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G = C2×C40order 80 = 24·5

Abelian group of type [2,40]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C40, SmallGroup(80,23)

Series: Derived Chief Lower central Upper central

C1 — C2×C40
C1C2C4C20C40 — C2×C40
C1 — C2×C40
C1 — C2×C40

Generators and relations for C2×C40
 G = < a,b | a2=b40=1, ab=ba >


Smallest permutation representation of C2×C40
Regular action on 80 points
Generators in S80
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 41)(40 42)
(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)

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

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

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

C2×C40 is a maximal subgroup of
C20.4C8  C20.8Q8  C408C4  C20.44D4  C406C4  C405C4  C40.6C4  D101C8  D205C4  D20.3C4  D407C2

80 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B5C5D8A···8H10A···10L20A···20P40A···40AF
order1222444455558···810···1020···2040···40
size1111111111111···11···11···11···1

80 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C5C8C10C10C20C20C40
kernelC2×C40C40C2×C20C20C2×C10C2×C8C10C8C2×C4C4C22C2
# reps1212248848832

Matrix representation of C2×C40 in GL3(𝔽41) generated by

100
010
0040
,
2700
040
003
G:=sub<GL(3,GF(41))| [1,0,0,0,1,0,0,0,40],[27,0,0,0,4,0,0,0,3] >;

C2×C40 in GAP, Magma, Sage, TeX

C_2\times C_{40}
% in TeX

G:=Group("C2xC40");
// GroupNames label

G:=SmallGroup(80,23);
// by ID

G=gap.SmallGroup(80,23);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-2,100,58]);
// Polycyclic

G:=Group<a,b|a^2=b^40=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C40 in TeX

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