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G = C5xD11order 110 = 2·5·11

Direct product of C5 and D11

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

Aliases: C5xD11, C55:2C2, C11:3C10, SmallGroup(110,4)

Series: Derived Chief Lower central Upper central

C1C11 — C5xD11
C1C11C55 — C5xD11
C11 — C5xD11
C1C5

Generators and relations for C5xD11
 G = < a,b,c | a5=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 28 in 8 conjugacy classes, 6 normal (all characteristic)
Quotients: C1, C2, C5, C10, D11, C5xD11
11C2
11C10

Smallest permutation representation of C5xD11
On 55 points
Generators in S55
(1 54 43 32 21)(2 55 44 33 22)(3 45 34 23 12)(4 46 35 24 13)(5 47 36 25 14)(6 48 37 26 15)(7 49 38 27 16)(8 50 39 28 17)(9 51 40 29 18)(10 52 41 30 19)(11 53 42 31 20)
(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)
(1 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)

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

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

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

35 conjugacy classes

class 1  2 5A5B5C5D10A10B10C10D11A···11E55A···55T
order1255551010101011···1155···55
size1111111111111112···22···2

35 irreducible representations

dim111122
type+++
imageC1C2C5C10D11C5xD11
kernelC5xD11C55D11C11C5C1
# reps1144520

Matrix representation of C5xD11 in GL2(F331) generated by

3230
0323
,
3301
206124
,
3300
2061
G:=sub<GL(2,GF(331))| [323,0,0,323],[330,206,1,124],[330,206,0,1] >;

C5xD11 in GAP, Magma, Sage, TeX

C_5\times D_{11}
% in TeX

G:=Group("C5xD11");
// GroupNames label

G:=SmallGroup(110,4);
// by ID

G=gap.SmallGroup(110,4);
# by ID

G:=PCGroup([3,-2,-5,-11,902]);
// Polycyclic

G:=Group<a,b,c|a^5=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5xD11 in TeX

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