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G = C6×C12order 72 = 23·32

Abelian group of type [6,12]

direct product, abelian, monomial

Aliases: C6×C12, SmallGroup(72,36)

Series: Derived Chief Lower central Upper central

C1 — C6×C12
C1C2C6C3×C6C3×C12 — C6×C12
C1 — C6×C12
C1 — C6×C12

Generators and relations for C6×C12
 G = < a,b | a6=b12=1, ab=ba >


Smallest permutation representation of C6×C12
Regular action on 72 points
Generators in S72
(1 35 51 39 70 22)(2 36 52 40 71 23)(3 25 53 41 72 24)(4 26 54 42 61 13)(5 27 55 43 62 14)(6 28 56 44 63 15)(7 29 57 45 64 16)(8 30 58 46 65 17)(9 31 59 47 66 18)(10 32 60 48 67 19)(11 33 49 37 68 20)(12 34 50 38 69 21)
(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)| (1,35,51,39,70,22)(2,36,52,40,71,23)(3,25,53,41,72,24)(4,26,54,42,61,13)(5,27,55,43,62,14)(6,28,56,44,63,15)(7,29,57,45,64,16)(8,30,58,46,65,17)(9,31,59,47,66,18)(10,32,60,48,67,19)(11,33,49,37,68,20)(12,34,50,38,69,21), (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( (1,35,51,39,70,22)(2,36,52,40,71,23)(3,25,53,41,72,24)(4,26,54,42,61,13)(5,27,55,43,62,14)(6,28,56,44,63,15)(7,29,57,45,64,16)(8,30,58,46,65,17)(9,31,59,47,66,18)(10,32,60,48,67,19)(11,33,49,37,68,20)(12,34,50,38,69,21), (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([(1,35,51,39,70,22),(2,36,52,40,71,23),(3,25,53,41,72,24),(4,26,54,42,61,13),(5,27,55,43,62,14),(6,28,56,44,63,15),(7,29,57,45,64,16),(8,30,58,46,65,17),(9,31,59,47,66,18),(10,32,60,48,67,19),(11,33,49,37,68,20),(12,34,50,38,69,21)], [(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)])

C6×C12 is a maximal subgroup of   C12.58D6  C6.Dic6  C12⋊Dic3  C6.11D12  C12.59D6

72 conjugacy classes

class 1 2A2B2C3A···3H4A4B4C4D6A···6X12A···12AF
order12223···344446···612···12
size11111···111111···11···1

72 irreducible representations

dim11111111
type+++
imageC1C2C2C3C4C6C6C12
kernelC6×C12C3×C12C62C2×C12C3×C6C12C2×C6C6
# reps1218416832

Matrix representation of C6×C12 in GL2(𝔽13) generated by

10
010
,
60
012
G:=sub<GL(2,GF(13))| [1,0,0,10],[6,0,0,12] >;

C6×C12 in GAP, Magma, Sage, TeX

C_6\times C_{12}
% in TeX

G:=Group("C6xC12");
// GroupNames label

G:=SmallGroup(72,36);
// by ID

G=gap.SmallGroup(72,36);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-2,180]);
// Polycyclic

G:=Group<a,b|a^6=b^12=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C6×C12 in TeX

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