C2xQ8xF5 - GroupNames

G = C₂xQ₈xF₅ order 320 = 2⁶·5

Direct product of C₂, Q₈ and F₅

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂xQ₈xF₅, D₁₀.₁₆C₂⁴, D₅:(C₄xQ₈), C₁₀:(C₄xQ₈), (Q₈xD₅):₁₀C₄, (Q₈xC₁₀):₉C₄, Dic₁₀:₇(C₂xC₄), D₁₀.₂₈(C₂xQ₈), C₄:F₅.₁₁C₂², (C₂xF₅).₅C₂³, C₂.₁₄(C₂³xF₅), C₄.₃₀(C₂²xF₅), (C₂xDic₁₀):₁₃C₄, D₅.₂(C₂²xQ₈), C₂₀.₃₀(C₂²xC₄), C₁₀.₁₃(C₂³xC₄), (C₄xD₅).₅₃C₂³, (C₄xF₅).₁₅C₂², D₁₀.₅₈(C₄oD₄), (Q₈xD₅).₁₄C₂², D₁₀.₄₈(C₂²xC₄), Dic₅.₄(C₂²xC₄), C₂².₅₉(C₂²xF₅), (C₂²xF₅).₂₅C₂², (C₂²xD₅).₂₈₅C₂³, C₅:(C₂xC₄xQ₈), (C₂xC₄xF₅).₆C₂, (C₅xQ₈):₇(C₂xC₄), (C₂xC₄:F₅).₇C₂, (C₂xQ₈xD₅).₁₃C₂, D₅.₃(C₂xC₄oD₄), (C₂xC₄).₉₃(C₂xF₅), (C₂xC₂₀).₇₄(C₂xC₄), (C₄xD₅).₃₅(C₂xC₄), (C₂xC₄xD₅).₂₂₀C₂², (C₂xDic₅).₈₂(C₂xC₄), (C₂xC₁₀).₁₀₃(C₂²xC₄), SmallGroup(320,1599)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₂xQ₈xF₅

Chief series

C₁ — C₅ — D₅ — D₁₀ — C₂xF₅ — C₂²xF₅ — C₂xC₄xF₅ — C₂xQ₈xF₅

Lower central

C₅ — C₁₀ — C₂xQ₈xF₅

Upper central

C₁ — C₂² — C₂xQ₈

Generators and relations for C₂xQ₈xF₅
G = < a,b,c,d,e | a²=b⁴=d⁵=e⁴=1, c²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d³ >

Subgroups: 890 in 298 conjugacy classes, 156 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₂xC₄, C₂xC₄, Q₈, Q₈, C₂³, D₅, D₅, C₁₀, C₁₀, C₄², C₄:C₄, C₂²xC₄, C₂xQ₈, C₂xQ₈, Dic₅, C₂₀, F₅, F₅, D₁₀, D₁₀, C₂xC₁₀, C₂xC₄², C₂xC₄:C₄, C₄xQ₈, C₂²xQ₈, Dic₁₀, C₄xD₅, C₂xDic₅, C₂xC₂₀, C₅xQ₈, C₂xF₅, C₂xF₅, C₂²xD₅, C₂xC₄xQ₈, C₄xF₅, C₄:F₅, C₂xDic₁₀, C₂xC₄xD₅, Q₈xD₅, Q₈xC₁₀, C₂²xF₅, C₂²xF₅, C₂xC₄xF₅, C₂xC₄:F₅, Q₈xF₅, C₂xQ₈xD₅, C₂xQ₈xF₅
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, Q₈, C₂³, C₂²xC₄, C₂xQ₈, C₄oD₄, C₂⁴, F₅, C₄xQ₈, C₂³xC₄, C₂²xQ₈, C₂xC₄oD₄, C₂xF₅, C₂xC₄xQ₈, C₂²xF₅, Q₈xF₅, C₂³xF₅, C₂xQ₈xF₅

Smallest permutation representation of C₂xQ₈xF₅
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

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

50 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	···	4F	4G	···	4N	4O	···	4AF	5	10A	10B	10C	20A	···	20F
order	1	2	2	2	2	2	2	2	4	···	4	4	···	4	4	···	4	5	10	10	10	20	···	20
size	1	1	1	1	5	5	5	5	2	···	2	5	···	5	10	···	10	4	4	4	4	8	···	8

50 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	4	4	4	8
type	+	+	+	+	+				-		+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	Q₈	C₄oD₄	F₅	C₂xF₅	C₂xF₅	Q₈xF₅
kernel	C₂xQ₈xF₅	C₂xC₄xF₅	C₂xC₄:F₅	Q₈xF₅	C₂xQ₈xD₅	C₂xDic₁₀	Q₈xD₅	Q₈xC₁₀	C₂xF₅	D₁₀	C₂xQ₈	C₂xC₄	Q₈	C₂
# reps	1	3	3	8	1	6	8	2	4	4	1	3	4	2

Matrix representation of C₂xQ₈xF₅ ►in GL₆(F₄₁)

40	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	0
0	0	0	0	0	40

40	2	0	0	0	0
40	1	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

9	23	0	0	0	0
0	32	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

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

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

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,23,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

C₂xQ₈xF₅ in GAP, Magma, Sage, TeX

C_2\times Q_8\times F_5

% in TeX

G:=Group("C2xQ8xF5");

// GroupNames label

G:=SmallGroup(320,1599);

// by ID

G=gap.SmallGroup(320,1599);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,297,136,6278,818]);

// Polycyclic

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

// generators/relations

G = C2xQ8xF5 order 320 = 26·5

Direct product of C2, Q8 and F5

G = C₂xQ₈xF₅ order 320 = 2⁶·5

Direct product of C₂, Q₈ and F₅