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G = C4×D9order 72 = 23·32

Direct product of C4 and D9

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

Aliases: C4×D9, C362C2, D18.C2, C4Dic9, C6.7D6, C12.5S3, C2.1D18, Dic92C2, C18.2C22, C91(C2×C4), C3.(C4×S3), SmallGroup(72,5)

Series: Derived Chief Lower central Upper central

C1C9 — C4×D9
C1C3C9C18D18 — C4×D9
C9 — C4×D9
C1C4

Generators and relations for C4×D9
 G = < a,b,c | a4=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
9C2
9C22
9C4
3S3
3S3
9C2×C4
3D6
3Dic3
3C4×S3

Character table of C4×D9

 class 12A2B2C34A4B4C4D69A9B9C12A12B18A18B18C36A36B36C36D36E36F
 size 119921199222222222222222
ρ1111111111111111111111111    trivial
ρ211-1-11-1-1111111-1-1111-1-1-1-1-1-1    linear of order 2
ρ311-1-1111-1-1111111111111111    linear of order 2
ρ411111-1-1-1-11111-1-1111-1-1-1-1-1-1    linear of order 2
ρ51-1-111i-ii-i-1111i-i-1-1-1ii-i-i-ii    linear of order 4
ρ61-11-11-iii-i-1111-ii-1-1-1-i-iiii-i    linear of order 4
ρ71-1-111-ii-ii-1111-ii-1-1-1-i-iiii-i    linear of order 4
ρ81-11-11i-i-ii-1111i-i-1-1-1ii-i-i-ii    linear of order 4
ρ92200222002-1-1-122-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1022002-2-2002-1-1-1-2-2-1-1-1111111    orthogonal lifted from D6
ρ112200-12200-1ζ9594ζ9792ζ989-1-1ζ9792ζ9594ζ989ζ989ζ9792ζ9792ζ9594ζ989ζ9594    orthogonal lifted from D9
ρ122200-1-2-200-1ζ9594ζ9792ζ98911ζ9792ζ9594ζ9899899792979295949899594    orthogonal lifted from D18
ρ132200-1-2-200-1ζ989ζ9594ζ979211ζ9594ζ989ζ97929792959495949899792989    orthogonal lifted from D18
ρ142200-12200-1ζ989ζ9594ζ9792-1-1ζ9594ζ989ζ9792ζ9792ζ9594ζ9594ζ989ζ9792ζ989    orthogonal lifted from D9
ρ152200-1-2-200-1ζ9792ζ989ζ959411ζ989ζ9792ζ95949594989989979295949792    orthogonal lifted from D18
ρ162200-12200-1ζ9792ζ989ζ9594-1-1ζ989ζ9792ζ9594ζ9594ζ989ζ989ζ9792ζ9594ζ9792    orthogonal lifted from D9
ρ172-2002-2i2i00-2-1-1-1-2i2i111ii-i-i-ii    complex lifted from C4×S3
ρ182-20022i-2i00-2-1-1-12i-2i111-i-iiii-i    complex lifted from C4×S3
ρ192-200-1-2i2i001ζ989ζ9594ζ9792i-i95949899792ζ43ζ9743ζ92ζ43ζ9543ζ94ζ4ζ954ζ94ζ4ζ984ζ9ζ4ζ974ζ92ζ43ζ9843ζ9    complex faithful
ρ202-200-1-2i2i001ζ9594ζ9792ζ989i-i97929594989ζ43ζ9843ζ9ζ43ζ9743ζ92ζ4ζ974ζ92ζ4ζ954ζ94ζ4ζ984ζ9ζ43ζ9543ζ94    complex faithful
ρ212-200-12i-2i001ζ9792ζ989ζ9594-ii98997929594ζ4ζ954ζ94ζ4ζ984ζ9ζ43ζ9843ζ9ζ43ζ9743ζ92ζ43ζ9543ζ94ζ4ζ974ζ92    complex faithful
ρ222-200-12i-2i001ζ989ζ9594ζ9792-ii95949899792ζ4ζ974ζ92ζ4ζ954ζ94ζ43ζ9543ζ94ζ43ζ9843ζ9ζ43ζ9743ζ92ζ4ζ984ζ9    complex faithful
ρ232-200-12i-2i001ζ9594ζ9792ζ989-ii97929594989ζ4ζ984ζ9ζ4ζ974ζ92ζ43ζ9743ζ92ζ43ζ9543ζ94ζ43ζ9843ζ9ζ4ζ954ζ94    complex faithful
ρ242-200-1-2i2i001ζ9792ζ989ζ9594i-i98997929594ζ43ζ9543ζ94ζ43ζ9843ζ9ζ4ζ984ζ9ζ4ζ974ζ92ζ4ζ954ζ94ζ43ζ9743ζ92    complex faithful

Smallest permutation representation of C4×D9
On 36 points
Generators in S36
(1 32 14 23)(2 33 15 24)(3 34 16 25)(4 35 17 26)(5 36 18 27)(6 28 10 19)(7 29 11 20)(8 30 12 21)(9 31 13 22)
(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 13)(2 12)(3 11)(4 10)(5 18)(6 17)(7 16)(8 15)(9 14)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)

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

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

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

C4×D9 is a maximal subgroup of   C8⋊D9  D365C2  D42D9  Q83D9  C18.D6  C12.11S4  D90.C2
C4×D9 is a maximal quotient of   C8⋊D9  Dic9⋊C4  D18⋊C4  C18.D6  D90.C2

Matrix representation of C4×D9 in GL2(𝔽17) generated by

130
013
,
04
414
,
315
414
G:=sub<GL(2,GF(17))| [13,0,0,13],[0,4,4,14],[3,4,15,14] >;

C4×D9 in GAP, Magma, Sage, TeX

C_4\times D_9
% in TeX

G:=Group("C4xD9");
// GroupNames label

G:=SmallGroup(72,5);
// by ID

G=gap.SmallGroup(72,5);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,26,803,138,1204]);
// Polycyclic

G:=Group<a,b,c|a^4=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×D9 in TeX
Character table of C4×D9 in TeX

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