D8:5F5 - GroupNames

G = D₈⋊₅F₅ order 320 = 2⁶·5

The semidirect product of D₈ and F₅ acting through Inn(D₈)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₈⋊₅F₅, Dic₂₀⋊₅C₄, C₅⋊C₈.₄D₄, (C₅×D₈)⋊₆C₄, C₅⋊₁(C₈○D₈), (C₈×F₅)⋊₂C₂, D₄.D₅⋊₁C₄, D₄.F₅⋊₁C₂, D₄.₁(C₂×F₅), C₂.₁₇(D₄×F₅), C₈.₁₄(C₂×F₅), C₄₀.₁₂(C₂×C₄), D₄⋊F₅⋊₁C₂, C₁₀.₁₆(C₄×D₄), D₁₀.Q₈⋊₃C₂, C₄.₃(C₂²×F₅), D₈⋊₃D₅.₅C₂, C₂₀.₃(C₂²×C₄), D₁₀.₁(C₄○D₄), C₄.F₅.₁C₂², D₅⋊C₈.₁₀C₂², Dic₅.₇₂(C₂×D₄), Dic₁₀.₁(C₂×C₄), (C₄×D₅).₂₅C₂³, (C₈×D₅).₂₂C₂², (C₄×F₅).₁₀C₂², D₄⋊₂D₅.₄C₂², C₅⋊₂C₈.₇(C₂×C₄), (C₅×D₄).₁(C₂×C₄), SmallGroup(320,1070)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — D₈⋊₅F₅

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₄×D₅ — D₅⋊C₈ — D₄.F₅ — D₈⋊₅F₅

Lower central

C₅ — C₁₀ — C₂₀ — D₈⋊₅F₅

Upper central

C₁ — C₂ — C₄ — D₈

Generators and relations for D₈⋊₅F₅
G = < a,b,c,d | a⁸=b²=c⁵=d⁴=1, bab=a^-1, ac=ca, ad=da, bc=cb, dbd^-1=a²b, dcd^-1=c³ >

Subgroups: 386 in 106 conjugacy classes, 40 normal (24 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₅, C₈, C₈, C₂×C₄, D₄, D₄, Q₈, D₅, C₁₀, C₁₀, C₄², C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₄○D₄, Dic₅, Dic₅, C₂₀, F₅, D₁₀, C₂×C₁₀, C₄×C₈, C₄≀C₂, C₈.C₄, C₈○D₄, C₄○D₈, C₅⋊₂C₈, C₄₀, C₅⋊C₈, C₅⋊C₈, Dic₁₀, C₄×D₅, C₂×Dic₅, C₅⋊D₄, C₅×D₄, C₂×F₅, C₈○D₈, C₈×D₅, Dic₂₀, D₄.D₅, C₅×D₈, D₅⋊C₈, C₄.F₅, C₄×F₅, C₂×C₅⋊C₈, C₂².F₅, D₄⋊₂D₅, C₈×F₅, D₁₀.Q₈, D₄⋊F₅, D₈⋊₃D₅, D₄.F₅, D₈⋊₅F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²×C₄, C₂×D₄, C₄○D₄, F₅, C₄×D₄, C₂×F₅, C₈○D₈, C₂²×F₅, D₄×F₅, D₈⋊₅F₅

Smallest permutation representation of D₈⋊₅F₅
►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 59)(2 58)(3 57)(4 64)(5 63)(6 62)(7 61)(8 60)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 40)(16 39)(17 28)(18 27)(19 26)(20 25)(21 32)(22 31)(23 30)(24 29)(41 73)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 72)(56 71)

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

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

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

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

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

35 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	4I	5	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L	8M	8N	10A	10B	10C	20	40A	40B
order	1	2	2	2	2	4	4	4	4	4	4	4	4	4	5	8	8	8	8	8	8	8	8	8	8	8	8	8	8	10	10	10	20	40	40
size	1	1	4	4	10	2	5	5	10	10	10	10	20	20	4	2	2	5	5	5	5	10	10	10	10	20	20	20	20	4	16	16	8	8	8

35 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	4	4	4	8	8
type	+	+	+	+	+	+				+			+	+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	C₄○D₄	C₈○D₈	F₅	C₂×F₅	C₂×F₅	D₄×F₅	D₈⋊₅F₅
kernel	D₈⋊₅F₅	C₈×F₅	D₁₀.Q₈	D₄⋊F₅	D₈⋊₃D₅	D₄.F₅	Dic₂₀	D₄.D₅	C₅×D₈	C₅⋊C₈	D₁₀	C₅	D₈	C₈	D₄	C₂	C₁
# reps	1	1	1	2	1	2	2	4	2	2	2	8	1	1	2	1	2

Matrix representation of D₈⋊₅F₅ ►in GL₆(𝔽₄₁)

3	0	0	0	0	0
34	14	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

14	19	0	0	0	0
7	27	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	40	0
0	0	0	0	0	40

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

1	0	0	0	0	0
25	32	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))| [3,34,0,0,0,0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[14,7,0,0,0,0,19,27,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[1,25,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

D₈⋊₅F₅ in GAP, Magma, Sage, TeX

D_8\rtimes_5F_5

% in TeX

G:=Group("D8:5F5");

// GroupNames label

G:=SmallGroup(320,1070);

// by ID

G=gap.SmallGroup(320,1070);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,136,851,438,102,6278,1595]);

// Polycyclic

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

// generators/relations

G = D8⋊5F5 order 320 = 26·5

The semidirect product of D8 and F5 acting through Inn(D8)

G = D₈⋊₅F₅ order 320 = 2⁶·5

The semidirect product of D₈ and F₅ acting through Inn(D₈)