direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C4×F11, D11⋊C20, C44⋊2C10, D22.C10, Dic11⋊2C10, (C4×D11)⋊C5, C11⋊C20⋊2C2, C11⋊1(C2×C20), (C2×F11).C2, C2.1(C2×F11), C22.2(C2×C10), C11⋊C5⋊1(C2×C4), (C4×C11⋊C5)⋊2C2, (C2×C11⋊C5).2C22, SmallGroup(440,8)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C11 — C22 — C2×C11⋊C5 — C2×F11 — C4×F11 |
C11 — C4×F11 |
Generators and relations for C4×F11
G = < a,b,c | a4=b11=c10=1, ab=ba, ac=ca, cbc-1=b6 >
(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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