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G = D38order 76 = 22·19

Dihedral group

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

Aliases: D38, C2×D19, C38⋊C2, C19⋊C22, sometimes denoted D76 or Dih38 or Dih76, SmallGroup(76,3)

Series: Derived Chief Lower central Upper central

C1C19 — D38
C1C19D19 — D38
C19 — D38
C1C2

Generators and relations for D38
 G = < a,b | a38=b2=1, bab=a-1 >

19C2
19C2
19C22

Character table of D38

 class 12A2B2C19A19B19C19D19E19F19G19H19I38A38B38C38D38E38F38G38H38I
 size 111919222222222222222222
ρ11111111111111111111111    trivial
ρ21-11-1111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311-1-1111111111111111111    linear of order 2
ρ41-1-11111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ52-200ζ1914195ζ1911198ζ1917192ζ1915194ζ1910199ζ1916193ζ1913196ζ1912197ζ19181919181919141951911198191719219151941910199191619319131961912197    orthogonal faithful
ρ62200ζ1913196ζ1917192ζ1910199ζ191819ζ1912197ζ1915194ζ1911198ζ1916193ζ1914195ζ1914195ζ1913196ζ1917192ζ1910199ζ191819ζ1912197ζ1915194ζ1911198ζ1916193    orthogonal lifted from D19
ρ72200ζ1914195ζ1911198ζ1917192ζ1915194ζ1910199ζ1916193ζ1913196ζ1912197ζ191819ζ191819ζ1914195ζ1911198ζ1917192ζ1915194ζ1910199ζ1916193ζ1913196ζ1912197    orthogonal lifted from D19
ρ82200ζ1911198ζ1910199ζ1912197ζ1914195ζ1916193ζ191819ζ1917192ζ1915194ζ1913196ζ1913196ζ1911198ζ1910199ζ1912197ζ1914195ζ1916193ζ191819ζ1917192ζ1915194    orthogonal lifted from D19
ρ92-200ζ1913196ζ1917192ζ1910199ζ191819ζ1912197ζ1915194ζ1911198ζ1916193ζ191419519141951913196191719219101991918191912197191519419111981916193    orthogonal faithful
ρ102-200ζ1910199ζ1916193ζ1915194ζ1911198ζ191819ζ1913196ζ1912197ζ1914195ζ191719219171921910199191619319151941911198191819191319619121971914195    orthogonal faithful
ρ112200ζ191819ζ1913196ζ1911198ζ1916193ζ1917192ζ1912197ζ1914195ζ1910199ζ1915194ζ1915194ζ191819ζ1913196ζ1911198ζ1916193ζ1917192ζ1912197ζ1914195ζ1910199    orthogonal lifted from D19
ρ122200ζ1916193ζ191819ζ1914195ζ1910199ζ1913196ζ1917192ζ1915194ζ1911198ζ1912197ζ1912197ζ1916193ζ191819ζ1914195ζ1910199ζ1913196ζ1917192ζ1915194ζ1911198    orthogonal lifted from D19
ρ132-200ζ1911198ζ1910199ζ1912197ζ1914195ζ1916193ζ191819ζ1917192ζ1915194ζ191319619131961911198191019919121971914195191619319181919171921915194    orthogonal faithful
ρ142-200ζ191819ζ1913196ζ1911198ζ1916193ζ1917192ζ1912197ζ1914195ζ1910199ζ191519419151941918191913196191119819161931917192191219719141951910199    orthogonal faithful
ρ152-200ζ1912197ζ1915194ζ191819ζ1917192ζ1914195ζ1911198ζ1916193ζ1913196ζ191019919101991912197191519419181919171921914195191119819161931913196    orthogonal faithful
ρ162200ζ1915194ζ1914195ζ1913196ζ1912197ζ1911198ζ1910199ζ191819ζ1917192ζ1916193ζ1916193ζ1915194ζ1914195ζ1913196ζ1912197ζ1911198ζ1910199ζ191819ζ1917192    orthogonal lifted from D19
ρ172-200ζ1915194ζ1914195ζ1913196ζ1912197ζ1911198ζ1910199ζ191819ζ1917192ζ191619319161931915194191419519131961912197191119819101991918191917192    orthogonal faithful
ρ182-200ζ1917192ζ1912197ζ1916193ζ1913196ζ1915194ζ1914195ζ1910199ζ191819ζ191119819111981917192191219719161931913196191519419141951910199191819    orthogonal faithful
ρ192-200ζ1916193ζ191819ζ1914195ζ1910199ζ1913196ζ1917192ζ1915194ζ1911198ζ191219719121971916193191819191419519101991913196191719219151941911198    orthogonal faithful
ρ202200ζ1917192ζ1912197ζ1916193ζ1913196ζ1915194ζ1914195ζ1910199ζ191819ζ1911198ζ1911198ζ1917192ζ1912197ζ1916193ζ1913196ζ1915194ζ1914195ζ1910199ζ191819    orthogonal lifted from D19
ρ212200ζ1910199ζ1916193ζ1915194ζ1911198ζ191819ζ1913196ζ1912197ζ1914195ζ1917192ζ1917192ζ1910199ζ1916193ζ1915194ζ1911198ζ191819ζ1913196ζ1912197ζ1914195    orthogonal lifted from D19
ρ222200ζ1912197ζ1915194ζ191819ζ1917192ζ1914195ζ1911198ζ1916193ζ1913196ζ1910199ζ1910199ζ1912197ζ1915194ζ191819ζ1917192ζ1914195ζ1911198ζ1916193ζ1913196    orthogonal lifted from D19

Smallest permutation representation of D38
On 38 points
Generators in S38
(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)
(1 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 20)

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

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

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

D38 is a maximal subgroup of   D76  C19⋊D4
D38 is a maximal quotient of   Dic38  D76  C19⋊D4

Matrix representation of D38 in GL2(𝔽37) generated by

036
111
,
119
3626
G:=sub<GL(2,GF(37))| [0,1,36,11],[11,36,9,26] >;

D38 in GAP, Magma, Sage, TeX

D_{38}
% in TeX

G:=Group("D38");
// GroupNames label

G:=SmallGroup(76,3);
// by ID

G=gap.SmallGroup(76,3);
# by ID

G:=PCGroup([3,-2,-2,-19,650]);
// Polycyclic

G:=Group<a,b|a^38=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D38 in TeX
Character table of D38 in TeX

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