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## G = (C3×C6).9D12order 432 = 24·33

### 2nd non-split extension by C3×C6 of D12 acting via D12/C3=D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊S3 — (C3×C6).9D12
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C6×C3⋊S3 — C33⋊9(C2×C4) — (C3×C6).9D12
 Lower central C33 — C3×C3⋊S3 — (C3×C6).9D12
 Upper central C1 — C2

Generators and relations for (C3×C6).9D12
G = < a,b,c,d | a3=b6=c12=1, d2=b3, ab=ba, cac-1=b2, dad-1=a-1, cbc-1=ab3, bd=db, dcd-1=c-1 >

Subgroups: 592 in 96 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C32⋊C4, S3×C6, C2×C3⋊S3, C4⋊Dic3, C3×C3⋊S3, C32×C6, S3×Dic3, C6.D6, C2×C32⋊C4, C3×C3⋊Dic3, C3×C32⋊C4, C6×C3⋊S3, C3⋊S3.Q8, C339(C2×C4), C6×C32⋊C4, (C3×C6).9D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, Dic6, D12, C2×Dic3, C4⋊Dic3, S3≀C2, C3⋊S3.Q8, C322D12, (C3×C6).9D12

Character table of (C3×C6).9D12

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 9 9 2 4 4 8 8 18 18 18 18 18 18 2 4 4 8 8 18 18 18 18 18 18 36 36 36 36 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 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 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 -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 -1 -1 -1 1 -1 1 linear of order 2 ρ5 1 -1 -1 1 1 1 1 1 1 -i -1 1 -i i i -1 -1 -1 -1 -1 -1 1 -1 -1 1 1 -i i i -i linear of order 4 ρ6 1 -1 -1 1 1 1 1 1 1 i -1 1 i -i -i -1 -1 -1 -1 -1 -1 1 -1 -1 1 1 i -i -i i linear of order 4 ρ7 1 -1 -1 1 1 1 1 1 1 i 1 -1 -i i -i -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 -i -i i i linear of order 4 ρ8 1 -1 -1 1 1 1 1 1 1 -i 1 -1 i -i i -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 i i -i -i linear of order 4 ρ9 2 2 2 2 -1 2 2 -1 -1 0 -2 -2 0 0 0 -1 2 2 -1 -1 -1 -1 1 1 1 1 0 0 0 0 orthogonal lifted from D6 ρ10 2 2 -2 -2 2 2 2 2 2 0 0 0 0 0 0 2 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -1 2 2 -1 -1 0 2 2 0 0 0 -1 2 2 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ12 2 2 -2 -2 -1 2 2 -1 -1 0 0 0 0 0 0 -1 2 2 -1 -1 1 1 -√3 √3 √3 -√3 0 0 0 0 orthogonal lifted from D12 ρ13 2 2 -2 -2 -1 2 2 -1 -1 0 0 0 0 0 0 -1 2 2 -1 -1 1 1 √3 -√3 -√3 √3 0 0 0 0 orthogonal lifted from D12 ρ14 2 -2 -2 2 -1 2 2 -1 -1 0 -2 2 0 0 0 1 -2 -2 1 1 1 -1 1 1 -1 -1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ15 2 -2 -2 2 -1 2 2 -1 -1 0 2 -2 0 0 0 1 -2 -2 1 1 1 -1 -1 -1 1 1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ16 2 -2 2 -2 2 2 2 2 2 0 0 0 0 0 0 -2 -2 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ17 2 -2 2 -2 -1 2 2 -1 -1 0 0 0 0 0 0 1 -2 -2 1 1 -1 1 √3 -√3 √3 -√3 0 0 0 0 symplectic lifted from Dic6, Schur index 2 ρ18 2 -2 2 -2 -1 2 2 -1 -1 0 0 0 0 0 0 1 -2 -2 1 1 -1 1 -√3 √3 -√3 √3 0 0 0 0 symplectic lifted from Dic6, Schur index 2 ρ19 4 4 0 0 4 1 -2 1 -2 0 0 0 2 2 0 4 1 -2 1 -2 0 0 0 0 0 0 -1 0 -1 0 orthogonal lifted from S3≀C2 ρ20 4 4 0 0 4 -2 1 -2 1 -2 0 0 0 0 -2 4 -2 1 -2 1 0 0 0 0 0 0 0 1 0 1 orthogonal lifted from S3≀C2 ρ21 4 4 0 0 4 1 -2 1 -2 0 0 0 -2 -2 0 4 1 -2 1 -2 0 0 0 0 0 0 1 0 1 0 orthogonal lifted from S3≀C2 ρ22 4 4 0 0 4 -2 1 -2 1 2 0 0 0 0 2 4 -2 1 -2 1 0 0 0 0 0 0 0 -1 0 -1 orthogonal lifted from S3≀C2 ρ23 4 -4 0 0 4 1 -2 1 -2 0 0 0 2i -2i 0 -4 -1 2 -1 2 0 0 0 0 0 0 -i 0 i 0 complex lifted from C3⋊S3.Q8 ρ24 4 -4 0 0 4 1 -2 1 -2 0 0 0 -2i 2i 0 -4 -1 2 -1 2 0 0 0 0 0 0 i 0 -i 0 complex lifted from C3⋊S3.Q8 ρ25 4 -4 0 0 4 -2 1 -2 1 -2i 0 0 0 0 2i -4 2 -1 2 -1 0 0 0 0 0 0 0 -i 0 i complex lifted from C3⋊S3.Q8 ρ26 4 -4 0 0 4 -2 1 -2 1 2i 0 0 0 0 -2i -4 2 -1 2 -1 0 0 0 0 0 0 0 i 0 -i complex lifted from C3⋊S3.Q8 ρ27 8 8 0 0 -4 -4 2 2 -1 0 0 0 0 0 0 -4 -4 2 2 -1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊2D12 ρ28 8 8 0 0 -4 2 -4 -1 2 0 0 0 0 0 0 -4 2 -4 -1 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊2D12 ρ29 8 -8 0 0 -4 -4 2 2 -1 0 0 0 0 0 0 4 4 -2 -2 1 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ30 8 -8 0 0 -4 2 -4 -1 2 0 0 0 0 0 0 4 -2 4 1 -2 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of (C3×C6).9D12
On 48 points
Generators in S48
(1 5 9)(3 11 7)(14 22 18)(16 20 24)(25 33 29)(27 31 35)(37 45 41)(39 43 47)
(1 20)(2 13 10 21 6 17)(3 22)(4 19 8 23 12 15)(5 24)(7 14)(9 16)(11 18)(25 37)(26 46 30 38 34 42)(27 39)(28 44 36 40 32 48)(29 41)(31 43)(33 45)(35 47)
(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 39 40 41 42 43 44 45 46 47 48)
(1 47 20 35)(2 46 21 34)(3 45 22 33)(4 44 23 32)(5 43 24 31)(6 42 13 30)(7 41 14 29)(8 40 15 28)(9 39 16 27)(10 38 17 26)(11 37 18 25)(12 48 19 36)

G:=sub<Sym(48)| (1,5,9)(3,11,7)(14,22,18)(16,20,24)(25,33,29)(27,31,35)(37,45,41)(39,43,47), (1,20)(2,13,10,21,6,17)(3,22)(4,19,8,23,12,15)(5,24)(7,14)(9,16)(11,18)(25,37)(26,46,30,38,34,42)(27,39)(28,44,36,40,32,48)(29,41)(31,43)(33,45)(35,47), (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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,47,20,35)(2,46,21,34)(3,45,22,33)(4,44,23,32)(5,43,24,31)(6,42,13,30)(7,41,14,29)(8,40,15,28)(9,39,16,27)(10,38,17,26)(11,37,18,25)(12,48,19,36)>;

G:=Group( (1,5,9)(3,11,7)(14,22,18)(16,20,24)(25,33,29)(27,31,35)(37,45,41)(39,43,47), (1,20)(2,13,10,21,6,17)(3,22)(4,19,8,23,12,15)(5,24)(7,14)(9,16)(11,18)(25,37)(26,46,30,38,34,42)(27,39)(28,44,36,40,32,48)(29,41)(31,43)(33,45)(35,47), (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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,47,20,35)(2,46,21,34)(3,45,22,33)(4,44,23,32)(5,43,24,31)(6,42,13,30)(7,41,14,29)(8,40,15,28)(9,39,16,27)(10,38,17,26)(11,37,18,25)(12,48,19,36) );

G=PermutationGroup([[(1,5,9),(3,11,7),(14,22,18),(16,20,24),(25,33,29),(27,31,35),(37,45,41),(39,43,47)], [(1,20),(2,13,10,21,6,17),(3,22),(4,19,8,23,12,15),(5,24),(7,14),(9,16),(11,18),(25,37),(26,46,30,38,34,42),(27,39),(28,44,36,40,32,48),(29,41),(31,43),(33,45),(35,47)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,47,20,35),(2,46,21,34),(3,45,22,33),(4,44,23,32),(5,43,24,31),(6,42,13,30),(7,41,14,29),(8,40,15,28),(9,39,16,27),(10,38,17,26),(11,37,18,25),(12,48,19,36)]])

Matrix representation of (C3×C6).9D12 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 12
,
 10 10 0 0 0 0 3 7 0 0 0 0 0 0 0 1 0 12 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0
,
 2 2 0 0 0 0 4 11 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 8 0 0 0 0 5 0 0 0 0 0 0 8

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[10,3,0,0,0,0,10,7,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,12,0,12,0],[2,4,0,0,0,0,2,11,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,8,0,8] >;

(C3×C6).9D12 in GAP, Magma, Sage, TeX

(C_3\times C_6)._9D_{12}
% in TeX

G:=Group("(C3xC6).9D12");
// GroupNames label

G:=SmallGroup(432,587);
// by ID

G=gap.SmallGroup(432,587);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,36,1684,1691,298,677,348,1027,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^12=1,d^2=b^3,a*b=b*a,c*a*c^-1=b^2,d*a*d^-1=a^-1,c*b*c^-1=a*b^3,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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