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G = C2×C43⋊C3order 258 = 2·3·43

Direct product of C2 and C43⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C2×C43⋊C3, C86⋊C3, C432C6, SmallGroup(258,2)

Series: Derived Chief Lower central Upper central

C1C43 — C2×C43⋊C3
C1C43C43⋊C3 — C2×C43⋊C3
C43 — C2×C43⋊C3
C1C2

Generators and relations for C2×C43⋊C3
 G = < a,b,c | a2=b43=c3=1, ab=ba, ac=ca, cbc-1=b6 >

43C3
43C6

Smallest permutation representation of C2×C43⋊C3
On 86 points
Generators in S86
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)
(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 81 82 83 84 85 86)
(2 37 7)(3 30 13)(4 23 19)(5 16 25)(6 9 31)(8 38 43)(10 24 12)(11 17 18)(14 39 36)(15 32 42)(20 40 29)(21 33 35)(22 26 41)(27 34 28)(45 80 50)(46 73 56)(47 66 62)(48 59 68)(49 52 74)(51 81 86)(53 67 55)(54 60 61)(57 82 79)(58 75 85)(63 83 72)(64 76 78)(65 69 84)(70 77 71)

G:=sub<Sym(86)| (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86), (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)(44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86), (2,37,7)(3,30,13)(4,23,19)(5,16,25)(6,9,31)(8,38,43)(10,24,12)(11,17,18)(14,39,36)(15,32,42)(20,40,29)(21,33,35)(22,26,41)(27,34,28)(45,80,50)(46,73,56)(47,66,62)(48,59,68)(49,52,74)(51,81,86)(53,67,55)(54,60,61)(57,82,79)(58,75,85)(63,83,72)(64,76,78)(65,69,84)(70,77,71)>;

G:=Group( (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86), (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)(44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86), (2,37,7)(3,30,13)(4,23,19)(5,16,25)(6,9,31)(8,38,43)(10,24,12)(11,17,18)(14,39,36)(15,32,42)(20,40,29)(21,33,35)(22,26,41)(27,34,28)(45,80,50)(46,73,56)(47,66,62)(48,59,68)(49,52,74)(51,81,86)(53,67,55)(54,60,61)(57,82,79)(58,75,85)(63,83,72)(64,76,78)(65,69,84)(70,77,71) );

G=PermutationGroup([[(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,61),(19,62),(20,63),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,71),(29,72),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(37,80),(38,81),(39,82),(40,83),(41,84),(42,85),(43,86)], [(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),(44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86)], [(2,37,7),(3,30,13),(4,23,19),(5,16,25),(6,9,31),(8,38,43),(10,24,12),(11,17,18),(14,39,36),(15,32,42),(20,40,29),(21,33,35),(22,26,41),(27,34,28),(45,80,50),(46,73,56),(47,66,62),(48,59,68),(49,52,74),(51,81,86),(53,67,55),(54,60,61),(57,82,79),(58,75,85),(63,83,72),(64,76,78),(65,69,84),(70,77,71)]])

34 conjugacy classes

class 1  2 3A3B6A6B43A···43N86A···86N
order12336643···4386···86
size11434343433···33···3

34 irreducible representations

dim111133
type++
imageC1C2C3C6C43⋊C3C2×C43⋊C3
kernelC2×C43⋊C3C43⋊C3C86C43C2C1
# reps11221414

Matrix representation of C2×C43⋊C3 in GL4(𝔽1033) generated by

1032000
0100
0010
0001
,
1000
05981451
0100
0010
,
837000
0100
0668741009
0701963958
G:=sub<GL(4,GF(1033))| [1032,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,598,1,0,0,145,0,1,0,1,0,0],[837,0,0,0,0,1,668,701,0,0,74,963,0,0,1009,958] >;

C2×C43⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_{43}\rtimes C_3
% in TeX

G:=Group("C2xC43:C3");
// GroupNames label

G:=SmallGroup(258,2);
// by ID

G=gap.SmallGroup(258,2);
# by ID

G:=PCGroup([3,-2,-3,-43,977]);
// Polycyclic

G:=Group<a,b,c|a^2=b^43=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
// generators/relations

Export

Subgroup lattice of C2×C43⋊C3 in TeX

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