D8:4Dic5 - GroupNames

G = D₈:₄Dic₅ order 320 = 2⁶·5

4^th semidirect product of D₈ and Dic₅ acting via Dic₅/C₁₀=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₈:₄Dic₅, Q₁₆:₄Dic₅, SD₁₆:₂Dic₅, (C₅xD₈):₁₂C₄, C₄oD₈.₄D₅, C₄₀:₈C₄:₂C₂, C₅:₈(C₈.₂₆D₄), C₄₀.₇₂(C₂xC₄), (C₅xQ₁₆):₁₂C₄, (C₅xSD₁₆):₆C₄, C₄.₂₁₈(D₄xD₅), C₅:₂C₈.₅₂D₄, C₄₀.₆C₄:₉C₂, C₈.₆(C₂xDic₅), C₄oD₄.₂₃D₁₀, C₂₀.₃₇₇(C₂xD₄), (C₂xC₈).₁₀₀D₁₀, C₁₀.₁₃₀(C₄xD₄), D₄.Dic₅:₄C₂, Q₈.₄(C₂xDic₅), D₄.₄(C₂xDic₅), C₂.₁₇(D₄xDic₅), D₄:₂Dic₅:₅C₂, (C₂xC₄₀).₄₅C₂², C₄.₈(C₂²xDic₅), (C₂xC₂₀).₄₆₈C₂³, C₂₀.₁₃₇(C₂²xC₄), C₂².₄(D₄:₂D₅), (C₄xDic₅).₆₃C₂², C₄.Dic₅.₂₃C₂², (C₅xC₄oD₈).₃C₂, (C₅xD₄).₂₅(C₂xC₄), (C₅xQ₈).₂₆(C₂xC₄), (C₂xC₁₀).₁₂(C₄oD₄), (C₅xC₄oD₄).₁₀C₂², (C₂xC₄).₅₅₅(C₂²xD₅), (C₂xC₅:₂C₈).₁₆₉C₂², SmallGroup(320,824)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — D₈:₄Dic₅

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂xC₂₀ — C₂xC₅:₂C₈ — D₄.Dic₅ — D₈:₄Dic₅

Lower central

C₅ — C₁₀ — C₂₀ — D₈:₄Dic₅

Upper central

C₁ — C₄ — C₂xC₄ — C₄oD₈

Generators and relations for D₈:₄Dic₅
G = < a,b,c,d | a⁸=b²=c¹⁰=1, d²=c⁵, bab=a^-1, ac=ca, dad^-1=a⁵, cbc^-1=a⁴b, dbd^-1=a²b, dcd^-1=c^-1 >

Subgroups: 278 in 104 conjugacy classes, 53 normal (29 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, C₈:C₄, C₄wrC₂, C₈.C₄, C₈oD₄, C₄oD₈, C₅:₂C₈, C₅:₂C₈, C₄₀, C₂xDic₅, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₅xD₄, C₅xQ₈, C₈.₂₆D₄, C₂xC₅:₂C₈, C₂xC₅:₂C₈, C₄.Dic₅, C₄.Dic₅, C₄xDic₅, C₂xC₄₀, C₅xD₈, C₅xSD₁₆, C₅xQ₁₆, C₅xC₄oD₄, C₄₀:₈C₄, C₄₀.₆C₄, D₄:₂Dic₅, D₄.Dic₅, C₅xC₄oD₈, D₈:₄Dic₅
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, D₅, C₂²xC₄, C₂xD₄, C₄oD₄, Dic₅, D₁₀, C₄xD₄, C₂xDic₅, C₂²xD₅, C₈.₂₆D₄, D₄xD₅, D₄:₂D₅, C₂²xDic₅, D₄xDic₅, D₈:₄Dic₅

Smallest permutation representation of D₈:₄Dic₅
►On 80 points

Generators in S₈₀

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

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

(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)

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

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

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

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

50 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	5A	5B	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	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	5	5	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	20	20	2	2	4	4	10	10	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

50 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+				+	+		+	-	-	-	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₅	C₄oD₄	D₁₀	Dic₅	Dic₅	Dic₅	D₁₀	C₈.₂₆D₄	D₄xD₅	D₄:₂D₅	D₈:₄Dic₅
kernel	D₈:₄Dic₅	C₄₀:₈C₄	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	2	2	2	8

Matrix representation of D₈:₄Dic₅ ►in GL₄(F₄₁) generated by

10	38	0	0
3	31	0	0
0	0	33	27
0	0	14	8

0	0	10	38
0	0	3	31
33	27	0	0
14	8	0	0

0	40	0	0
1	7	0	0
0	0	0	1
0	0	40	34

19	32	0	0
22	22	0	0
0	0	7	1
0	0	34	34

G:=sub<GL(4,GF(41))| [10,3,0,0,38,31,0,0,0,0,33,14,0,0,27,8],[0,0,33,14,0,0,27,8,10,3,0,0,38,31,0,0],[0,1,0,0,40,7,0,0,0,0,0,40,0,0,1,34],[19,22,0,0,32,22,0,0,0,0,7,34,0,0,1,34] >;

D₈:₄Dic₅ in GAP, Magma, Sage, TeX

D_8\rtimes_4{\rm Dic}_5

% in TeX

G:=Group("D8:4Dic5");

// GroupNames label

G:=SmallGroup(320,824);

// by ID

G=gap.SmallGroup(320,824);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,136,851,438,102,12550]);

// 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,d*a*d^-1=a^5,c*b*c^-1=a^4*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;

// generators/relations

G = D8:4Dic5 order 320 = 26·5

4th semidirect product of D8 and Dic5 acting via Dic5/C10=C2

G = D₈:₄Dic₅ order 320 = 2⁶·5

4^th semidirect product of D₈ and Dic₅ acting via Dic₅/C₁₀=C₂