D8:5Dic5 - GroupNames

G = D₈:₅Dic₅ order 320 = 2⁶·5

The semidirect product of D₈ and Dic₅ acting through Inn(D₈)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₈:₅Dic₅, Q₁₆:₅Dic₅, SD₁₆:₃Dic₅, (C₅xD₈):₉C₄, C₅:₈(C₈oD₈), (C₅xQ₁₆):₉C₄, C₄oD₈.₅D₅, C₄₀.₆₁(C₂xC₄), (C₈xDic₅):₂C₂, (C₅xSD₁₆):₈C₄, C₄.₂₁₇(D₄xD₅), C₅:₂C₈.₃₅D₄, C₄₀.₆C₄:₈C₂, C₄oD₄.₂₂D₁₀, C₁₀.₁₂₉(C₄xD₄), (C₂xC₈).₂₅₄D₁₀, C₂₀.₃₇₆(C₂xD₄), D₄.Dic₅:₃C₂, Q₈.₃(C₂xDic₅), D₄.₃(C₂xDic₅), C₈.₁₁(C₂xDic₅), C₂.₁₆(D₄xDic₅), D₄:₂Dic₅:₄C₂, (C₂xC₄₀).₄₄C₂², C₄.₇(C₂²xDic₅), C₂₀.₁₃₆(C₂²xC₄), (C₂xC₂₀).₄₆₇C₂³, C₂².₃(D₄:₂D₅), C₄.Dic₅.₂₂C₂², (C₄xDic₅).₂₇₇C₂², (C₅xC₄oD₈).₂C₂, (C₅xD₄).₂₄(C₂xC₄), (C₅xQ₈).₂₅(C₂xC₄), (C₅xC₄oD₄).₉C₂², (C₂xC₁₀).₁₁(C₄oD₄), (C₂xC₄).₅₅₄(C₂²xD₅), (C₂xC₅:₂C₈).₂₈₈C₂², SmallGroup(320,823)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — D₈:₅Dic₅

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂xC₂₀ — C₂xC₅:₂C₈ — D₄.Dic₅ — D₈:₅Dic₅

Lower central

C₅ — C₁₀ — C₂₀ — D₈:₅Dic₅

Upper central

C₁ — C₄ — C₂xC₄ — C₄oD₈

Generators and relations for D₈:₅Dic₅
G = < a,b,c,d | a⁸=b²=c¹⁰=1, d²=c⁵, bab=a^-1, ac=ca, ad=da, cbc^-1=a⁴b, dbd^-1=a²b, dcd^-1=c^-1 >

Subgroups: 278 in 106 conjugacy classes, 53 normal (31 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, C₁₀, C₁₀, C₄², C₂xC₈, C₂xC₈, M₄(2), D₈, SD₁₆, Q₁₆, C₄oD₄, Dic₅, C₂₀, C₂₀, C₂xC₁₀, C₂xC₁₀, C₄xC₈, C₄wrC₂, C₈.C₄, C₈oD₄, C₄oD₈, C₅:₂C₈, C₅:₂C₈, C₄₀, C₂xDic₅, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₅xD₄, C₅xQ₈, C₈oD₈, C₂xC₅:₂C₈, C₂xC₅:₂C₈, C₄.Dic₅, C₄.Dic₅, C₄xDic₅, C₂xC₄₀, C₅xD₈, C₅xSD₁₆, C₅xQ₁₆, C₅xC₄oD₄, C₈xDic₅, C₄₀.₆C₄, D₄:₂Dic₅, D₄.Dic₅, C₅xC₄oD₈, D₈:₅Dic₅
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, D₅, C₂²xC₄, C₂xD₄, C₄oD₄, Dic₅, D₁₀, C₄xD₄, C₂xDic₅, C₂²xD₅, C₈oD₈, D₄xD₅, D₄:₂D₅, C₂²xDic₅, D₄xDic₅, D₈:₅Dic₅

Smallest permutation representation of D₈:₅Dic₅
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+				+	+		+	-	-	-	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₅	C₄oD₄	D₁₀	Dic₅	Dic₅	Dic₅	D₁₀	C₈oD₈	D₄xD₅	D₄:₂D₅	D₈:₅Dic₅
kernel	D₈:₅Dic₅	C₈xDic₅	C₄₀.₆C₄	D₄:₂Dic₅	D₄.Dic₅	C₅xC₄oD₈	C₅xD₈	C₅xSD₁₆	C₅xQ₁₆	C₅:₂C₈	C₄oD₈	C₂xC₁₀	C₂xC₈	D₈	SD₁₆	Q₁₆	C₄oD₄	C₅	C₄	C₂²	C₁
# reps	1	1	1	2	2	1	2	4	2	2	2	2	2	2	4	2	4	8	2	2	8

Matrix representation of D₈:₅Dic₅ ►in GL₄(F₄₁) generated by

14	26	0	0
0	3	0	0
0	0	1	0
0	0	0	1

3	15	0	0
35	38	0	0
0	0	40	0
0	0	0	40

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

9	5	0	0
0	40	0	0
0	0	1	0
0	0	6	40

G:=sub<GL(4,GF(41))| [14,0,0,0,26,3,0,0,0,0,1,0,0,0,0,1],[3,35,0,0,15,38,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,40,1,0,0,0,0,0,40,0,0,1,6],[9,0,0,0,5,40,0,0,0,0,1,6,0,0,0,40] >;

D₈:₅Dic₅ in GAP, Magma, Sage, TeX

D_8\rtimes_5{\rm Dic}_5

% in TeX

G:=Group("D8:5Dic5");

// GroupNames label

G:=SmallGroup(320,823);

// by ID

G=gap.SmallGroup(320,823);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,136,851,438,102,12550]);

// Polycyclic

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

// generators/relations

G = D8:5Dic5 order 320 = 26·5

The semidirect product of D8 and Dic5 acting through Inn(D8)

G = D₈:₅Dic₅ order 320 = 2⁶·5

The semidirect product of D₈ and Dic₅ acting through Inn(D₈)