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G = C6xD11order 132 = 22·3·11

Direct product of C6 and D11

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

Aliases: C6xD11, C22:C6, C66:2C2, C33:3C22, C11:(C2xC6), SmallGroup(132,7)

Series: Derived Chief Lower central Upper central

C1C11 — C6xD11
C1C11C33C3xD11 — C6xD11
C11 — C6xD11
C1C6

Generators and relations for C6xD11
 G = < a,b,c | a6=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 80 in 20 conjugacy classes, 14 normal (10 characteristic)
Quotients: C1, C2, C3, C22, C6, C2xC6, D11, D22, C3xD11, C6xD11
11C2
11C2
11C22
11C6
11C6
11C2xC6

Smallest permutation representation of C6xD11
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)]])

C6xD11 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+++++
imageC1C2C2C3C6C6D11D22C3xD11C6xD11
kernelC6xD11C3xD11C66D22D11C22C6C3C2C1
# reps121242551010

Matrix representation of C6xD11 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] >;

C6xD11 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 C6xD11 in TeX

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