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## G = C4×D11order 88 = 23·11

### Direct product of C4 and D11

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

 Derived series C1 — C11 — C4×D11
 Chief series C1 — C11 — C22 — D22 — C4×D11
 Lower central C11 — C4×D11
 Upper central C1 — C4

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

Character table of C4×D11

 class 1 2A 2B 2C 4A 4B 4C 4D 11A 11B 11C 11D 11E 22A 22B 22C 22D 22E 44A 44B 44C 44D 44E 44F 44G 44H 44I 44J size 1 1 11 11 1 1 11 11 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 -1 -1 1 i -i -i i 1 1 1 1 1 -1 -1 -1 -1 -1 i -i -i -i -i -i i i i i linear of order 4 ρ6 1 -1 1 -1 -i i -i i 1 1 1 1 1 -1 -1 -1 -1 -1 -i i i i i i -i -i -i -i linear of order 4 ρ7 1 -1 1 -1 i -i i -i 1 1 1 1 1 -1 -1 -1 -1 -1 i -i -i -i -i -i i i i i linear of order 4 ρ8 1 -1 -1 1 -i i i -i 1 1 1 1 1 -1 -1 -1 -1 -1 -i i i i i i -i -i -i -i linear of order 4 ρ9 2 2 0 0 2 2 0 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ116+ζ115 ζ117+ζ114 ζ119+ζ112 ζ1110+ζ11 ζ118+ζ113 ζ118+ζ113 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 orthogonal lifted from D11 ρ10 2 2 0 0 -2 -2 0 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ118+ζ113 ζ119+ζ112 ζ1110+ζ11 ζ116+ζ115 ζ117+ζ114 -ζ117-ζ114 -ζ117-ζ114 -ζ118-ζ113 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 -ζ118-ζ113 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 orthogonal lifted from D22 ρ11 2 2 0 0 -2 -2 0 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ119+ζ112 ζ116+ζ115 ζ118+ζ113 ζ117+ζ114 ζ1110+ζ11 -ζ1110-ζ11 -ζ1110-ζ11 -ζ119-ζ112 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 -ζ119-ζ112 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 orthogonal lifted from D22 ρ12 2 2 0 0 -2 -2 0 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ1110+ζ11 ζ118+ζ113 ζ117+ζ114 ζ119+ζ112 ζ116+ζ115 -ζ116-ζ115 -ζ116-ζ115 -ζ1110-ζ11 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 -ζ1110-ζ11 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 orthogonal lifted from D22 ρ13 2 2 0 0 2 2 0 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ119+ζ112 ζ116+ζ115 ζ118+ζ113 ζ117+ζ114 ζ1110+ζ11 ζ1110+ζ11 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 orthogonal lifted from D11 ρ14 2 2 0 0 2 2 0 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ117+ζ114 ζ1110+ζ11 ζ116+ζ115 ζ118+ζ113 ζ119+ζ112 ζ119+ζ112 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 orthogonal lifted from D11 ρ15 2 2 0 0 2 2 0 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ118+ζ113 ζ119+ζ112 ζ1110+ζ11 ζ116+ζ115 ζ117+ζ114 ζ117+ζ114 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 orthogonal lifted from D11 ρ16 2 2 0 0 -2 -2 0 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ116+ζ115 ζ117+ζ114 ζ119+ζ112 ζ1110+ζ11 ζ118+ζ113 -ζ118-ζ113 -ζ118-ζ113 -ζ116-ζ115 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 -ζ116-ζ115 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 orthogonal lifted from D22 ρ17 2 2 0 0 2 2 0 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ1110+ζ11 ζ118+ζ113 ζ117+ζ114 ζ119+ζ112 ζ116+ζ115 ζ116+ζ115 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 orthogonal lifted from D11 ρ18 2 2 0 0 -2 -2 0 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ117+ζ114 ζ1110+ζ11 ζ116+ζ115 ζ118+ζ113 ζ119+ζ112 -ζ119-ζ112 -ζ119-ζ112 -ζ117-ζ114 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 -ζ117-ζ114 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 orthogonal lifted from D22 ρ19 2 -2 0 0 -2i 2i 0 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 -ζ117-ζ114 -ζ1110-ζ11 -ζ116-ζ115 -ζ118-ζ113 -ζ119-ζ112 ζ43ζ119+ζ43ζ112 ζ4ζ119+ζ4ζ112 ζ4ζ117+ζ4ζ114 ζ4ζ116+ζ4ζ115 ζ4ζ118+ζ4ζ113 ζ4ζ1110+ζ4ζ11 ζ43ζ117+ζ43ζ114 ζ43ζ116+ζ43ζ115 ζ43ζ118+ζ43ζ113 ζ43ζ1110+ζ43ζ11 complex faithful ρ20 2 -2 0 0 -2i 2i 0 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 -ζ118-ζ113 -ζ119-ζ112 -ζ1110-ζ11 -ζ116-ζ115 -ζ117-ζ114 ζ43ζ117+ζ43ζ114 ζ4ζ117+ζ4ζ114 ζ4ζ118+ζ4ζ113 ζ4ζ1110+ζ4ζ11 ζ4ζ116+ζ4ζ115 ζ4ζ119+ζ4ζ112 ζ43ζ118+ζ43ζ113 ζ43ζ1110+ζ43ζ11 ζ43ζ116+ζ43ζ115 ζ43ζ119+ζ43ζ112 complex faithful ρ21 2 -2 0 0 2i -2i 0 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 -ζ1110-ζ11 -ζ118-ζ113 -ζ117-ζ114 -ζ119-ζ112 -ζ116-ζ115 ζ4ζ116+ζ4ζ115 ζ43ζ116+ζ43ζ115 ζ43ζ1110+ζ43ζ11 ζ43ζ117+ζ43ζ114 ζ43ζ119+ζ43ζ112 ζ43ζ118+ζ43ζ113 ζ4ζ1110+ζ4ζ11 ζ4ζ117+ζ4ζ114 ζ4ζ119+ζ4ζ112 ζ4ζ118+ζ4ζ113 complex faithful ρ22 2 -2 0 0 -2i 2i 0 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 -ζ1110-ζ11 -ζ118-ζ113 -ζ117-ζ114 -ζ119-ζ112 -ζ116-ζ115 ζ43ζ116+ζ43ζ115 ζ4ζ116+ζ4ζ115 ζ4ζ1110+ζ4ζ11 ζ4ζ117+ζ4ζ114 ζ4ζ119+ζ4ζ112 ζ4ζ118+ζ4ζ113 ζ43ζ1110+ζ43ζ11 ζ43ζ117+ζ43ζ114 ζ43ζ119+ζ43ζ112 ζ43ζ118+ζ43ζ113 complex faithful ρ23 2 -2 0 0 2i -2i 0 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 -ζ118-ζ113 -ζ119-ζ112 -ζ1110-ζ11 -ζ116-ζ115 -ζ117-ζ114 ζ4ζ117+ζ4ζ114 ζ43ζ117+ζ43ζ114 ζ43ζ118+ζ43ζ113 ζ43ζ1110+ζ43ζ11 ζ43ζ116+ζ43ζ115 ζ43ζ119+ζ43ζ112 ζ4ζ118+ζ4ζ113 ζ4ζ1110+ζ4ζ11 ζ4ζ116+ζ4ζ115 ζ4ζ119+ζ4ζ112 complex faithful ρ24 2 -2 0 0 -2i 2i 0 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 -ζ116-ζ115 -ζ117-ζ114 -ζ119-ζ112 -ζ1110-ζ11 -ζ118-ζ113 ζ43ζ118+ζ43ζ113 ζ4ζ118+ζ4ζ113 ζ4ζ116+ζ4ζ115 ζ4ζ119+ζ4ζ112 ζ4ζ1110+ζ4ζ11 ζ4ζ117+ζ4ζ114 ζ43ζ116+ζ43ζ115 ζ43ζ119+ζ43ζ112 ζ43ζ1110+ζ43ζ11 ζ43ζ117+ζ43ζ114 complex faithful ρ25 2 -2 0 0 2i -2i 0 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 -ζ116-ζ115 -ζ117-ζ114 -ζ119-ζ112 -ζ1110-ζ11 -ζ118-ζ113 ζ4ζ118+ζ4ζ113 ζ43ζ118+ζ43ζ113 ζ43ζ116+ζ43ζ115 ζ43ζ119+ζ43ζ112 ζ43ζ1110+ζ43ζ11 ζ43ζ117+ζ43ζ114 ζ4ζ116+ζ4ζ115 ζ4ζ119+ζ4ζ112 ζ4ζ1110+ζ4ζ11 ζ4ζ117+ζ4ζ114 complex faithful ρ26 2 -2 0 0 -2i 2i 0 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 -ζ119-ζ112 -ζ116-ζ115 -ζ118-ζ113 -ζ117-ζ114 -ζ1110-ζ11 ζ43ζ1110+ζ43ζ11 ζ4ζ1110+ζ4ζ11 ζ4ζ119+ζ4ζ112 ζ4ζ118+ζ4ζ113 ζ4ζ117+ζ4ζ114 ζ4ζ116+ζ4ζ115 ζ43ζ119+ζ43ζ112 ζ43ζ118+ζ43ζ113 ζ43ζ117+ζ43ζ114 ζ43ζ116+ζ43ζ115 complex faithful ρ27 2 -2 0 0 2i -2i 0 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 -ζ119-ζ112 -ζ116-ζ115 -ζ118-ζ113 -ζ117-ζ114 -ζ1110-ζ11 ζ4ζ1110+ζ4ζ11 ζ43ζ1110+ζ43ζ11 ζ43ζ119+ζ43ζ112 ζ43ζ118+ζ43ζ113 ζ43ζ117+ζ43ζ114 ζ43ζ116+ζ43ζ115 ζ4ζ119+ζ4ζ112 ζ4ζ118+ζ4ζ113 ζ4ζ117+ζ4ζ114 ζ4ζ116+ζ4ζ115 complex faithful ρ28 2 -2 0 0 2i -2i 0 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 -ζ117-ζ114 -ζ1110-ζ11 -ζ116-ζ115 -ζ118-ζ113 -ζ119-ζ112 ζ4ζ119+ζ4ζ112 ζ43ζ119+ζ43ζ112 ζ43ζ117+ζ43ζ114 ζ43ζ116+ζ43ζ115 ζ43ζ118+ζ43ζ113 ζ43ζ1110+ζ43ζ11 ζ4ζ117+ζ4ζ114 ζ4ζ116+ζ4ζ115 ζ4ζ118+ζ4ζ113 ζ4ζ1110+ζ4ζ11 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

 34 0 0 0 88 0 0 0 88
,
 1 0 0 0 0 1 0 88 71
,
 88 0 0 0 0 1 0 1 0
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

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