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## G = C22×D11order 88 = 23·11

### Direct product of C22 and D11

Aliases: C22×D11, C11⋊C23, C22⋊C22, (C2×C22)⋊3C2, SmallGroup(88,11)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×D11
G = < a,b,c,d | a2=b2=c11=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Character table of C22×D11

 class 1 2A 2B 2C 2D 2E 2F 2G 11A 11B 11C 11D 11E 22A 22B 22C 22D 22E 22F 22G 22H 22I 22J 22K 22L 22M 22N 22O size 1 1 1 1 11 11 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 -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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 -2 2 -2 0 0 0 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 -ζ118-ζ113 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 -ζ116-ζ115 -ζ117-ζ114 -ζ118-ζ113 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 -ζ119-ζ112 -ζ1110-ζ11 ζ117+ζ114 -ζ117-ζ114 orthogonal lifted from D22 ρ10 2 2 -2 -2 0 0 0 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ116+ζ115 -ζ116-ζ115 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 ζ1110+ζ11 -ζ118-ζ113 -ζ116-ζ115 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 ζ117+ζ114 ζ119+ζ112 -ζ118-ζ113 ζ118+ζ113 orthogonal lifted from D22 ρ11 2 -2 2 -2 0 0 0 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 -ζ117-ζ114 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 -ζ118-ζ113 -ζ119-ζ112 -ζ117-ζ114 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 -ζ1110-ζ11 -ζ116-ζ115 ζ119+ζ112 -ζ119-ζ112 orthogonal lifted from D22 ρ12 2 -2 -2 2 0 0 0 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 -ζ117-ζ114 -ζ117-ζ114 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 -ζ118-ζ113 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 -ζ119-ζ112 orthogonal lifted from D22 ρ13 2 -2 2 -2 0 0 0 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 -ζ1110-ζ11 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 -ζ119-ζ112 -ζ116-ζ115 -ζ1110-ζ11 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 -ζ118-ζ113 -ζ117-ζ114 ζ116+ζ115 -ζ116-ζ115 orthogonal lifted from D22 ρ14 2 2 2 2 0 0 0 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ1110+ζ11 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ119+ζ112 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ116+ζ115 orthogonal lifted from D11 ρ15 2 -2 -2 2 0 0 0 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 -ζ116-ζ115 -ζ116-ζ115 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 -ζ1110-ζ11 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 -ζ118-ζ113 orthogonal lifted from D22 ρ16 2 2 -2 -2 0 0 0 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ118+ζ113 -ζ118-ζ113 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 ζ116+ζ115 -ζ117-ζ114 -ζ118-ζ113 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 ζ119+ζ112 ζ1110+ζ11 -ζ117-ζ114 ζ117+ζ114 orthogonal lifted from D22 ρ17 2 -2 -2 2 0 0 0 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 -ζ119-ζ112 -ζ119-ζ112 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 -ζ117-ζ114 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 -ζ1110-ζ11 orthogonal lifted from D22 ρ18 2 2 -2 -2 0 0 0 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ119+ζ112 -ζ119-ζ112 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 ζ117+ζ114 -ζ1110-ζ11 -ζ119-ζ112 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 ζ116+ζ115 ζ118+ζ113 -ζ1110-ζ11 ζ1110+ζ11 orthogonal lifted from D22 ρ19 2 2 -2 -2 0 0 0 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ1110+ζ11 -ζ1110-ζ11 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 ζ119+ζ112 -ζ116-ζ115 -ζ1110-ζ11 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 ζ118+ζ113 ζ117+ζ114 -ζ116-ζ115 ζ116+ζ115 orthogonal lifted from D22 ρ20 2 -2 -2 2 0 0 0 0 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 -ζ1110-ζ11 -ζ1110-ζ11 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 -ζ119-ζ112 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 -ζ116-ζ115 orthogonal lifted from D22 ρ21 2 2 2 2 0 0 0 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ117+ζ114 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ118+ζ113 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ119+ζ112 orthogonal lifted from D11 ρ22 2 2 -2 -2 0 0 0 0 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ117+ζ114 -ζ117-ζ114 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 ζ118+ζ113 -ζ119-ζ112 -ζ117-ζ114 -ζ116-ζ115 -ζ118-ζ113 -ζ1110-ζ11 ζ1110+ζ11 ζ116+ζ115 -ζ119-ζ112 ζ119+ζ112 orthogonal lifted from D22 ρ23 2 2 2 2 0 0 0 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ118+ζ113 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ116+ζ115 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ117+ζ114 orthogonal lifted from D11 ρ24 2 -2 2 -2 0 0 0 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 -ζ116-ζ115 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 -ζ1110-ζ11 -ζ118-ζ113 -ζ116-ζ115 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 -ζ117-ζ114 -ζ119-ζ112 ζ118+ζ113 -ζ118-ζ113 orthogonal lifted from D22 ρ25 2 -2 2 -2 0 0 0 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 -ζ119-ζ112 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 -ζ117-ζ114 -ζ1110-ζ11 -ζ119-ζ112 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 -ζ116-ζ115 -ζ118-ζ113 ζ1110+ζ11 -ζ1110-ζ11 orthogonal lifted from D22 ρ26 2 2 2 2 0 0 0 0 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ116+ζ115 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ1110+ζ11 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ118+ζ113 orthogonal lifted from D11 ρ27 2 -2 -2 2 0 0 0 0 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 -ζ118-ζ113 -ζ118-ζ113 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 -ζ116-ζ115 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 -ζ117-ζ114 orthogonal lifted from D22 ρ28 2 2 2 2 0 0 0 0 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ119+ζ112 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ117+ζ114 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ1110+ζ11 orthogonal lifted from D11

Smallest permutation representation of C22×D11
On 44 points
Generators in S44
(1 43)(2 44)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)
(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 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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 44)(11 43)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 33)(20 32)(21 31)(22 30)

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

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

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

C22×D11 is a maximal subgroup of   D22⋊C4
C22×D11 is a maximal quotient of   D445C2  D42D11  D44⋊C2

Matrix representation of C22×D11 in GL3(𝔽23) generated by

 1 0 0 0 22 0 0 0 22
,
 22 0 0 0 1 0 0 0 1
,
 1 0 0 0 0 1 0 22 4
,
 22 0 0 0 0 22 0 22 0
G:=sub<GL(3,GF(23))| [1,0,0,0,22,0,0,0,22],[22,0,0,0,1,0,0,0,1],[1,0,0,0,0,22,0,1,4],[22,0,0,0,0,22,0,22,0] >;

C22×D11 in GAP, Magma, Sage, TeX

C_2^2\times D_{11}
% in TeX

G:=Group("C2^2xD11");
// GroupNames label

G:=SmallGroup(88,11);
// by ID

G=gap.SmallGroup(88,11);
# by ID

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

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

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