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## G = C4×F11order 440 = 23·5·11

### Direct product of C4 and F11

Aliases: C4×F11, D11⋊C20, C442C10, D22.C10, Dic112C10, (C4×D11)⋊C5, C11⋊C202C2, C111(C2×C20), (C2×F11).C2, C2.1(C2×F11), C22.2(C2×C10), C11⋊C51(C2×C4), (C4×C11⋊C5)⋊2C2, (C2×C11⋊C5).2C22, SmallGroup(440,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C4×F11
 Chief series C1 — C11 — C22 — C2×C11⋊C5 — C2×F11 — C4×F11
 Lower central C11 — C4×F11
 Upper central C1 — C4

Generators and relations for C4×F11
G = < a,b,c | a4=b11=c10=1, ab=ba, ac=ca, cbc-1=b6 >

Smallest permutation representation of C4×F11
On 44 points
Generators in S44
(1 34 12 23)(2 35 13 24)(3 36 14 25)(4 37 15 26)(5 38 16 27)(6 39 17 28)(7 40 18 29)(8 41 19 30)(9 42 20 31)(10 43 21 32)(11 44 22 33)
(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)
(2 3 5 9 6 11 10 8 4 7)(13 14 16 20 17 22 21 19 15 18)(24 25 27 31 28 33 32 30 26 29)(35 36 38 42 39 44 43 41 37 40)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 10A ··· 10L 11 20A ··· 20P 22 44A 44B order 1 2 2 2 4 4 4 4 5 5 5 5 10 ··· 10 11 20 ··· 20 22 44 44 size 1 1 11 11 1 1 11 11 11 11 11 11 11 ··· 11 10 11 ··· 11 10 10 10

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 10 10 10 type + + + + + + image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 F11 C2×F11 C4×F11 kernel C4×F11 C11⋊C20 C4×C11⋊C5 C2×F11 F11 C4×D11 Dic11 C44 D22 D11 C4 C2 C1 # reps 1 1 1 1 4 4 4 4 4 16 1 1 2

Matrix representation of C4×F11 in GL10(𝔽661)

 555 0 0 0 0 0 0 0 0 0 0 555 0 0 0 0 0 0 0 0 0 0 555 0 0 0 0 0 0 0 0 0 0 555 0 0 0 0 0 0 0 0 0 0 555 0 0 0 0 0 0 0 0 0 0 555 0 0 0 0 0 0 0 0 0 0 555 0 0 0 0 0 0 0 0 0 0 555 0 0 0 0 0 0 0 0 0 0 555 0 0 0 0 0 0 0 0 0 0 555
,
 0 0 0 0 0 0 0 0 0 660 1 0 0 0 0 0 0 0 0 660 0 1 0 0 0 0 0 0 0 660 0 0 1 0 0 0 0 0 0 660 0 0 0 1 0 0 0 0 0 660 0 0 0 0 1 0 0 0 0 660 0 0 0 0 0 1 0 0 0 660 0 0 0 0 0 0 1 0 0 660 0 0 0 0 0 0 0 1 0 660 0 0 0 0 0 0 0 0 1 660
,
 1 0 0 0 0 660 0 0 0 0 0 0 0 0 0 660 1 0 0 0 0 1 0 0 0 660 0 0 0 0 0 0 0 0 0 660 0 1 0 0 0 0 1 0 0 660 0 0 0 0 0 0 0 0 0 660 0 0 1 0 0 0 0 1 0 660 0 0 0 0 0 0 0 0 0 660 0 0 0 1 0 0 0 0 1 660 0 0 0 0 0 0 0 0 0 660 0 0 0 0

G:=sub<GL(10,GF(661))| [555,0,0,0,0,0,0,0,0,0,0,555,0,0,0,0,0,0,0,0,0,0,555,0,0,0,0,0,0,0,0,0,0,555,0,0,0,0,0,0,0,0,0,0,555,0,0,0,0,0,0,0,0,0,0,555,0,0,0,0,0,0,0,0,0,0,555,0,0,0,0,0,0,0,0,0,0,555,0,0,0,0,0,0,0,0,0,0,555,0,0,0,0,0,0,0,0,0,0,555],[0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,660,660,660,660,660,660,660,660,660,660],[1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,660,660,660,660,660,660,660,660,660,660,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0] >;

C4×F11 in GAP, Magma, Sage, TeX

C_4\times F_{11}
% in TeX

G:=Group("C4xF11");
// GroupNames label

G:=SmallGroup(440,8);
// by ID

G=gap.SmallGroup(440,8);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-11,106,10004,2264]);
// Polycyclic

G:=Group<a,b,c|a^4=b^11=c^10=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
// generators/relations

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