₁₀.₁₈D₈

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₂²×D₅ — C₂×C₄×D₅ — C₂×C₄⋊F₅ — D₁₀.₁₈D₈

Lower central

C₅ — C₁₀ — C₂₀ — D₁₀.₁₈D₈

Upper central

C₁ — C₂² — C₂×C₄ — C₄⋊C₄

Generators and relations for D₁₀.₁₈D₈
G = < a,b,c,d | a¹⁰=b²=c⁸=1, d²=a^-1b, bab=a^-1, cac^-1=dad^-1=a³, cbc^-1=dbd^-1=a²b, dcd^-1=a⁴bc^-1 >

Subgroups: 498 in 114 conjugacy classes, 46 normal (42 characteristic)
C₁, C₂^[×3], C₂^[×4], C₄^[×2], C₄^[×6], C₂², C₂²^[×6], C₅, C₈^[×2], C₂×C₄, C₂×C₄^[×13], C₂³, D₅^[×4], C₁₀^[×3], C₄⋊C₄, C₄⋊C₄^[×5], C₂×C₈^[×4], C₂²×C₄^[×3], Dic₅^[×2], Dic₅, C₂₀^[×2], C₂₀, F₅^[×2], D₁₀^[×6], C₂×C₁₀, C₂×C₄⋊C₄^[×2], C₂²×C₈, C₅⋊C₈^[×2], C₄×D₅^[×4], C₄×D₅^[×2], C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₂×F₅^[×4], C₂²×D₅, C₂².₄Q₁₆, C₁₀.D₄, C₄⋊Dic₅, C₅×C₄⋊C₄, D₅⋊C₈^[×2], D₅⋊C₈, C₄⋊F₅^[×2], C₄⋊F₅, C₂×C₅⋊C₈, C₂×C₄×D₅, C₂×C₄×D₅, C₂²×F₅, D₅×C₄⋊C₄, C₂×D₅⋊C₈, C₂×C₄⋊F₅, D₁₀.₁₈D₈
Quotients: C₁, C₂^[×3], C₄^[×6], C₂², C₂×C₄^[×3], D₄^[×3], Q₈, C₄², C₂²⋊C₄^[×3], C₄⋊C₄^[×3], D₈, SD₁₆^[×2], Q₁₆, F₅, C₂.C₄², D₄⋊C₄^[×2], Q₈⋊C₄^[×2], C₄.Q₈, C₂.D₈, C₂×F₅, C₂².₄Q₁₆, C₄×F₅, C₄⋊F₅, C₂²⋊F₅, D₁₀.₃Q₈, D₂₀⋊C₄, Q₈⋊F₅, D₁₀.₁₈D₈

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

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	4	4	8	8
type	+	+	+	+					+	-	+	+	+		-	+	+			+	+	-
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	Q₈	D₄	D₄	D₈	SD₁₆	Q₁₆	F₅	C₂×F₅	C₄×F₅	C₄⋊F₅	C₂²⋊F₅	D₂₀⋊C₄	Q₈⋊F₅
kernel	D₁₀.₁₈D₈	D₅×C₄⋊C₄	C₂×D₅⋊C₈	C₂×C₄⋊F₅	C₄⋊Dic₅	C₅×C₄⋊C₄	D₅⋊C₈	C₄⋊F₅	C₄×D₅	C₄×D₅	C₂×Dic₅	C₂²×D₅	D₁₀	D₁₀	D₁₀	C₄⋊C₄	C₂×C₄	C₄	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	2	2	4	4	1	1	1	1	2	4	2	1	1	2	2	2	1	1

Matrix representation of D₁₀.₁₈D₈ ►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

15	26	0	0	0	0
15	15	0	0	0	0
0	0	7	0	14	14
0	0	14	14	0	7
0	0	27	34	27	0
0	0	34	7	7	34

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

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],[15,15,0,0,0,0,26,15,0,0,0,0,0,0,7,14,27,34,0,0,0,14,34,7,0,0,14,0,27,7,0,0,14,7,0,34],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

D₁₀.₁₈D₈ in GAP, Magma, Sage, TeX

D_{10}._{18}D_8

% in TeX

G:=Group("D10.18D8");

// GroupNames label

G:=SmallGroup(320,212);

// by ID

G=gap.SmallGroup(320,212);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,1684,851,102,6278,3156]);

// Polycyclic

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

// generators/relations

G = D10.18D8 order 320 = 26·5

7th non-split extension by D10 of D8 acting via D8/D4=C2

G = D₁₀.₁₈D₈ order 320 = 2⁶·5

7^th non-split extension by D₁₀ of D₈ acting via D₈/D₄=C₂