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## G = C11×F5order 220 = 22·5·11

### Direct product of C11 and F5

Aliases: C11×F5, C5⋊C44, C552C4, D5.C22, (D5×C11).2C2, SmallGroup(220,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C11×F5
 Chief series C1 — C5 — D5 — D5×C11 — C11×F5
 Lower central C5 — C11×F5
 Upper central C1 — C11

Generators and relations for C11×F5
G = < a,b,c | a11=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C11×F5
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 34 32 13 50)(2 35 33 14 51)(3 36 23 15 52)(4 37 24 16 53)(5 38 25 17 54)(6 39 26 18 55)(7 40 27 19 45)(8 41 28 20 46)(9 42 29 21 47)(10 43 30 22 48)(11 44 31 12 49)
(12 44 31 49)(13 34 32 50)(14 35 33 51)(15 36 23 52)(16 37 24 53)(17 38 25 54)(18 39 26 55)(19 40 27 45)(20 41 28 46)(21 42 29 47)(22 43 30 48)

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

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

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

55 conjugacy classes

 class 1 2 4A 4B 5 11A ··· 11J 22A ··· 22J 44A ··· 44T 55A ··· 55J order 1 2 4 4 5 11 ··· 11 22 ··· 22 44 ··· 44 55 ··· 55 size 1 5 5 5 4 1 ··· 1 5 ··· 5 5 ··· 5 4 ··· 4

55 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + image C1 C2 C4 C11 C22 C44 F5 C11×F5 kernel C11×F5 D5×C11 C55 F5 D5 C5 C11 C1 # reps 1 1 2 10 10 20 1 10

Matrix representation of C11×F5 in GL4(𝔽661) generated by

 457 0 0 0 0 457 0 0 0 0 457 0 0 0 0 457
,
 660 660 660 660 1 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 1 0 0 660 660 660 660
G:=sub<GL(4,GF(661))| [457,0,0,0,0,457,0,0,0,0,457,0,0,0,0,457],[660,1,0,0,660,0,1,0,660,0,0,1,660,0,0,0],[1,0,0,660,0,0,1,660,0,0,0,660,0,1,0,660] >;

C11×F5 in GAP, Magma, Sage, TeX

C_{11}\times F_5
% in TeX

G:=Group("C11xF5");
// GroupNames label

G:=SmallGroup(220,9);
// by ID

G=gap.SmallGroup(220,9);
# by ID

G:=PCGroup([4,-2,-11,-2,-5,88,1411,139]);
// Polycyclic

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

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