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G = D5×C11order 110 = 2·5·11

Direct product of C11 and D5

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

Aliases: D5×C11, C5⋊C22, C553C2, SmallGroup(110,3)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C11
C1C5C55 — D5×C11
C5 — D5×C11
C1C11

Generators and relations for D5×C11
 G = < a,b,c | a11=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C22

Smallest permutation representation of D5×C11
On 55 points
Generators in S55
(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 50 28 36 14)(2 51 29 37 15)(3 52 30 38 16)(4 53 31 39 17)(5 54 32 40 18)(6 55 33 41 19)(7 45 23 42 20)(8 46 24 43 21)(9 47 25 44 22)(10 48 26 34 12)(11 49 27 35 13)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 12)(11 13)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 45)(43 46)(44 47)

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

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

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

D5×C11 is a maximal subgroup of   C11⋊F5

44 conjugacy classes

class 1  2 5A5B11A···11J22A···22J55A···55T
order125511···1122···2255···55
size15221···15···52···2

44 irreducible representations

dim111122
type+++
imageC1C2C11C22D5D5×C11
kernelD5×C11C55D5C5C11C1
# reps111010220

Matrix representation of D5×C11 in GL2(𝔽331) generated by

1800
0180
,
330118
330117
,
3300
3301
G:=sub<GL(2,GF(331))| [180,0,0,180],[330,330,118,117],[330,330,0,1] >;

D5×C11 in GAP, Magma, Sage, TeX

D_5\times C_{11}
% in TeX

G:=Group("D5xC11");
// GroupNames label

G:=SmallGroup(110,3);
// by ID

G=gap.SmallGroup(110,3);
# by ID

G:=PCGroup([3,-2,-11,-5,794]);
// Polycyclic

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

Export

Subgroup lattice of D5×C11 in TeX

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