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G = C22×D9order 72 = 23·32

Direct product of C22 and D9

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

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

C1C9 — C22×D9
C1C3C9D9D18 — C22×D9
C9 — C22×D9
C1C22

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 >

9C2
9C2
9C2
9C2
9C22
9C22
9C22
9C22
9C22
9C22
3S3
3S3
3S3
3S3
9C23
3D6
3D6
3D6
3D6
3D6
3D6
3C22×S3

Character table of C22×D9

 class 12A2B2C2D2E2F2G36A6B6C9A9B9C18A18B18C18D18E18F18G18H18I
 size 111199992222222222222222
ρ1111111111111111111111111    trivial
ρ21-1-111-1-111-11-1111-1-1111-1-1-1-1    linear of order 2
ρ31-11-1-1-1111-1-11111-11-1-1-111-1-1    linear of order 2
ρ411-1-1-11-1111-1-11111-1-1-1-1-1-111    linear of order 2
ρ511-1-11-11-111-1-11111-1-1-1-1-1-111    linear of order 2
ρ61-11-111-1-11-1-11111-11-1-1-111-1-1    linear of order 2
ρ71-1-11-111-11-11-1111-1-1111-1-1-1-1    linear of order 2
ρ81111-1-1-1-11111111111111111    linear of order 2
ρ922-2-2000022-2-2-1-1-1-1111111-1-1    orthogonal lifted from D6
ρ102-2-2200002-22-2-1-1-111-1-1-11111    orthogonal lifted from D6
ρ11222200002222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ122-22-200002-2-22-1-1-11-1111-1-111    orthogonal lifted from D6
ρ132-2-220000-11-11ζ9594ζ9792ζ9899899594ζ9594ζ989ζ9792989979297929594    orthogonal lifted from D18
ρ1422220000-1-1-1-1ζ989ζ9594ζ9792ζ9792ζ989ζ989ζ9792ζ9594ζ9792ζ9594ζ9594ζ989    orthogonal lifted from D9
ρ152-22-20000-111-1ζ9594ζ9792ζ989989ζ959495949899792ζ989ζ979297929594    orthogonal lifted from D18
ρ162-2-220000-11-11ζ989ζ9594ζ97929792989ζ989ζ9792ζ9594979295949594989    orthogonal lifted from D18
ρ172-22-20000-111-1ζ989ζ9594ζ97929792ζ98998997929594ζ9792ζ95949594989    orthogonal lifted from D18
ρ1822220000-1-1-1-1ζ9792ζ989ζ9594ζ9594ζ9792ζ9792ζ9594ζ989ζ9594ζ989ζ989ζ9792    orthogonal lifted from D9
ρ1922-2-20000-1-111ζ9792ζ989ζ9594ζ95949792979295949899594989ζ989ζ9792    orthogonal lifted from D18
ρ2022220000-1-1-1-1ζ9594ζ9792ζ989ζ989ζ9594ζ9594ζ989ζ9792ζ989ζ9792ζ9792ζ9594    orthogonal lifted from D9
ρ2122-2-20000-1-111ζ9594ζ9792ζ989ζ9899594959498997929899792ζ9792ζ9594    orthogonal lifted from D18
ρ2222-2-20000-1-111ζ989ζ9594ζ9792ζ97929899899792959497929594ζ9594ζ989    orthogonal lifted from D18
ρ232-22-20000-111-1ζ9792ζ989ζ95949594ζ979297929594989ζ9594ζ9899899792    orthogonal lifted from D18
ρ242-2-220000-11-11ζ9792ζ989ζ959495949792ζ9792ζ9594ζ98995949899899792    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

1800
010
001
,
100
0180
0018
,
100
0714
052
,
100
01417
0125
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

Export

Subgroup lattice of C22×D9 in TeX
Character table of C22×D9 in TeX

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