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G = C42⋊C9order 144 = 24·32

The semidirect product of C42 and C9 acting via C9/C3=C3

metabelian, soluble, monomial, A-group

Aliases: C42⋊C9, (C4×C12).C3, (C2×C6).1A4, C3.(C42⋊C3), C22.(C3.A4), SmallGroup(144,3)

Series: Derived Chief Lower central Upper central

C1C42 — C42⋊C9
C1C22C42C4×C12 — C42⋊C9
C42 — C42⋊C9
C1C3

Generators and relations for C42⋊C9
 G = < a,b,c | a4=b4=c9=1, ab=ba, cac-1=ab-1, cbc-1=a-1b2 >

3C2
3C4
3C4
3C6
16C9
3C2×C4
3C12
3C12
3C2×C12
4C3.A4

Character table of C42⋊C9

 class 123A3B4A4B4C4D6A6B9A9B9C9D9E9F12A12B12C12D12E12F12G12H
 size 131133333316161616161633333333
ρ1111111111111111111111111    trivial
ρ21111111111ζ32ζ3ζ3ζ32ζ32ζ311111111    linear of order 3
ρ31111111111ζ3ζ32ζ32ζ3ζ3ζ3211111111    linear of order 3
ρ411ζ3ζ321111ζ3ζ32ζ9ζ95ζ92ζ97ζ94ζ98ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 9
ρ511ζ32ζ31111ζ32ζ3ζ98ζ94ζ97ζ92ζ95ζ9ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 9
ρ611ζ32ζ31111ζ32ζ3ζ92ζ9ζ94ζ95ζ98ζ97ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 9
ρ711ζ3ζ321111ζ3ζ32ζ94ζ92ζ98ζ9ζ97ζ95ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 9
ρ811ζ32ζ31111ζ32ζ3ζ95ζ97ζ9ζ98ζ92ζ94ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 9
ρ911ζ3ζ321111ζ3ζ32ζ97ζ98ζ95ζ94ζ9ζ92ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 9
ρ103333-1-1-1-133000000-1-1-1-1-1-1-1-1    orthogonal lifted from A4
ρ1133-3+3-3/2-3-3-3/2-1-1-1-1-3+3-3/2-3-3-3/2000000ζ65ζ65ζ6ζ6ζ6ζ6ζ65ζ65    complex lifted from C3.A4
ρ1233-3-3-3/2-3+3-3/2-1-1-1-1-3-3-3/2-3+3-3/2000000ζ6ζ6ζ65ζ65ζ65ζ65ζ6ζ6    complex lifted from C3.A4
ρ133-133-1-2i11-1+2i-1-1000000-1-2i1-1-2i11-1+2i1-1+2i    complex lifted from C42⋊C3
ρ143-1331-1+2i-1-2i1-1-10000001-1+2i1-1+2i-1-2i1-1-2i1    complex lifted from C42⋊C3
ρ153-1331-1-2i-1+2i1-1-10000001-1-2i1-1-2i-1+2i1-1+2i1    complex lifted from C42⋊C3
ρ163-133-1+2i11-1-2i-1-1000000-1+2i1-1+2i11-1-2i1-1-2i    complex lifted from C42⋊C3
ρ173-1-3-3-3/2-3+3-3/2-1-2i11-1+2iζ6ζ6500000043ζ3232ζ3243ζ33ζ3ζ34ζ33ζ324ζ3232    complex faithful
ρ183-1-3-3-3/2-3+3-3/21-1+2i-1-2i1ζ6ζ65000000ζ324ζ3232ζ34ζ3343ζ33ζ343ζ3232ζ32    complex faithful
ρ193-1-3+3-3/2-3-3-3/21-1-2i-1+2i1ζ65ζ6000000ζ343ζ33ζ3243ζ32324ζ3232ζ324ζ33ζ3    complex faithful
ρ203-1-3+3-3/2-3-3-3/2-1-2i11-1+2iζ65ζ600000043ζ33ζ343ζ3232ζ32ζ324ζ3232ζ34ζ33    complex faithful
ρ213-1-3-3-3/2-3+3-3/21-1-2i-1+2i1ζ6ζ65000000ζ3243ζ3232ζ343ζ334ζ33ζ34ζ3232ζ32    complex faithful
ρ223-1-3+3-3/2-3-3-3/2-1+2i11-1-2iζ65ζ60000004ζ33ζ34ζ3232ζ32ζ3243ζ3232ζ343ζ33    complex faithful
ρ233-1-3+3-3/2-3-3-3/21-1+2i-1-2i1ζ65ζ6000000ζ34ζ33ζ324ζ323243ζ3232ζ3243ζ33ζ3    complex faithful
ρ243-1-3-3-3/2-3+3-3/2-1+2i11-1-2iζ6ζ650000004ζ3232ζ324ζ33ζ3ζ343ζ33ζ3243ζ3232    complex faithful

Smallest permutation representation of C42⋊C9
On 36 points
Generators in S36
(2 11 26 33)(3 34 27 12)(5 14 20 36)(6 28 21 15)(8 17 23 30)(9 31 24 18)
(1 32 25 10)(2 26)(3 34 27 12)(4 35 19 13)(5 20)(6 28 21 15)(7 29 22 16)(8 23)(9 31 24 18)(11 33)(14 36)(17 30)
(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)

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

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

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

C42⋊C9 is a maximal subgroup of   C42⋊D9  C42⋊C18  C422C18  C9×C42⋊C3  C42⋊3- 1+2  C122.C3
C42⋊C9 is a maximal quotient of   C2.(C42⋊C9)  C42⋊C27

Matrix representation of C42⋊C9 in GL3(𝔽13) generated by

500
010
008
,
500
050
0012
,
009
100
010
G:=sub<GL(3,GF(13))| [5,0,0,0,1,0,0,0,8],[5,0,0,0,5,0,0,0,12],[0,1,0,0,0,1,9,0,0] >;

C42⋊C9 in GAP, Magma, Sage, TeX

C_4^2\rtimes C_9
% in TeX

G:=Group("C4^2:C9");
// GroupNames label

G:=SmallGroup(144,3);
// by ID

G=gap.SmallGroup(144,3);
# by ID

G:=PCGroup([6,-3,-3,-2,2,-2,2,18,326,230,2379,69,2164,3893]);
// Polycyclic

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

Export

Subgroup lattice of C42⋊C9 in TeX
Character table of C42⋊C9 in TeX

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