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## G = F5×D11order 440 = 23·5·11

### Direct product of F5 and D11

Aliases: F5×D11, D55⋊C4, D5.1D22, C55⋊(C2×C4), C5⋊(C4×D11), C11⋊F5⋊C2, (C5×D11)⋊C4, (C11×F5)⋊C2, C111(C2×F5), (D5×D11).C2, (D5×C11).C22, SmallGroup(440,43)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C55 — F5×D11
 Chief series C1 — C11 — C55 — D5×C11 — D5×D11 — F5×D11
 Lower central C55 — F5×D11
 Upper central C1

Generators and relations for F5×D11
G = < a,b,c,d | a5=b4=c11=d2=1, bab-1=a3, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

5C2
11C2
55C2
5C4
55C4
55C22
11D5
11C10
5C22
5D11
55C2×C4
11F5
11D10
5D22
5C44
11C2×F5

Smallest permutation representation of F5×D11
On 55 points
Generators in S55
(1 21 32 43 54)(2 22 33 44 55)(3 12 23 34 45)(4 13 24 35 46)(5 14 25 36 47)(6 15 26 37 48)(7 16 27 38 49)(8 17 28 39 50)(9 18 29 40 51)(10 19 30 41 52)(11 20 31 42 53)
(12 23 45 34)(13 24 46 35)(14 25 47 36)(15 26 48 37)(16 27 49 38)(17 28 50 39)(18 29 51 40)(19 30 52 41)(20 31 53 42)(21 32 54 43)(22 33 55 44)
(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)| (1,21,32,43,54)(2,22,33,44,55)(3,12,23,34,45)(4,13,24,35,46)(5,14,25,36,47)(6,15,26,37,48)(7,16,27,38,49)(8,17,28,39,50)(9,18,29,40,51)(10,19,30,41,52)(11,20,31,42,53), (12,23,45,34)(13,24,46,35)(14,25,47,36)(15,26,48,37)(16,27,49,38)(17,28,50,39)(18,29,51,40)(19,30,52,41)(20,31,53,42)(21,32,54,43)(22,33,55,44), (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( (1,21,32,43,54)(2,22,33,44,55)(3,12,23,34,45)(4,13,24,35,46)(5,14,25,36,47)(6,15,26,37,48)(7,16,27,38,49)(8,17,28,39,50)(9,18,29,40,51)(10,19,30,41,52)(11,20,31,42,53), (12,23,45,34)(13,24,46,35)(14,25,47,36)(15,26,48,37)(16,27,49,38)(17,28,50,39)(18,29,51,40)(19,30,52,41)(20,31,53,42)(21,32,54,43)(22,33,55,44), (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([[(1,21,32,43,54),(2,22,33,44,55),(3,12,23,34,45),(4,13,24,35,46),(5,14,25,36,47),(6,15,26,37,48),(7,16,27,38,49),(8,17,28,39,50),(9,18,29,40,51),(10,19,30,41,52),(11,20,31,42,53)], [(12,23,45,34),(13,24,46,35),(14,25,47,36),(15,26,48,37),(16,27,49,38),(17,28,50,39),(18,29,51,40),(19,30,52,41),(20,31,53,42),(21,32,54,43),(22,33,55,44)], [(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 2A 2B 2C 4A 4B 4C 4D 5 10 11A ··· 11E 22A ··· 22E 44A ··· 44J 55A ··· 55E order 1 2 2 2 4 4 4 4 5 10 11 ··· 11 22 ··· 22 44 ··· 44 55 ··· 55 size 1 5 11 55 5 5 55 55 4 44 2 ··· 2 10 ··· 10 10 ··· 10 8 ··· 8

35 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 4 8 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 D11 D22 C4×D11 F5 C2×F5 F5×D11 kernel F5×D11 C11×F5 C11⋊F5 D5×D11 C5×D11 D55 F5 D5 C5 D11 C11 C1 # reps 1 1 1 1 2 2 5 5 10 1 1 5

Matrix representation of F5×D11 in GL6(𝔽661)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 245 0 0 2 1 1 245 0 0 1 2 1 245 0 0 634 634 634 657
,
 106 0 0 0 0 0 0 106 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 660 660 660 416 0 0 0 0 0 1
,
 218 219 0 0 0 0 660 660 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 219 0 0 0 0 0 660 0 0 0 0 0 0 660 0 0 0 0 0 0 660 0 0 0 0 0 0 660 0 0 0 0 0 0 660

G:=sub<GL(6,GF(661))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,1,634,0,0,1,1,2,634,0,0,1,1,1,634,0,0,245,245,245,657],[106,0,0,0,0,0,0,106,0,0,0,0,0,0,0,1,660,0,0,0,0,0,660,0,0,0,1,0,660,0,0,0,0,0,416,1],[218,660,0,0,0,0,219,660,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,219,660,0,0,0,0,0,0,660,0,0,0,0,0,0,660,0,0,0,0,0,0,660,0,0,0,0,0,0,660] >;

F5×D11 in GAP, Magma, Sage, TeX

F_5\times D_{11}
% in TeX

G:=Group("F5xD11");
// GroupNames label

G:=SmallGroup(440,43);
// by ID

G=gap.SmallGroup(440,43);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-11,26,168,173,10004]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=c^11=d^2=1,b*a*b^-1=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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