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G = A4×D11  order 264 = 23·3·11

Direct product of A4 and D11

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

Aliases: A4×D11, C11⋊(C2×A4), (C2×C22)⋊C6, (C22×D11)⋊C3, C22⋊(C3×D11), (C11×A4)⋊2C2, SmallGroup(264,33)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C22 — A4×D11
Chief series
C1 — C11 — C2×C22 — C11×A4 — A4×D11
Lower central
C2×C22 — A4×D11
Upper central
C1

Generators and relations for A4×D11
 G = < a,b,c,d,e | a2=b2=c3=d11=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

C1
3C2
11C2
33C2
4C3
C11
C22
33C22
33C22
44C6
D11
3C22
3D11
4C33
11C23
A4
C2×C22
3D22
3D22
4C3×D11
11C2×A4
C22×D11
C11×A4
A4×D11

Character table of A4×D11

 class 12A2B2C3A3B6A6B11A11B11C11D11E22A22B22C22D22E33A33B33C33D33E33F33G33H33I33J
 size 13113344444422222666668888888888
ρ11111111111111111111111111111    trivial
ρ211-1-111-1-111111111111111111111    linear of order 2
ρ31111ζ3ζ32ζ32ζ31111111111ζ3ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3    linear of order 3
ρ411-1-1ζ32ζ3ζ65ζ61111111111ζ32ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32    linear of order 6
ρ51111ζ32ζ3ζ3ζ321111111111ζ32ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32    linear of order 3
ρ611-1-1ζ3ζ32ζ6ζ651111111111ζ3ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3    linear of order 6
ρ722002200ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ1110+ζ11ζ117+ζ114ζ116+ζ115ζ117+ζ114ζ119+ζ112ζ1110+ζ11ζ118+ζ113ζ118+ζ113ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ1110+ζ11ζ117+ζ114ζ116+ζ115ζ119+ζ112ζ1110+ζ11ζ117+ζ114    orthogonal lifted from D11
ρ822002200ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ117+ζ114ζ116+ζ115ζ119+ζ112ζ116+ζ115ζ118+ζ113ζ117+ζ114ζ1110+ζ11ζ1110+ζ11ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ117+ζ114ζ116+ζ115ζ119+ζ112ζ118+ζ113ζ117+ζ114ζ116+ζ115    orthogonal lifted from D11
ρ922002200ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ118+ζ113ζ1110+ζ11ζ117+ζ114ζ1110+ζ11ζ116+ζ115ζ118+ζ113ζ119+ζ112ζ119+ζ112ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ118+ζ113ζ1110+ζ11ζ117+ζ114ζ116+ζ115ζ118+ζ113ζ1110+ζ11    orthogonal lifted from D11
ρ1022002200ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ116+ζ115ζ119+ζ112ζ118+ζ113ζ119+ζ112ζ1110+ζ11ζ116+ζ115ζ117+ζ114ζ117+ζ114ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ116+ζ115ζ119+ζ112ζ118+ζ113ζ1110+ζ11ζ116+ζ115ζ119+ζ112    orthogonal lifted from D11
ρ1122002200ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ119+ζ112ζ118+ζ113ζ1110+ζ11ζ118+ζ113ζ117+ζ114ζ119+ζ112ζ116+ζ115ζ116+ζ115ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ119+ζ112ζ118+ζ113ζ1110+ζ11ζ117+ζ114ζ119+ζ112ζ118+ζ113    orthogonal lifted from D11
ρ122200-1-√-3-1+√-300ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ116+ζ115ζ119+ζ112ζ118+ζ113ζ119+ζ112ζ1110+ζ11ζ116+ζ115ζ117+ζ114ζ32ζ117+ζ32ζ114ζ3ζ117+ζ3ζ114ζ3ζ118+ζ3ζ113ζ3ζ1110+ζ3ζ11ζ3ζ116+ζ3ζ115ζ3ζ119+ζ3ζ112ζ32ζ118+ζ32ζ113ζ32ζ1110+ζ32ζ11ζ32ζ116+ζ32ζ115ζ32ζ119+ζ32ζ112    complex lifted from C3×D11
ρ132200-1+√-3-1-√-300ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ117+ζ114ζ116+ζ115ζ119+ζ112ζ116+ζ115ζ118+ζ113ζ117+ζ114ζ1110+ζ11ζ3ζ1110+ζ3ζ11ζ32ζ1110+ζ32ζ11ζ32ζ119+ζ32ζ112ζ32ζ118+ζ32ζ113ζ32ζ117+ζ32ζ114ζ32ζ116+ζ32ζ115ζ3ζ119+ζ3ζ112ζ3ζ118+ζ3ζ113ζ3ζ117+ζ3ζ114ζ3ζ116+ζ3ζ115    complex lifted from C3×D11
ρ142200-1+√-3-1-√-300ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ118+ζ113ζ1110+ζ11ζ117+ζ114ζ1110+ζ11ζ116+ζ115ζ118+ζ113ζ119+ζ112ζ3ζ119+ζ3ζ112ζ32ζ119+ζ32ζ112ζ32ζ117+ζ32ζ114ζ32ζ116+ζ32ζ115ζ32ζ118+ζ32ζ113ζ32ζ1110+ζ32ζ11ζ3ζ117+ζ3ζ114ζ3ζ116+ζ3ζ115ζ3ζ118+ζ3ζ113ζ3ζ1110+ζ3ζ11    complex lifted from C3×D11
ρ152200-1+√-3-1-√-300ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ116+ζ115ζ119+ζ112ζ118+ζ113ζ119+ζ112ζ1110+ζ11ζ116+ζ115ζ117+ζ114ζ3ζ117+ζ3ζ114ζ32ζ117+ζ32ζ114ζ32ζ118+ζ32ζ113ζ32ζ1110+ζ32ζ11ζ32ζ116+ζ32ζ115ζ32ζ119+ζ32ζ112ζ3ζ118+ζ3ζ113ζ3ζ1110+ζ3ζ11ζ3ζ116+ζ3ζ115ζ3ζ119+ζ3ζ112    complex lifted from C3×D11
ρ162200-1+√-3-1-√-300ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ1110+ζ11ζ117+ζ114ζ116+ζ115ζ117+ζ114ζ119+ζ112ζ1110+ζ11ζ118+ζ113ζ3ζ118+ζ3ζ113ζ32ζ118+ζ32ζ113ζ32ζ116+ζ32ζ115ζ32ζ119+ζ32ζ112ζ32ζ1110+ζ32ζ11ζ32ζ117+ζ32ζ114ζ3ζ116+ζ3ζ115ζ3ζ119+ζ3ζ112ζ3ζ1110+ζ3ζ11ζ3ζ117+ζ3ζ114    complex lifted from C3×D11
ρ172200-1-√-3-1+√-300ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ119+ζ112ζ118+ζ113ζ1110+ζ11ζ118+ζ113ζ117+ζ114ζ119+ζ112ζ116+ζ115ζ32ζ116+ζ32ζ115ζ3ζ116+ζ3ζ115ζ3ζ1110+ζ3ζ11ζ3ζ117+ζ3ζ114ζ3ζ119+ζ3ζ112ζ3ζ118+ζ3ζ113ζ32ζ1110+ζ32ζ11ζ32ζ117+ζ32ζ114ζ32ζ119+ζ32ζ112ζ32ζ118+ζ32ζ113    complex lifted from C3×D11
ρ182200-1-√-3-1+√-300ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ118+ζ113ζ1110+ζ11ζ117+ζ114ζ1110+ζ11ζ116+ζ115ζ118+ζ113ζ119+ζ112ζ32ζ119+ζ32ζ112ζ3ζ119+ζ3ζ112ζ3ζ117+ζ3ζ114ζ3ζ116+ζ3ζ115ζ3ζ118+ζ3ζ113ζ3ζ1110+ζ3ζ11ζ32ζ117+ζ32ζ114ζ32ζ116+ζ32ζ115ζ32ζ118+ζ32ζ113ζ32ζ1110+ζ32ζ11    complex lifted from C3×D11
ρ192200-1-√-3-1+√-300ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ117+ζ114ζ116+ζ115ζ119+ζ112ζ116+ζ115ζ118+ζ113ζ117+ζ114ζ1110+ζ11ζ32ζ1110+ζ32ζ11ζ3ζ1110+ζ3ζ11ζ3ζ119+ζ3ζ112ζ3ζ118+ζ3ζ113ζ3ζ117+ζ3ζ114ζ3ζ116+ζ3ζ115ζ32ζ119+ζ32ζ112ζ32ζ118+ζ32ζ113ζ32ζ117+ζ32ζ114ζ32ζ116+ζ32ζ115    complex lifted from C3×D11
ρ202200-1+√-3-1-√-300ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ119+ζ112ζ118+ζ113ζ1110+ζ11ζ118+ζ113ζ117+ζ114ζ119+ζ112ζ116+ζ115ζ3ζ116+ζ3ζ115ζ32ζ116+ζ32ζ115ζ32ζ1110+ζ32ζ11ζ32ζ117+ζ32ζ114ζ32ζ119+ζ32ζ112ζ32ζ118+ζ32ζ113ζ3ζ1110+ζ3ζ11ζ3ζ117+ζ3ζ114ζ3ζ119+ζ3ζ112ζ3ζ118+ζ3ζ113    complex lifted from C3×D11
ρ212200-1-√-3-1+√-300ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ1110+ζ11ζ117+ζ114ζ116+ζ115ζ117+ζ114ζ119+ζ112ζ1110+ζ11ζ118+ζ113ζ32ζ118+ζ32ζ113ζ3ζ118+ζ3ζ113ζ3ζ116+ζ3ζ115ζ3ζ119+ζ3ζ112ζ3ζ1110+ζ3ζ11ζ3ζ117+ζ3ζ114ζ32ζ116+ζ32ζ115ζ32ζ119+ζ32ζ112ζ32ζ1110+ζ32ζ11ζ32ζ117+ζ32ζ114    complex lifted from C3×D11
ρ223-1-31000033333-1-1-1-1-10000000000    orthogonal lifted from C2×A4
ρ233-13-1000033333-1-1-1-1-10000000000    orthogonal lifted from A4
ρ246-20000003ζ118+3ζ1133ζ116+3ζ1153ζ119+3ζ1123ζ1110+3ζ113ζ117+3ζ114-ζ116-ζ115-ζ117-ζ114-ζ119-ζ112-ζ1110-ζ11-ζ118-ζ1130000000000    orthogonal faithful
ρ256-20000003ζ116+3ζ1153ζ1110+3ζ113ζ117+3ζ1143ζ119+3ζ1123ζ118+3ζ113-ζ1110-ζ11-ζ118-ζ113-ζ117-ζ114-ζ119-ζ112-ζ116-ζ1150000000000    orthogonal faithful
ρ266-20000003ζ1110+3ζ113ζ119+3ζ1123ζ118+3ζ1133ζ117+3ζ1143ζ116+3ζ115-ζ119-ζ112-ζ116-ζ115-ζ118-ζ113-ζ117-ζ114-ζ1110-ζ110000000000    orthogonal faithful
ρ276-20000003ζ119+3ζ1123ζ117+3ζ1143ζ116+3ζ1153ζ118+3ζ1133ζ1110+3ζ11-ζ117-ζ114-ζ1110-ζ11-ζ116-ζ115-ζ118-ζ113-ζ119-ζ1120000000000    orthogonal faithful
ρ286-20000003ζ117+3ζ1143ζ118+3ζ1133ζ1110+3ζ113ζ116+3ζ1153ζ119+3ζ112-ζ118-ζ113-ζ119-ζ112-ζ1110-ζ11-ζ116-ζ115-ζ117-ζ1140000000000    orthogonal faithful

Smallest permutation representation of A4×D11
►On 44 points
Generators in S44
(1 21)(2 22)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)

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

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

Matrix representation of A4×D11 ►in GL5(𝔽67)

10000
01000
006600
00656652
003601
,
10000
01000
00666552
000660
000361
,
370000
037000
00666652
00100
00001
,
241000
6211000
00100
00010
00001
,
2949000
238000
006600
000660
000066

G:=sub<GL(5,GF(67))| [1,0,0,0,0,0,1,0,0,0,0,0,66,65,36,0,0,0,66,0,0,0,0,52,1],[1,0,0,0,0,0,1,0,0,0,0,0,66,0,0,0,0,65,66,36,0,0,52,0,1],[37,0,0,0,0,0,37,0,0,0,0,0,66,1,0,0,0,66,0,0,0,0,52,0,1],[24,62,0,0,0,1,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[29,2,0,0,0,49,38,0,0,0,0,0,66,0,0,0,0,0,66,0,0,0,0,0,66] >;
 

A4×D11 in GAP, Magma, Sage, TeX 

A_4\times D_{11}
 
% in TeX
 
G:=Group("A4xD11");
 
// GroupNames label
 
G:=SmallGroup(264,33);
 
// by ID
 
G=gap.SmallGroup(264,33);
 
# by ID
 
G:=PCGroup([5,-2,-3,-2,2,-11,142,68,6004]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^11=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

Export

Subgroup lattice of A4×D11 in TeX
Character table of A4×D11 in TeX

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