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## G = C22×D9order 72 = 23·32

### Direct product of C22 and D9

Aliases: C22×D9, C9⋊C23, C18⋊C22, C6.11D6, (C2×C18)⋊3C2, (C2×C6).4S3, C3.(C22×S3), SmallGroup(72,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C22×D9
 Chief series C1 — C3 — C9 — D9 — D18 — C22×D9
 Lower central C9 — C22×D9
 Upper central C1 — C22

Generators and relations for C22×D9
G = < a,b,c,d | a2=b2=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Character table of C22×D9

 class 1 2A 2B 2C 2D 2E 2F 2G 3 6A 6B 6C 9A 9B 9C 18A 18B 18C 18D 18E 18F 18G 18H 18I size 1 1 1 1 9 9 9 9 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 -2 -2 0 0 0 0 2 2 -2 -2 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ10 2 -2 -2 2 0 0 0 0 2 -2 2 -2 -1 -1 -1 1 1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ11 2 2 2 2 0 0 0 0 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 -2 2 -2 0 0 0 0 2 -2 -2 2 -1 -1 -1 1 -1 1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ13 2 -2 -2 2 0 0 0 0 -1 1 -1 1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ98-ζ9 -ζ95-ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 orthogonal lifted from D18 ρ14 2 2 2 2 0 0 0 0 -1 -1 -1 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ15 2 -2 2 -2 0 0 0 0 -1 1 1 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ98-ζ9 ζ95+ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 -ζ95-ζ94 orthogonal lifted from D18 ρ16 2 -2 -2 2 0 0 0 0 -1 1 -1 1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ97-ζ92 -ζ98-ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 orthogonal lifted from D18 ρ17 2 -2 2 -2 0 0 0 0 -1 1 1 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ97-ζ92 ζ98+ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 -ζ98-ζ9 orthogonal lifted from D18 ρ18 2 2 2 2 0 0 0 0 -1 -1 -1 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ19 2 2 -2 -2 0 0 0 0 -1 -1 1 1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ95-ζ94 -ζ98-ζ9 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D18 ρ20 2 2 2 2 0 0 0 0 -1 -1 -1 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ21 2 2 -2 -2 0 0 0 0 -1 -1 1 1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ98-ζ9 -ζ97-ζ92 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D18 ρ22 2 2 -2 -2 0 0 0 0 -1 -1 1 1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ97-ζ92 -ζ95-ζ94 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D18 ρ23 2 -2 2 -2 0 0 0 0 -1 1 1 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ95-ζ94 ζ97+ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 -ζ97-ζ92 orthogonal lifted from D18 ρ24 2 -2 -2 2 0 0 0 0 -1 1 -1 1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ95-ζ94 -ζ97-ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 orthogonal lifted from D18

Smallest permutation representation of C22×D9
On 36 points
Generators in S36
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)
(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)
(1 31)(2 30)(3 29)(4 28)(5 36)(6 35)(7 34)(8 33)(9 32)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(18 27)

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

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

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

C22×D9 is a maximal subgroup of   D18⋊C4  D9⋊A4
C22×D9 is a maximal quotient of   D365C2  D42D9  Q83D9

Matrix representation of C22×D9 in GL3(𝔽19) generated by

 18 0 0 0 1 0 0 0 1
,
 1 0 0 0 18 0 0 0 18
,
 1 0 0 0 7 14 0 5 2
,
 1 0 0 0 14 17 0 12 5
G:=sub<GL(3,GF(19))| [18,0,0,0,1,0,0,0,1],[1,0,0,0,18,0,0,0,18],[1,0,0,0,7,5,0,14,2],[1,0,0,0,14,12,0,17,5] >;

C22×D9 in GAP, Magma, Sage, TeX

C_2^2\times D_9
% in TeX

G:=Group("C2^2xD9");
// GroupNames label

G:=SmallGroup(72,17);
// by ID

G=gap.SmallGroup(72,17);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,803,138,1204]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^9=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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