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G = C22×F11  order 440 = 23·5·11

Direct product of C22 and F11

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C22×F11, D22⋊3C10, C11⋊C5⋊C23, C22⋊(C2×C10), D11⋊(C2×C10), C11⋊(C22×C10), (C22×D11)⋊C5, (C2×C22)⋊2C10, (C2×C11⋊C5)⋊C22, (C22×C11⋊C5)⋊2C2, SmallGroup(440,42)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C22×F11
 G = < a,b,c,d | a2=b2=c11=d10=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c6 >

Subgroups: 334 in 64 conjugacy classes, 37 normal (8 characteristic)
C1, C2, C2, C22, C22, C5, C23, C10, C11, C2×C10, D11, C22, C22×C10, D22, C2×C22, C11⋊C5, C22×D11, F11, C2×C11⋊C5, C2×F11, C22×C11⋊C5, C22×F11
Quotients: C1, C2, C22, C5, C23, C10, C2×C10, C22×C10, F11, C2×F11, C22×F11

Smallest permutation representation of C22×F11
►On 44 points
Generators in S44
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G5A5B5C5D10A···10AB 11 22A22B22C
order12222222555510···1011222222
size1111111111111111111111···1110101010

44 irreducible representations

dim1111111010
type+++++
imageC1C2C2C5C10C10F11C2×F11
kernelC22×F11C2×F11C22×C11⋊C5C22×D11D22C2×C22C22C2
# reps161424413

Matrix representation of C22×F11 ►in GL11(𝔽331)

3300000000000
01000000000
00100000000
00010000000
00001000000
00000100000
00000010000
00000001000
00000000100
00000000010
00000000001
,
3300000000000
0330000000000
0033000000000
0003300000000
0000330000000
0000033000000
0000003300000
0000000330000
0000000033000
0000000003300
0000000000330
,
10000000000
0000000000330
0100000000330
0010000000330
0001000000330
0000100000330
0000010000330
0000001000330
0000000100330
0000000010330
0000000001330
,
2070000000000
0000003300000
0330000000000
0000000330000
0033000000000
0000000033000
0003300000000
0000000003300
0000330000000
0000000000330
0000033000000

G:=sub<GL(11,GF(331))| [330,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1],[330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,330],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,330,330,330,330,330,330,330,330,330,330],[207,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0,0,0,0,0,0,0,0,0,0,0,0,330,0] >;
 

C22×F11 in GAP, Magma, Sage, TeX 

C_2^2\times F_{11}
 
% in TeX
 
G:=Group("C2^2xF11");
 
// GroupNames label
 
G:=SmallGroup(440,42);
 
// by ID
 
G=gap.SmallGroup(440,42);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-5,-11,10004,1144]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^11=d^10=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^6>;
 
// generators/relations
 

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