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G = D44order 88 = 23·11

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D44, C4⋊D11, C111D4, C441C2, D221C2, C2.4D22, C22.3C22, sometimes denoted D88 or Dih44 or Dih88, SmallGroup(88,5)

Series: Derived Chief Lower central Upper central

C1C22 — D44
C1C11C22D22 — D44
C11C22 — D44
C1C2C4

Generators and relations for D44
 G = < a,b | a44=b2=1, bab=a-1 >

22C2
22C2
11C22
11C22
2D11
2D11
11D4

Character table of D44

 class 12A2B2C411A11B11C11D11E22A22B22C22D22E44A44B44C44D44E44F44G44H44I44J
 size 112222222222222222222222222
ρ11111111111111111111111111    trivial
ρ211-11-11111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311-1-1111111111111111111111    linear of order 2
ρ4111-1-11111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ52-200022222-2-2-2-2-20000000000    orthogonal lifted from D4
ρ622002ζ119112ζ116115ζ118113ζ111011ζ117114ζ119112ζ118113ζ116115ζ111011ζ117114ζ117114ζ119112ζ118113ζ118113ζ119112ζ117114ζ111011ζ116115ζ116115ζ111011    orthogonal lifted from D11
ρ722002ζ117114ζ111011ζ116115ζ119112ζ118113ζ117114ζ116115ζ111011ζ119112ζ118113ζ118113ζ117114ζ116115ζ116115ζ117114ζ118113ζ119112ζ111011ζ111011ζ119112    orthogonal lifted from D11
ρ822002ζ116115ζ117114ζ119112ζ118113ζ111011ζ116115ζ119112ζ117114ζ118113ζ111011ζ111011ζ116115ζ119112ζ119112ζ116115ζ111011ζ118113ζ117114ζ117114ζ118113    orthogonal lifted from D11
ρ92200-2ζ117114ζ111011ζ116115ζ119112ζ118113ζ117114ζ116115ζ111011ζ119112ζ118113118113117114116115116115117114118113119112111011111011119112    orthogonal lifted from D22
ρ102200-2ζ119112ζ116115ζ118113ζ111011ζ117114ζ119112ζ118113ζ116115ζ111011ζ117114117114119112118113118113119112117114111011116115116115111011    orthogonal lifted from D22
ρ1122002ζ118113ζ119112ζ111011ζ117114ζ116115ζ118113ζ111011ζ119112ζ117114ζ116115ζ116115ζ118113ζ111011ζ111011ζ118113ζ116115ζ117114ζ119112ζ119112ζ117114    orthogonal lifted from D11
ρ122200-2ζ111011ζ118113ζ117114ζ116115ζ119112ζ111011ζ117114ζ118113ζ116115ζ119112119112111011117114117114111011119112116115118113118113116115    orthogonal lifted from D22
ρ132200-2ζ118113ζ119112ζ111011ζ117114ζ116115ζ118113ζ111011ζ119112ζ117114ζ116115116115118113111011111011118113116115117114119112119112117114    orthogonal lifted from D22
ρ142200-2ζ116115ζ117114ζ119112ζ118113ζ111011ζ116115ζ119112ζ117114ζ118113ζ111011111011116115119112119112116115111011118113117114117114118113    orthogonal lifted from D22
ρ1522002ζ111011ζ118113ζ117114ζ116115ζ119112ζ111011ζ117114ζ118113ζ116115ζ119112ζ119112ζ111011ζ117114ζ117114ζ111011ζ119112ζ116115ζ118113ζ118113ζ116115    orthogonal lifted from D11
ρ162-2000ζ111011ζ118113ζ117114ζ116115ζ1191121110111171141181131161151191124ζ1194ζ112ζ4ζ11104ζ11ζ4ζ1174ζ1144ζ1174ζ1144ζ11104ζ11ζ4ζ1194ζ11243ζ11643ζ115ζ43ζ11843ζ11343ζ11843ζ113ζ43ζ11643ζ115    orthogonal faithful
ρ172-2000ζ111011ζ118113ζ117114ζ116115ζ119112111011117114118113116115119112ζ4ζ1194ζ1124ζ11104ζ114ζ1174ζ114ζ4ζ1174ζ114ζ4ζ11104ζ114ζ1194ζ112ζ43ζ11643ζ11543ζ11843ζ113ζ43ζ11843ζ11343ζ11643ζ115    orthogonal faithful
ρ182-2000ζ118113ζ119112ζ111011ζ117114ζ116115118113111011119112117114116115ζ43ζ11643ζ115ζ43ζ11843ζ1134ζ11104ζ11ζ4ζ11104ζ1143ζ11843ζ11343ζ11643ζ1154ζ1174ζ1144ζ1194ζ112ζ4ζ1194ζ112ζ4ζ1174ζ114    orthogonal faithful
ρ192-2000ζ116115ζ117114ζ119112ζ118113ζ1110111161151191121171141181131110114ζ11104ζ11ζ43ζ11643ζ115ζ4ζ1194ζ1124ζ1194ζ11243ζ11643ζ115ζ4ζ11104ζ11ζ43ζ11843ζ113ζ4ζ1174ζ1144ζ1174ζ11443ζ11843ζ113    orthogonal faithful
ρ202-2000ζ117114ζ111011ζ116115ζ119112ζ11811311711411611511101111911211811343ζ11843ζ113ζ4ζ1174ζ11443ζ11643ζ115ζ43ζ11643ζ1154ζ1174ζ114ζ43ζ11843ζ1134ζ1194ζ1124ζ11104ζ11ζ4ζ11104ζ11ζ4ζ1194ζ112    orthogonal faithful
ρ212-2000ζ119112ζ116115ζ118113ζ111011ζ117114119112118113116115111011117114ζ4ζ1174ζ1144ζ1194ζ11243ζ11843ζ113ζ43ζ11843ζ113ζ4ζ1194ζ1124ζ1174ζ114ζ4ζ11104ζ11ζ43ζ11643ζ11543ζ11643ζ1154ζ11104ζ11    orthogonal faithful
ρ222-2000ζ116115ζ117114ζ119112ζ118113ζ111011116115119112117114118113111011ζ4ζ11104ζ1143ζ11643ζ1154ζ1194ζ112ζ4ζ1194ζ112ζ43ζ11643ζ1154ζ11104ζ1143ζ11843ζ1134ζ1174ζ114ζ4ζ1174ζ114ζ43ζ11843ζ113    orthogonal faithful
ρ232-2000ζ117114ζ111011ζ116115ζ119112ζ118113117114116115111011119112118113ζ43ζ11843ζ1134ζ1174ζ114ζ43ζ11643ζ11543ζ11643ζ115ζ4ζ1174ζ11443ζ11843ζ113ζ4ζ1194ζ112ζ4ζ11104ζ114ζ11104ζ114ζ1194ζ112    orthogonal faithful
ρ242-2000ζ119112ζ116115ζ118113ζ111011ζ1171141191121181131161151110111171144ζ1174ζ114ζ4ζ1194ζ112ζ43ζ11843ζ11343ζ11843ζ1134ζ1194ζ112ζ4ζ1174ζ1144ζ11104ζ1143ζ11643ζ115ζ43ζ11643ζ115ζ4ζ11104ζ11    orthogonal faithful
ρ252-2000ζ118113ζ119112ζ111011ζ117114ζ11611511811311101111911211711411611543ζ11643ζ11543ζ11843ζ113ζ4ζ11104ζ114ζ11104ζ11ζ43ζ11843ζ113ζ43ζ11643ζ115ζ4ζ1174ζ114ζ4ζ1194ζ1124ζ1194ζ1124ζ1174ζ114    orthogonal faithful

Smallest permutation representation of D44
On 44 points
Generators in S44
(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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 24)(22 23)

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

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

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

D44 is a maximal subgroup of
C8⋊D11  D88  D4⋊D11  Q8⋊D11  D445C2  D4×D11  D44⋊C2  C3⋊D44  D132  D44⋊C5  C5⋊D44  D220
D44 is a maximal quotient of
C8⋊D11  D88  Dic44  C44⋊C4  D22⋊C4  C3⋊D44  D132  C5⋊D44  D220

Matrix representation of D44 in GL2(𝔽43) generated by

401
420
,
3829
145
G:=sub<GL(2,GF(43))| [40,42,1,0],[38,14,29,5] >;

D44 in GAP, Magma, Sage, TeX

D_{44}
% in TeX

G:=Group("D44");
// GroupNames label

G:=SmallGroup(88,5);
// by ID

G=gap.SmallGroup(88,5);
# by ID

G:=PCGroup([4,-2,-2,-2,-11,49,21,1283]);
// Polycyclic

G:=Group<a,b|a^44=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D44 in TeX
Character table of D44 in TeX

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