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G = C4×D11order 88 = 23·11

Direct product of C4 and D11

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×D11, C442C2, D22.C2, C4Dic11, C2.1D22, Dic112C2, C22.2C22, C111(C2×C4), SmallGroup(88,4)

Series: Derived Chief Lower central Upper central

C1C11 — C4×D11
C1C11C22D22 — C4×D11
C11 — C4×D11
C1C4

Generators and relations for C4×D11
 G = < a,b,c | a4=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

11C2
11C2
11C22
11C4
11C2×C4

Character table of C4×D11

 class 12A2B2C4A4B4C4D11A11B11C11D11E22A22B22C22D22E44A44B44C44D44E44F44G44H44I44J
 size 11111111111122222222222222222222
ρ11111111111111111111111111111    trivial
ρ211-1-111-1-111111111111111111111    linear of order 2
ρ31111-1-1-1-11111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ411-1-1-1-1111111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ51-1-11i-i-ii11111-1-1-1-1-1i-i-i-i-i-iiiii    linear of order 4
ρ61-11-1-ii-ii11111-1-1-1-1-1-iiiiii-i-i-i-i    linear of order 4
ρ71-11-1i-ii-i11111-1-1-1-1-1i-i-i-i-i-iiiii    linear of order 4
ρ81-1-11-iii-i11111-1-1-1-1-1-iiiiii-i-i-i-i    linear of order 4
ρ922002200ζ118113ζ116115ζ119112ζ111011ζ117114ζ116115ζ117114ζ119112ζ111011ζ118113ζ118113ζ118113ζ116115ζ119112ζ111011ζ117114ζ116115ζ119112ζ111011ζ117114    orthogonal lifted from D11
ρ102200-2-200ζ117114ζ118113ζ111011ζ116115ζ119112ζ118113ζ119112ζ111011ζ116115ζ117114117114117114118113111011116115119112118113111011116115119112    orthogonal lifted from D22
ρ112200-2-200ζ111011ζ119112ζ118113ζ117114ζ116115ζ119112ζ116115ζ118113ζ117114ζ111011111011111011119112118113117114116115119112118113117114116115    orthogonal lifted from D22
ρ122200-2-200ζ116115ζ111011ζ117114ζ119112ζ118113ζ111011ζ118113ζ117114ζ119112ζ116115116115116115111011117114119112118113111011117114119112118113    orthogonal lifted from D22
ρ1322002200ζ111011ζ119112ζ118113ζ117114ζ116115ζ119112ζ116115ζ118113ζ117114ζ111011ζ111011ζ111011ζ119112ζ118113ζ117114ζ116115ζ119112ζ118113ζ117114ζ116115    orthogonal lifted from D11
ρ1422002200ζ119112ζ117114ζ116115ζ118113ζ111011ζ117114ζ111011ζ116115ζ118113ζ119112ζ119112ζ119112ζ117114ζ116115ζ118113ζ111011ζ117114ζ116115ζ118113ζ111011    orthogonal lifted from D11
ρ1522002200ζ117114ζ118113ζ111011ζ116115ζ119112ζ118113ζ119112ζ111011ζ116115ζ117114ζ117114ζ117114ζ118113ζ111011ζ116115ζ119112ζ118113ζ111011ζ116115ζ119112    orthogonal lifted from D11
ρ162200-2-200ζ118113ζ116115ζ119112ζ111011ζ117114ζ116115ζ117114ζ119112ζ111011ζ118113118113118113116115119112111011117114116115119112111011117114    orthogonal lifted from D22
ρ1722002200ζ116115ζ111011ζ117114ζ119112ζ118113ζ111011ζ118113ζ117114ζ119112ζ116115ζ116115ζ116115ζ111011ζ117114ζ119112ζ118113ζ111011ζ117114ζ119112ζ118113    orthogonal lifted from D11
ρ182200-2-200ζ119112ζ117114ζ116115ζ118113ζ111011ζ117114ζ111011ζ116115ζ118113ζ119112119112119112117114116115118113111011117114116115118113111011    orthogonal lifted from D22
ρ192-200-2i2i00ζ119112ζ117114ζ116115ζ118113ζ111011117114111011116115118113119112ζ43ζ11943ζ112ζ4ζ1194ζ112ζ4ζ1174ζ114ζ4ζ1164ζ115ζ4ζ1184ζ113ζ4ζ11104ζ11ζ43ζ11743ζ114ζ43ζ11643ζ115ζ43ζ11843ζ113ζ43ζ111043ζ11    complex faithful
ρ202-200-2i2i00ζ117114ζ118113ζ111011ζ116115ζ119112118113119112111011116115117114ζ43ζ11743ζ114ζ4ζ1174ζ114ζ4ζ1184ζ113ζ4ζ11104ζ11ζ4ζ1164ζ115ζ4ζ1194ζ112ζ43ζ11843ζ113ζ43ζ111043ζ11ζ43ζ11643ζ115ζ43ζ11943ζ112    complex faithful
ρ212-2002i-2i00ζ116115ζ111011ζ117114ζ119112ζ118113111011118113117114119112116115ζ4ζ1164ζ115ζ43ζ11643ζ115ζ43ζ111043ζ11ζ43ζ11743ζ114ζ43ζ11943ζ112ζ43ζ11843ζ113ζ4ζ11104ζ11ζ4ζ1174ζ114ζ4ζ1194ζ112ζ4ζ1184ζ113    complex faithful
ρ222-200-2i2i00ζ116115ζ111011ζ117114ζ119112ζ118113111011118113117114119112116115ζ43ζ11643ζ115ζ4ζ1164ζ115ζ4ζ11104ζ11ζ4ζ1174ζ114ζ4ζ1194ζ112ζ4ζ1184ζ113ζ43ζ111043ζ11ζ43ζ11743ζ114ζ43ζ11943ζ112ζ43ζ11843ζ113    complex faithful
ρ232-2002i-2i00ζ117114ζ118113ζ111011ζ116115ζ119112118113119112111011116115117114ζ4ζ1174ζ114ζ43ζ11743ζ114ζ43ζ11843ζ113ζ43ζ111043ζ11ζ43ζ11643ζ115ζ43ζ11943ζ112ζ4ζ1184ζ113ζ4ζ11104ζ11ζ4ζ1164ζ115ζ4ζ1194ζ112    complex faithful
ρ242-200-2i2i00ζ118113ζ116115ζ119112ζ111011ζ117114116115117114119112111011118113ζ43ζ11843ζ113ζ4ζ1184ζ113ζ4ζ1164ζ115ζ4ζ1194ζ112ζ4ζ11104ζ11ζ4ζ1174ζ114ζ43ζ11643ζ115ζ43ζ11943ζ112ζ43ζ111043ζ11ζ43ζ11743ζ114    complex faithful
ρ252-2002i-2i00ζ118113ζ116115ζ119112ζ111011ζ117114116115117114119112111011118113ζ4ζ1184ζ113ζ43ζ11843ζ113ζ43ζ11643ζ115ζ43ζ11943ζ112ζ43ζ111043ζ11ζ43ζ11743ζ114ζ4ζ1164ζ115ζ4ζ1194ζ112ζ4ζ11104ζ11ζ4ζ1174ζ114    complex faithful
ρ262-200-2i2i00ζ111011ζ119112ζ118113ζ117114ζ116115119112116115118113117114111011ζ43ζ111043ζ11ζ4ζ11104ζ11ζ4ζ1194ζ112ζ4ζ1184ζ113ζ4ζ1174ζ114ζ4ζ1164ζ115ζ43ζ11943ζ112ζ43ζ11843ζ113ζ43ζ11743ζ114ζ43ζ11643ζ115    complex faithful
ρ272-2002i-2i00ζ111011ζ119112ζ118113ζ117114ζ116115119112116115118113117114111011ζ4ζ11104ζ11ζ43ζ111043ζ11ζ43ζ11943ζ112ζ43ζ11843ζ113ζ43ζ11743ζ114ζ43ζ11643ζ115ζ4ζ1194ζ112ζ4ζ1184ζ113ζ4ζ1174ζ114ζ4ζ1164ζ115    complex faithful
ρ282-2002i-2i00ζ119112ζ117114ζ116115ζ118113ζ111011117114111011116115118113119112ζ4ζ1194ζ112ζ43ζ11943ζ112ζ43ζ11743ζ114ζ43ζ11643ζ115ζ43ζ11843ζ113ζ43ζ111043ζ11ζ4ζ1174ζ114ζ4ζ1164ζ115ζ4ζ1184ζ113ζ4ζ11104ζ11    complex faithful

Smallest permutation representation of C4×D11
On 44 points
Generators in S44
(1 43 21 32)(2 44 22 33)(3 34 12 23)(4 35 13 24)(5 36 14 25)(6 37 15 26)(7 38 16 27)(8 39 17 28)(9 40 18 29)(10 41 19 30)(11 42 20 31)
(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)| (1,43,21,32)(2,44,22,33)(3,34,12,23)(4,35,13,24)(5,36,14,25)(6,37,15,26)(7,38,16,27)(8,39,17,28)(9,40,18,29)(10,41,19,30)(11,42,20,31), (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( (1,43,21,32)(2,44,22,33)(3,34,12,23)(4,35,13,24)(5,36,14,25)(6,37,15,26)(7,38,16,27)(8,39,17,28)(9,40,18,29)(10,41,19,30)(11,42,20,31), (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([(1,43,21,32),(2,44,22,33),(3,34,12,23),(4,35,13,24),(5,36,14,25),(6,37,15,26),(7,38,16,27),(8,39,17,28),(9,40,18,29),(10,41,19,30),(11,42,20,31)], [(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)])

C4×D11 is a maximal subgroup of   C88⋊C2  D445C2  D42D11  D44⋊C2  D33⋊C4  D552C4
C4×D11 is a maximal quotient of   C88⋊C2  Dic11⋊C4  D22⋊C4  D33⋊C4  D552C4

Matrix representation of C4×D11 in GL3(𝔽89) generated by

3400
0880
0088
,
100
001
08871
,
8800
001
010
G:=sub<GL(3,GF(89))| [34,0,0,0,88,0,0,0,88],[1,0,0,0,0,88,0,1,71],[88,0,0,0,0,1,0,1,0] >;

C4×D11 in GAP, Magma, Sage, TeX

C_4\times D_{11}
% in TeX

G:=Group("C4xD11");
// GroupNames label

G:=SmallGroup(88,4);
// by ID

G=gap.SmallGroup(88,4);
# by ID

G:=PCGroup([4,-2,-2,-2,-11,21,1283]);
// Polycyclic

G:=Group<a,b,c|a^4=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×D11 in TeX
Character table of C4×D11 in TeX

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