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G = C4×F11  order 440 = 23·5·11

Direct product of C4 and F11

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

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 >

C1
C2
11C2
11C2
11C5
C11
C4
11C4
11C22
11C10
11C10
11C10
D11
C22
D11
C11⋊C5
11C2×C4
11C20
11C2×C10
11C20
D22
C44
Dic11
C2×C11⋊C5
F11
F11
11C2×C20
C4×D11
C2×F11
C11⋊C20
C4×C11⋊C5
C4×F11

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 2A2B2C4A4B4C4D5A5B5C5D10A···10L 11 20A···20P 22 44A44B
order12224444555510···101120···20224444
size1111111111111111111111···111011···11101010

44 irreducible representations

dim1111111111101010
type++++++
imageC1C2C2C2C4C5C10C10C10C20F11C2×F11C4×F11
kernelC4×F11C11⋊C20C4×C11⋊C5C2×F11F11C4×D11Dic11C44D22D11C4C2C1
# reps11114444416112

Matrix representation of C4×F11 ►in GL10(𝔽661)

555000000000
055500000000
005550000000
000555000000
000055500000
000005550000
000000555000
000000055500
000000005550
000000000555
,
000000000660
100000000660
010000000660
001000000660
000100000660
000010000660
000001000660
000000100660
000000010660
000000001660
,
100006600000
000006601000
010006600000
000006600100
001006600000
000006600010
000106600000
000006600001
000016600000
000006600000

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
 

Export

Subgroup lattice of C4×F11 in TeX

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