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G = C2×C42order 84 = 22·3·7

Abelian group of type [2,42]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C42, SmallGroup(84,15)

Series: Derived Chief Lower central Upper central

C1 — C2×C42
C1C7C21C42 — C2×C42
C1 — C2×C42
C1 — C2×C42

Generators and relations for C2×C42
 G = < a,b | a2=b42=1, ab=ba >


Smallest permutation representation of C2×C42
Regular action on 84 points
Generators in S84
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 82)(14 83)(15 84)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)
(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)

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

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

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

C2×C42 is a maximal subgroup of   C217D4  C21.A4

84 conjugacy classes

class 1 2A2B2C3A3B6A···6F7A···7F14A···14R21A···21L42A···42AJ
order1222336···67···714···1421···2142···42
size1111111···11···11···11···11···1

84 irreducible representations

dim11111111
type++
imageC1C2C3C6C7C14C21C42
kernelC2×C42C42C2×C14C14C2×C6C6C22C2
# reps13266181236

Matrix representation of C2×C42 in GL2(𝔽43) generated by

420
042
,
30
040
G:=sub<GL(2,GF(43))| [42,0,0,42],[3,0,0,40] >;

C2×C42 in GAP, Magma, Sage, TeX

C_2\times C_{42}
% in TeX

G:=Group("C2xC42");
// GroupNames label

G:=SmallGroup(84,15);
// by ID

G=gap.SmallGroup(84,15);
# by ID

G:=PCGroup([4,-2,-2,-3,-7]);
// Polycyclic

G:=Group<a,b|a^2=b^42=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C42 in TeX

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