C2xD40:C2 - GroupNames

G = C₂xD₄₀:C₂ order 320 = 2⁶·5

Direct product of C₂ and D₄₀:C₂

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

Aliases: C₂xD₄₀:C₂, C₄₀:₄C₂³, SD₁₆:₈D₁₀, D₂₀:₂C₂³, C₂₀.₆C₂⁴, D₄₀:₂₀C₂², (C₂xC₈):₁₀D₁₀, C₄.₄₃(D₄xD₅), C₈:₄(C₂²xD₅), (C₂xD₄₀):₂₆C₂, C₅:₂C₈:₂C₂³, (C₂xQ₈):₂₁D₁₀, (C₂xSD₁₆):₄D₅, (C₄xD₅).₁₅D₄, C₂₀.₈₁(C₂xD₄), (D₄xD₅):₆C₂², Q₈:D₅:₈C₂², (C₅xQ₈):₂C₂³, Q₈:₂(C₂²xD₅), C₄.₆(C₂³xD₅), C₁₀:₃(C₈:C₂²), (C₂xC₄₀):₁₃C₂², D₄:D₅:₁₀C₂², (C₁₀xSD₁₆):₅C₂, D₁₀.₈₄(C₂xD₄), C₈:D₅:₈C₂², D₄.₄(C₂²xD₅), (C₅xD₄).₄C₂³, (C₄xD₅).₃C₂³, (C₂xD₄).₁₈₂D₁₀, (C₂xD₂₀):₃₃C₂², Dic₅.₉₅(C₂xD₄), (Q₈xC₁₀):₁₈C₂², Q₈:₂D₅:₅C₂², (C₅xSD₁₆):₈C₂², C₂².₁₃₉(D₄xD₅), (C₂xC₂₀).₅₂₃C₂³, (C₂xDic₅).₂₄₈D₄, (C₂²xD₅).₁₃₅D₄, C₁₀.₁₀₇(C₂²xD₄), (D₄xC₁₀).₁₆₄C₂², (C₂xD₄xD₅):₂₃C₂, C₅:₃(C₂xC₈:C₂²), C₂.₈₀(C₂xD₄xD₅), (C₂xD₄:D₅):₂₇C₂, (C₂xC₈:D₅):₄C₂, (C₂xQ₈:D₅):₂₆C₂, (C₂xC₅:₂C₈):₁₅C₂², (C₂xQ₈:₂D₅):₁₄C₂, (C₂xC₁₀).₃₉₆(C₂xD₄), (C₂xC₄xD₅).₁₆₅C₂², (C₂xC₄).₆₁₂(C₂²xD₅), SmallGroup(320,1431)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — C₂xD₄₀:C₂

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₄xD₅ — C₂xC₄xD₅ — C₂xD₄xD₅ — C₂xD₄₀:C₂

Lower central

C₅ — C₁₀ — C₂₀ — C₂xD₄₀:C₂

Upper central

C₁ — C₂² — C₂xC₄ — C₂xSD₁₆

Generators and relations for C₂xD₄₀:C₂
G = < a,b,c,d | a²=b⁴⁰=c²=d²=1, ab=ba, ac=ca, ad=da, cbc=b^-1, dbd=b¹¹, cd=dc >

Subgroups: 1374 in 298 conjugacy classes, 103 normal (33 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, D₅, C₁₀, C₁₀, C₁₀, C₂xC₈, C₂xC₈, M₄(2), D₈, SD₁₆, SD₁₆, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, Dic₅, C₂₀, C₂₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₂xM₄(2), C₂xD₈, C₂xSD₁₆, C₂xSD₁₆, C₈:C₂², C₂²xD₄, C₂xC₄oD₄, C₅:₂C₈, C₄₀, C₄xD₅, C₄xD₅, D₂₀, D₂₀, C₂xDic₅, C₅:D₄, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₅xD₄, C₅xQ₈, C₅xQ₈, C₂²xD₅, C₂²xD₅, C₂²xC₁₀, C₂xC₈:C₂², C₈:D₅, D₄₀, C₂xC₅:₂C₈, D₄:D₅, Q₈:D₅, C₂xC₄₀, C₅xSD₁₆, C₂xC₄xD₅, C₂xC₄xD₅, C₂xD₂₀, C₂xD₂₀, D₄xD₅, D₄xD₅, Q₈:₂D₅, Q₈:₂D₅, C₂xC₅:D₄, D₄xC₁₀, Q₈xC₁₀, C₂³xD₅, C₂xC₈:D₅, C₂xD₄₀, D₄₀:C₂, C₂xD₄:D₅, C₂xQ₈:D₅, C₁₀xSD₁₆, C₂xD₄xD₅, C₂xQ₈:₂D₅, C₂xD₄₀:C₂
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂xD₄, C₂⁴, D₁₀, C₈:C₂², C₂²xD₄, C₂²xD₅, C₂xC₈:C₂², D₄xD₅, C₂³xD₅, D₄₀:C₂, C₂xD₄xD₅, C₂xD₄₀:C₂

Smallest permutation representation of C₂xD₄₀:C₂
►On 80 points

Generators in S₈₀

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

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

(1 11)(2 22)(3 33)(5 15)(6 26)(7 37)(9 19)(10 30)(13 23)(14 34)(17 27)(18 38)(21 31)(25 35)(29 39)(41 71)(43 53)(44 64)(45 75)(47 57)(48 68)(49 79)(51 61)(52 72)(55 65)(56 76)(59 69)(60 80)(63 73)(67 77)

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

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

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

50 conjugacy classes

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

50 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

D₁₀

C₈:C₂²

D₄xD₅

D₄₀:C₂

kernel

C₂xD₄₀:C₂

C₂xC₈:D₅

C₂xD₄₀

D₄₀:C₂

C₂xD₄:D₅

C₂xQ₈:D₅

C₁₀xSD₁₆

C₂xD₄xD₅

C₂xQ₈:₂D₅

C₄xD₅

C₂xDic₅

C₂²xD₅

C₂xSD₁₆

C₂xC₈

SD₁₆

C₂xD₄

C₂xQ₈

C₁₀

C₄

C₂²

C₂

# reps

Matrix representation of C₂xD₄₀:C₂ ►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

7	35	0	0	0	0
7	0	0	0	0	0
0	0	26	26	26	26
0	0	15	34	15	34
0	0	28	28	0	0
0	0	13	24	0	0

40	1	0	0	0	0
0	1	0	0	0	0
0	0	0	40	0	39
0	0	40	0	39	0
0	0	0	0	0	1
0	0	0	0	1	0

40	0	0	0	0	0
0	40	0	0	0	0
0	0	1	0	2	0
0	0	0	1	0	2
0	0	0	0	40	0
0	0	0	0	0	40

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],[7,7,0,0,0,0,35,0,0,0,0,0,0,0,26,15,28,13,0,0,26,34,28,24,0,0,26,15,0,0,0,0,26,34,0,0],[40,0,0,0,0,0,1,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,39,0,1,0,0,39,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,2,0,40,0,0,0,0,2,0,40] >;

C₂xD₄₀:C₂ in GAP, Magma, Sage, TeX

C_2\times D_{40}\rtimes C_2

% in TeX

G:=Group("C2xD40:C2");

// GroupNames label

G:=SmallGroup(320,1431);

// by ID

G=gap.SmallGroup(320,1431);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,1123,185,136,438,235,102,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^40=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^11,c*d=d*c>;

// generators/relations

G = C2xD40:C2 order 320 = 26·5

Direct product of C2 and D40:C2

G = C₂xD₄₀:C₂ order 320 = 2⁶·5

Direct product of C₂ and D₄₀:C₂