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G = C2×C40order 80 = 24·5

Abelian group of type [2,40]

Aliases: C2×C40, SmallGroup(80,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C40
 Chief series C1 — C2 — C4 — C20 — C40 — C2×C40
 Lower central C1 — C2×C40
 Upper central 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 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 8A ··· 8H 10A ··· 10L 20A ··· 20P 40A ··· 40AF order 1 2 2 2 4 4 4 4 5 5 5 5 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 1 1 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C4 C5 C8 C10 C10 C20 C20 C40 kernel C2×C40 C40 C2×C20 C20 C2×C10 C2×C8 C10 C8 C2×C4 C4 C22 C2 # reps 1 2 1 2 2 4 8 8 4 8 8 32

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

 1 0 0 0 1 0 0 0 40
,
 27 0 0 0 4 0 0 0 3
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

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