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G = C2×C36order 72 = 23·32

Abelian group of type [2,36]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C36, SmallGroup(72,9)

Series: Derived Chief Lower central Upper central

C1 — C2×C36
C1C3C6C18C36 — C2×C36
C1 — C2×C36
C1 — C2×C36

Generators and relations for C2×C36
 G = < a,b | a2=b36=1, ab=ba >


Smallest permutation representation of C2×C36
Regular action on 72 points
Generators in S72
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)
(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)

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

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

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

C2×C36 is a maximal subgroup of   C4.Dic9  Dic9⋊C4  C4⋊Dic9  D18⋊C4  D365C2

72 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A···6F9A···9F12A···12H18A···18R36A···36X
order12223344446···69···912···1218···1836···36
size11111111111···11···11···11···11···1

72 irreducible representations

dim111111111111
type+++
imageC1C2C2C3C4C6C6C9C12C18C18C36
kernelC2×C36C36C2×C18C2×C12C18C12C2×C6C2×C4C6C4C22C2
# reps12124426812624

Matrix representation of C2×C36 in GL2(𝔽37) generated by

360
036
,
240
07
G:=sub<GL(2,GF(37))| [36,0,0,36],[24,0,0,7] >;

C2×C36 in GAP, Magma, Sage, TeX

C_2\times C_{36}
% in TeX

G:=Group("C2xC36");
// GroupNames label

G:=SmallGroup(72,9);
// by ID

G=gap.SmallGroup(72,9);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-3,60,102]);
// Polycyclic

G:=Group<a,b|a^2=b^36=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C36 in TeX

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