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G = C6×D11order 132 = 22·3·11

Direct product of C6 and D11

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C6×D11, C22⋊C6, C662C2, C333C22, C11⋊(C2×C6), SmallGroup(132,7)

Series: Derived Chief Lower central Upper central

C1C11 — C6×D11
C1C11C33C3×D11 — C6×D11
C11 — C6×D11
C1C6

Generators and relations for C6×D11
 G = < a,b,c | a6=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

11C2
11C2
11C22
11C6
11C6
11C2×C6

Smallest permutation representation of C6×D11
On 66 points
Generators in S66
(1 54 32 43 21 65)(2 55 33 44 22 66)(3 45 23 34 12 56)(4 46 24 35 13 57)(5 47 25 36 14 58)(6 48 26 37 15 59)(7 49 27 38 16 60)(8 50 28 39 17 61)(9 51 29 40 18 62)(10 52 30 41 19 63)(11 53 31 42 20 64)
(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)
(1 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 44)(11 43)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 55)(20 54)(21 53)(22 52)(23 62)(24 61)(25 60)(26 59)(27 58)(28 57)(29 56)(30 66)(31 65)(32 64)(33 63)

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

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

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

C6×D11 is a maximal subgroup of   C33⋊D4  C3⋊D44

42 conjugacy classes

class 1 2A2B2C3A3B6A6B6C6D6E6F11A···11E22A···22E33A···33J66A···66J
order12223366666611···1122···2233···3366···66
size1111111111111111112···22···22···22···2

42 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6D11D22C3×D11C6×D11
kernelC6×D11C3×D11C66D22D11C22C6C3C2C1
# reps121242551010

Matrix representation of C6×D11 in GL2(𝔽43) generated by

70
07
,
4224
2815
,
2835
2815
G:=sub<GL(2,GF(43))| [7,0,0,7],[42,28,24,15],[28,28,35,15] >;

C6×D11 in GAP, Magma, Sage, TeX

C_6\times D_{11}
% in TeX

G:=Group("C6xD11");
// GroupNames label

G:=SmallGroup(132,7);
// by ID

G=gap.SmallGroup(132,7);
# by ID

G:=PCGroup([4,-2,-2,-3,-11,1923]);
// Polycyclic

G:=Group<a,b,c|a^6=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C6×D11 in TeX

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